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### Stress Testing

After completing this reading, you should be able to:

• Describe the rationale for the use of stress testing as a risk management tool.
• Identify key aspects of stress testing governance, including choice of scenarios, regulatory specifications, model building, stress-testing coverage, capital and liquidity stress testing and reverse stress testing.
• Describe the relationship between stress testing and other risk measures, particularly in enterprise-wide stress testing.
• Explain the importance of stressed inputs and their importance in stressed VaR and stressed ES.
• Identify the advantages and disadvantages of stressed risk metrics.
• Describe the key elements of effective governance over stress testing.
• Describe the responsibilities of the board of directors and senior management in stress testing activities.
• Identify elements of clear and comprehensive policies, procedures, and documentation on stress testing.
• Identify areas of validation and independent review of stress tests that require attention from a governance perspective.
• Describe the important role of the internal audit in stress testing governance and control.
• Describe the Basel stress testing principles for banks regarding the implementation of stress testing.

## Why Use Stress Testing As A Risk Management Tool?

• Stress testing serves to warn a firm’s management of potential adverse events arising from the firm’s risk exposure and goes further to give estimates of the amount of capital needed to absorb losses that may result from such events.
• Stress tests help to avoid any form of complacency that may creep in after an extended period of stability and profitability. It serves to remind management that losses could still occur, and adequate plans have to be put in place in readiness for every eventuality. This way, a firm is able to avoid issues like underpricing of products, something that could prove financially fatal.
• Stress testing is a key risk management tool during periods of expansion when a firm introduces new products into the market. For such products, there may be very limited loss data or none at all, and hypothetical stress testing helps to come up with reliable loss estimates.
• Under pillar 1 of Basel II, stress testing is a requirement for all banks using the Internal Models Approach (IMA) to model market risk and the internal ratings-based approach to model credit risk. These banks have to employ stress testing to determine the level of capital they are required to have.
• Stress testing supplements other risk management tools, helping banks to mitigate risks through measures such as hedging and insurance. By itself, stress testing cannot address all risk management weaknesses, nor can it provide a one-stop solution.

## Key Aspects of Stress-testing Governance

### Stress-testing Coverage

• It’s important to include all risks, liabilities, and business lines in stress-testing activities. If some areas of the business are left out, stress-testing can give false hopes, making a firm believe it’s safe while in the real sense it’s not.
• Stress tests should be conducted over a range of time horizons to capture both short-term and long-term risks.
• Stress tests should capture the interrelationships between various risks and business lines. There should be efforts to single out risk drivers that feature prominently throughout the organization.
• Stress tests should be carried out at the portfolio level, department level, and even for the enterprise as a whole.

### Stress-testing Types and Approaches

• During scenario analysis, the scenario tested should be in line with the direction and long-term strategy set by the board of directors. The range of scenarios should be wide to incorporate sufficiently severe outcomes, including those that have the potential to challenge the firm as a going concern.
• While stress-testing at departmental levels cares should be made to make similar assumptions across the board to ensure the reliability and relevance of enterprise-wide, aggregated results.
• Scenarios tested should be comprised of both specific and systematic risks. Also, there’s need to stress-test hypothetical scenarios – those that have a positive probability of occurrence, even if similar events have never been observed in the past.

### Capital and Liquidity Stress-testing

• When stress-testing for capital and liquidity adequacy, the firm’s overall strategy and annual planning cycle should be taken into account. Results should be refreshed in light of emerging strategic decisions that have the potential to affect the financial stability of the firm significantly.
• Stress tests for capital and liquidity accuracy should seek to reveal how cash flows – both income and expenses – would be affected, especially in an environment where multiple risks materialize simultaneously.
• For entities with subsidiaries, tests should assess the likelihood of individual subsidiaries facing capital/liquidity crises on their own, regardless of the financial stability of the parent company.
• Stress tests should simulate the effects of capital problems and illiquidity pressures happening at the same time. Also, there should be efforts to show if one of these problems can exacerbate the other.
• Stress tests should help the firm to come up with contingency plans to deal with crippling capital/liquidity problems. The tests should also make clear the specific steps that could be taken, as well as those that cannot be taken in the face of the aforementioned problems.

## The Relationship between Stress Testing and Other Risk measures

Stress testing plays a large role in the assessment of banks’ capital adequacy as a result of the 2007-2009 financial crisis. However, banks continue to use other risk-management tools such as VaR in conjunction with stress testing.

### Similarities between Enterprise-wide stress-testing and Economic Capital/VaR

• Both stress testing and EC-VaR are used by financial institutions for purposes of risk management.
• For both enterprise-wide stress tests and VaR, the inputs used are scenarios and data. Both involve the translation of scenarios into loss estimates.

### Differences

• While stress test losses take an accounting view, EC losses take a market view.
• While stress tests examine a long period, typically spanning several years, EC models focus on losses at a given point in time, say, the loss in value at the end of year $$t$$.
• While stress testing largely doesn’t involve probabilities, EC models are founded on probabilities. For example, a 99.9% VaR can be viewed as a 1-in-1,000 event.
• Stress testing focuses on the conditional expected loss (EL) while EC-VaR measures the unconditional unexpected loss

#### Stress Testing

Measure of unexpected loss

Measure of expected loss

Typically unconditional or TTC (through-the-cycle)

Typically conditional or PIT (point-in-time)

Typically one-year horizon

Typically beyond one-year horizon

Typically doesn’t identify causal risks

Reveals the causal (underlying) risks

Economic value-based

Accounting-based

Corresponds to a specific probabilistic loss level, not a macroeconomic scenario

Corresponds to a macroeconomic scenario, not a specific probabilistic loss level

Focus on a very large number of scenarios

Focus on just a few scenarios

## Using VaR Models in Stress Tests

One of the approaches that have been used to incorporate stress tests in VaR models involves trying to assess whether the stress test loss is part of the loss distribution developed in the VaR/EC estimation. This way, a hypothetical/historical stress scenario can be associated with a given probability.

For example, suppose a scenario’s loss magnitude translates into a $${ 95 }^{ th }$$ percentile loss on the VaR loss distribution. In such a case, the $${ 95 }^{ th }$$ percentile loss in the EC model could be taken as an approximation to the stressed loss for market risk.

A different approach used to incorporate stress testing in VaR involves the use of stressed inputs. According to Basel market rules released in 2011, banks should calculate a stressed VaR based on the 10-day, 99th percentile, one-tailed confidence interval value-at-risk measure of the current portfolio with model inputs calibrated to historical data from a continuous 12-month period of significant financial stress relevant to the bank’s portfolio.

The goal is to replicate a VaR calculation that would be generated based on a bank’s current portfolio if the relevant market factors were to be subjected to stress.

The market risk framework also demands the incorporation of stress tests. Banks that use the internal models approach, in particular, are required to have a rigorous stress testing program in place. In the revisions to Basel Ill (BCBS 2011b), stressed parameters are required to determine the default risk capital charge for counterparty credit risk.

Stressed inputs are also important in the capital charge for credit valuation adjustment (CVA).

### Advantages and Disadvantages of Stressed Risk Metrics

A risk metric with a stressed input is usually more conservative, meaning that the capital set aside is likely to be sufficient even if a shock/stress event occurs.

On the downside, however, such a risk metric loses its responsiveness to current market conditions. Instead, the metric primarily depends on portfolio composition.

## Key Elements of Effective Governance

Stress testing is a broad term used to describe a firm’s examination of its risks and the potential impact of stressful events on its financial condition. Stress testing seeks to lay bare all the financial circumstances a firm may find itself in. It helps to understand the range of potential risk exposures.

The importance of stress testing was highlighted in post-analysis of the 2007/2008 financial crises. At many firms, stress testing was insufficiently integrated into their risk management practices. The few that had shown more commitment to stress testing did not examine sufficiently severe scenarios, and test results had little or no consideration in decision making.

Most importantly, the crisis brought to the fore the key role of governance in stress testing. In summary, the following points stand out:

• Stress testing is the ultimate responsibility of the management body of an institution. The buck stops with senior management and the board of directors. The management should have an in-depth understanding of the institution’s risk profile and the potential impact of stress events.
• The stress testing programme should be guided by clear, well-detailed policies that outline the procedure to follow from start to finish, as well as describing the role played by various employees.
• There should be consistent documentation of stress testing results and methodologies
• There should be an independent review of, not just the testing procedure but also the results of the programme.

The ensuing learning outcomes seek to break down these key elements even further.

## Responsibilities of The Board of Directors and Senior Management in Stress Testing Activities

For effective operation of stress testing, the board of directors and senior management should have distinct responsibilities. What’s more, there should be some shared responsibilities, although a few roles can be set aside exclusively for one of the two groups.

### Responsibilities of the Board of Directors

1. The buck stops with the board: The board of directors is “ultimately” responsible for a firm’s stress tests. Even if board members don’t immerse themselves into the technical details of stress tests, they should ensure that they stay sufficiently knowledgeable about stress testing procedures and interpretation of results.
2. Continuous involvement: Board members should regularly receive summary information on stress tests, including results from every scenario. Members should then evaluate these results to ensure they take into account the firm’s risk appetite and overall strategy.
3. Continuous review: Board members should regularly review stress testing reports with a view to not just critic key assumptions but also supplement the information with their own views that better reflect the overall goals of the firm.
4. Integrating stress testing results into decision making: The board should make key decisions on investment, capital, and liquidity based on stress test results along with other information. While doing this, the board should proceed with a certain level of caution in cognizance of the fact that stress tests are subject to assumptions and a host of limitations.
5. Formulating stress-testing guidelines: It’s the responsibility of the board to come up with guidelines on stress testing, such as the risk tolerance level (risk appetite).

### Responsibilities of Senior Management

1. Implementation oversight: Senior management has the mandate to ensure that stress testing guidelines authorized by the board are implemented to the letter. This involves establishing policies and procedures that help to implement the board’s guidelines.
2. Regularly reporting to the board: Senior management should keep the board up-to-date on all matters to do with stress testing, including test designs, emerging issues, and compliance with stress-testing policies.
3. Coordinating and Integrating stress testing across the firm: Members of senior management have the responsibility to propagate widespread knowledge on stress tests across the firm, making sure that all departments understand its importance.
4. Identifying grey areas: Senior management should seek to identify inconsistencies, contradictions, and possible gaps in stress tests with a view to making improvements to the whole process.
5. Ensuring stress tests have a sufficient range: In consultations with the board of directors, senior management has to ensure that stress testing activities are sufficiently severe to gauge the firm’s preparation for all possible scenarios, including low-frequency high-impact events.
6. Using stress tests to assess the effectiveness of risk mitigation strategies: Stress tests should help the management to assess just how effective risk mitigation strategies are. If such strategies are effective, significantly severe events will not cause significant financial strain. If the tests predict significant financial turmoil, it could be that the hedging strategies adopted are ineffective.
7. Updating stress tests to reflect emerging risks: As time goes, an institution will gradually gain exposure to new risks, either as a result of market-wide trends or its own investment activities. It’s the responsibility of senior management to develop new stress-testing techniques that reflect the institution’s new risk profile.

## Policies, Procedures, and Documentations on Stress Testing

Stress testing policies and procedures should:

• Describe the overall goals of stress-testing activities
• Outline stress-testing practices articulately and in a manner that ensures the tests are sufficiently severe
• Outline the roles and responsibilities of everyone across the firm, particularly the management body (board of directors and senior management).
• Provide guidance on validation and independent review
• Specify the frequency of stress-testing activities
• Provide guidance on the process followed while choosing the appropriately stressful conditions for tests
• Outline how stress-testing results are put to use and by whom
• Be regularly updated in line with changes in the risk exposure of the firm

### Documentation

A firm should ensure that its stress-testing activities are well documented, including clear and precise details about key assumptions, methodologies, limitations, uncertainties, and test results. Documentation has several advantages:

• It gives management the chance to identify trends in risk exposures
• It promotes transparency by allowing third parties to scrutinize an institution’s stress-testing practices and even compare firms
• It provides for continuity of stress tests even in the face of significant changes in management. The new leadership easily “learns the ropes.”
• It helps stress test developers to further improve the process by forcing them to think clearly about their methods, assumptions, and choices.

## Areas of Validation and Independent Review for Stress Tests That Require Attention from a Governance Perspective

Validation and independent review of stress-testing activities help to ensure integrity in the process as a whole. Independent review of these activities is meant to give assurance that stress tests are conducted in accordance with laid down policies and procedures, and that results are used to advance the firm’s financial stability.

Areas of Validation and Independent Review for Stress Tests That Require Attention from a Governance Perspective include:

• Ensuring that validation and independent review are conducted on an ongoing basis;
• Ensuring that subjective or qualitative aspects of a stress test are also validated and reviewed, even if they cannot be tested in quantitative terms;
• Acknowledging limitations in stress testing;
• Ensuring that stress-testing standards are upheld;
• Acknowledging data weaknesses or limitations, if any;
• Ensuring that there’s sufficient independence in both validation and review of stress tests;
• Ensuring that third-party models used in stress-testing activities are validated and reviewed to determine if they are fit for the purpose at hand;
• Ensuring that stress tests results are implemented in a rigorous manner, and verifying that any departure from the recommended actions is backed up by solid reasons;

## Role of the Internal Audit In Stress Testing Governance and Control

Internal audit should:

• Independently evaluate the performance, integrity, and reliability of stress-testing activities;
• Ensure that stress tests across the organization are conducted in a sound manner and remain current in terms of the scenarios tested;
• Assess the skills and expertise of the staff involved in stress-testing activities;
• Check that approved changes to stress-testing policies and procedures are implemented and appropriately documented;
• Evaluate the independent review and validation exercises;

To accomplish all the above, internal audit staff must be well qualified. They should be well-grounded in stress-testing techniques and technical expertise so as to be able to differentiate between good and inappropriate practices.

## Question 1

Hardik and Simriti compare and contrast stress testing with economic capital and value at risk measures. Which of the following statements regarding differences between the two types of risk measures is most accurate?

1. Stress tests tend to calculate losses from the perspective of the market, while EC/VaR methods compute losses based on an accounting point of view
2. While stress tests focus on unconditional scenarios, EC/VaR methods focus on conditional scenarios
3. While stress tests examine a long period, typically spanning several years, EC models focus on losses at a given point in time, say, the loss in value at the end of year $$t$$.
4. Stress tests tend to use cardinal probabilities while EC/VaR methods use ordinal arrangements

The correct answer is C.

A is inaccurate. Stress tests tend to calculate losses from the perspective of accounting, while EC/VaR methods compute losses based on a market point of view.

B is inaccurate. While stress tests focus on conditional scenarios, EC/VaR methods focus on unconditional scenarios.

D is also inaccurate. Stress tests do not focus on probabilities. Instead, they focus on ordinal arrangements like “severe,” “more severe,” and “extremely severe.” EC/VaR methods, on the other hand, focus on cardinal probabilities. For instance, a 95% VaR loss could be interpreted as 5-in-100 events.

## Question 2

One of the approaches used to incorporate stress testing in VaR involves the use of stressed inputs. Which of the following statements most accurately represents a genuine disadvantage of relying on risk metrics that incorporate stressed inputs?

1. The metrics are usually more conservative (less aggressive)
2. The metrics are usually less conservative (more aggressive)
3. The capital set aside, as informed by the risk metrics, is likely to be insufficient
4. The risk metrics primarily depend on portfolio composition and are not responsive to emerging risks or current market conditions.

The correct answer is D.

The most common disadvantage of using stressed risk metrics is that they do not respond to current issues in the market. As such, significant shocks in the market can “catch the firm unawares” and result in extensive financial turmoil.

## Question 3

Sarah Wayne, FRM, works at Capital Bank, based in U.S. The bank owns a portfolio of corporate bonds and also has significant equity stakes in several medium-size companies across the United States. She was recently requested to head a risk management department subcommittee tasked with stress testing. The aim is to establish how well prepared the bank is for destabilizing events. Which of the following scenario analysis options would be best for the purpose at hand?

1. Hypothetical scenario analysis
2. Historical scenario analysis
3. Forward-looking hypothetical scenario analysis and historical scenario analysis
4. Cannot tell based on the given information

The correct answer is C.

Scenario analyses should be dynamic and forward-looking. This implies that historical scenario analysis and forward-looking hypothetical scenario analysis should be combined. Pure historical scenarios can give valuable insights into impact but can underestimate the confluence of events that are yet to occur. What’s more, historical scenario analyses are backward-looking and hence neglect recent developments (risk exposures) and current vulnerabilities of an institution. As such, scenario design should take into account both specific and systematic changes in the present and near future.

## Question 4

Senior management should be responsible for which of the following tasks?

1. Ensuring that stress testing policies and procedures are followed to the letter
2. Assessing the skills and expertise of the staff involved in stress-testing activities
3. Evaluating the independent review and validation exercises
4. Making key decisions on investment, capital, and liquidity based on stress test results along with any other information available.
5. Propagating widespread knowledge on stress tests across the firm, and making sure that all departments understand its importance
1. I, II, and IV
2. I and V
3. III and IV
4. V only

The correct answer is B.

Roles II and III belong to internal audit. Role IV belongs to the board of directors.


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## April 7, 2014

### On a Topological Topos

#### Posted by Emily Riehl

Guest post by Sean Moss

In this post I shall discuss the paper “On a Topological Topos” by Peter Johnstone. The basic problem is that algebraic topology needs a “convenient category of spaces” in which to work: the category $\mathcal{T}$ of topological spaces has few good categorical properties beyond having all small limits and colimits. Ideally we would like a subcategory, containing most spaces of interest, which is at least cartesian closed, so that there is a useful notion of function space for any pair of objects. A popular choice for a “convenient category” is the full subcategory of $\mathcal{T}$ consisting of compactly-generated spaces. Another approach is to weaken the notion of topological space, i.e. to embed $\mathcal{T}$ into a larger category, hopefully with better categorical properties.

A topos is a category with enough good properties (including cartesian closedness) that it acts like the category of sets. Thus a topos acts like a mathematical universe with ‘sets’, ‘functions’ and its own internal logic for manipulating them. It is exciting to think that if a “convenient topos of spaces” could be found, then its logical aspects could be applied to the study of its objects. The set-like nature of toposes might make it seem unlikely that this can happen. For instance, every topos is balanced, but the category of topological spaces is famously not. However, sheaves (the objects in Grothendieck toposes) originate from geometry and already behave somewhat like generalized spaces.

I shall begin by elaborating on this observation about Grothendieck toposes, and briefly examine some previous attempts at a “topological topos”. I shall explore the idea of replacing open sets with convergent sequences and see how this leads us to Johnstone’s topos. Finally I shall describe how Johnstone’s topos is a good setting for homotopy theory.

I would to thank Emily Riehl for organizing the Kan Extension Seminar and for much useful feedback. Thanks go also to the other seminar participants for being a constant source of interesting perspectives, and to Peter Johnstone, for his advice and for writing this incredible paper.

### Are there toposes of spaces?

We shall need to be flexible about what we mean by “space”. For the rest of this post I shall try to use the term “topological space” in a strict technical sense (a set of points plus specified open sets), whereas “space” will be a nebulous concept. The idea is that spaces have existence regardless of having been implemented as a topological space or not, and may naturally have more (or perhaps less) structure. Topology merely forms one setting for their rigorous study. In topology we can only detect the topological properties of spaces. For example, $\mathbb{R}$ and $(0,1)$ are isomorphic as topological spaces, but they are far from being the same space: consider how different their implementations as, say, metric spaces are. Some spaces are naturally considered as having algebraic or smooth structure. The type of question one wishes to ask about a space will bear upon the type of object as which it should be implemented.

An extremely important class of toposes consists of the Grothendieck toposes, which are categories of sheaves on a site. A site is a small category together with a Grothendieck coverage (also known as a Grothendieck topology). Informally, the Grothendieck coverage tells us how some objects can be “covered” by maps coming out of other objects. In the special case where the site is a topological space, the objects are open sets and the coverage tells us that an open set is covered by the inclusions of any family of open sets whose union is all of that open set. A sheaf on a site is then a contravariant $\mathrm{Set}$-valued functor on the underlying category (a presheaf) which satisfies a “unique patching” condition with respect to each covering sieve.

In the following two senses, a Grothendieck topos always behaves like a category of spaces:

(A) One way to describe the properties of a space is to consider the maps into that space. This is the idea behind the homotopy groups, where we consider (homotopy classes of) maps from the $n$-sphere into a space. Given a small category $\mathcal{C}$, each object is determined by knowing all the arrows into it and how these arrows “restrict” along other arrows, i.e. precisely the data of the representable presheaf. A non-representable presheaf can be viewed as a generalized object of $\mathcal{C}$, which is testable by the ‘classical’ objects of $\mathcal{C}$: it is described entirely by what the maps into it from objects of $\mathcal{C}$ ought to be. If we have in mind that $\mathcal{C}$ is some category of spaces, with some sense in which some spaces are covered by families of maps out of other spaces, (i.e. we have a Grothendieck coverage), then we should be able to patch maps into these generalized spaces together. So the topos of sheaves on this site is a setting in which we may be able to implement certain spaces, if we wish to study their properties testable by objects of $\mathcal{C}$.

(B) The category of presheaves on a small category $\mathcal{C}$ is its free cocompletion. Intuitively, it is the category of objects obtained by formally gluing together objects of $\mathcal{C}$. The use of the word “gluing” is itself a spatial metaphor. CW-complexes are built out of gluing together cells - simplicial sets are instructions for carrying out this gluing. Manifolds are built from gluing together open subsets of Euclidean space. Purely formal ‘gluing’ is not quite sufficient: the Yoneda embedding of $\mathcal{C}$ into its presheaves typically does not preserve any colimits already in $\mathcal{C}$. But if $\mathcal{C}$ is a category of spaces, its objects are not neutral with respect to each other: there may be a suitable Grothendieck coverage on $\mathcal{C}$ which tells us how some objects can cover others. The topos of sheaves is then the category of objects obtained by formally gluing objects of $\mathcal{C}$ in a way that respects these coverings. This is strongly connected with the preservation of colimits by the embedding of $\mathcal{C}$ into the sheaves. Colimits in the presheaf topos are constructed pointwise; to get the sheaf colimit one applies the reflection into the category of sheaves (“sheafification”) to the presheaf colimit. The more covers imposed on $\mathcal{C}$, the more work is done by the sheafification, so the closer we end up to the original colimit.

### Are there toposes in topology?

It is far from clear that we can choose a site for which the space-like behaviour of sheaves accords with the usual topological intuition. If we want to use a topological topos for homotopy theory, then ideally it should contain objects that we can recognize as the CW-complexes, and we should be able to construct them via more or less the usual colimits.

### Attempt 1: The “gros topos” of Giraud

The idea is to take sheaves on the ‘site’ of topological spaces, where covers are given by families of open inclusions of subspaces whose union is the whole space. We do not automatically get a topos unless the site is small, so instead take some small, full subcategory $\mathcal{C}$ of $\mathcal{T}$, which is closed under open subspaces. The gros topos is the topos of sheaves for this site.

The Yoneda embedding exhibits $\mathcal{C}$ as a subcategory, and in fact we can ‘embed’ $\mathcal{T}$ via the functor $X \mapsto \hom_\mathcal{T}(-,X)$, this will be full and faithful on a fairly large subcategory. By (B) one may like to consider the gros topos as the category of spaces glued together from objects of $\mathcal{C}$. This turns out not to be useful, since the site does not have enough covers for colimits to agree with those in $\mathcal{T}$. Moreover the site is so large that calculations are difficult.

### Attempt 2: Lawvere’s topos

We use observation (A). Motivated by the use of paths in homotopy theory, we take $M$ to be the full subcategory of $\mathcal{T}$ whose only object is the closed unit interval $I$. So $M$ is the monoid of continuous endomorphisms of $I$. Lawvere’s topos $\mathcal{L}$ is the topos of sheaves on $M$ with respect to the canonical Grothendieck coverage (the largest Grothendieck coverage on $M$ for which $\hom_M(-,I)$ is a sheaf).

Then an object $X$ of $\mathcal{L}$ is a set $X(I)$ of paths, together with, for any continuous $\gamma\colon I \to I$, a reparametrization map $X(\gamma) \colon X(I) \to X(I)$, where this assignment is functorial. The points of such a space are given by natural transformations $1 \to X$, i.e. ‘constant paths’ or paths which are fixed by every reparametrization. We can see which point a path visits at time $t$ by reparametrizing that path by the constant map $I \to I$ with value $t$. A word of caution: a given object in $\mathcal{L}$ may have distinct paths which agree on points for all time.

This site is much easier to calculate with than the gros site (once we have a handle on the canonical coverage). Again there is a functor $P \colon \mathcal{T} \to \mathcal{L}$ given by $X \mapsto \hom_\mathcal{T}(I,X)$, which is full and faithful on a fairly large subcategory (including CW-complexes). However, it is still the case that the site could do with more covers: the functor $P$ does not preserve all the colimits used to build up CW-complexes. By observation (B), an object of $\mathcal{L}$ is obtained by gluing together copies of the unit interval $I$, so it is possible to construct the circle $S^1$ out of copies of $I$, but we cannot do this in the usual way. The coequalizer of $I$ by its endpoints in $\mathcal{L}$ is not $S^1$, but a “signet-ring”: it is a circle with a ‘lump’, through which a path can cross only if waits there for non-zero time. We cannot solve this problem by adding in more covers, because the coverage is already canonical (adding in more covers will evict the representable $\hom_\mathcal{T}(-,I)$ from the topos).

The key idea in Johnstone’s topos is to replace paths with convergent sequences. Given a topological space $X$, a convergent sequence in $X$ is a function from $a \colon \mathbb{N}\cup\{\infty\} \to X$ such that whenever $U \subseteq X$ is an open set containing $a_\infty$, then there exists an $N$ such that $a_n \in U$ for all $n \gt N$. The convergent sequences are precisely the continuous maps out of $\mathbb{N}\cup\{\infty\}$ when we give it the topology that makes it the one-point compactification of the discrete space $\mathbb{N}$ - we denote this topological space by $\mathbb{N}^+$.

### Convergent sequences as primitive

It is a basic theorem in general topology that, given a function $f\colon X \to Y$ between topological spaces, if it is continuous then it preserves convergent sequences. The converse is not true for general topological spaces, but it is true whenever $Y$ is a sequential space. Given a topological space $X$, A set $U \subseteq X$ is sequentially open if for any convergent sequence $(a_n)$, with $a_\infty \in U$, $(a_n)$ is eventually in $U$. (Clearly any open subset is sequentially open.) A topological space is then said to be sequential if all of its sequentially open sets are open. The sequential spaces include all first-countable spaces and in fact they can be characterized as the topological quotients of metrizable spaces, so they certainly include all CW-complexes.

The notion of convergent sequence is arguably more intuitive than that of open set. For example, each convergent sequence gives you concrete data about the nearness of some family of points to another point, whereas open sets only give you such data when the topology (or at least a neighbourhood basis) is considered as a whole. It would be compelling to define a continuous function as one that preserves convergent sequences. This motivates the study of subsequential spaces.

A subsequential space consists of a set $X$ (of points) and family of “convergent sequences”: a specified subset of the set of functions $\mathbb{N}\cup\{\infty\} \to X$, such that:

1. for every point $x \in X$, the constant sequence $(x)$ converges to $x$;
2. if $(x_n)$ converges to $x$, then so does every subsequence of $(x_n)$;
3. if $(x_n)$ is a sequence and $x$ is a point such that every subsequence of $(x_n)$ contains a (sub)subsequence converging to $x$, then $(x_n)$ converges to $x$.

The third axiom is the general form of intertwining two or more sequences with the same limit or changing a finite initial segment of a sequence. Note that there is no ‘Hausdorff‘-style condition on the convergent sequences: a sequence may converge to more than one limit. A continuous map between subsequential spaces $X \to Y$ is a function from the points of $X$ to the points of $Y$ that preserves convergence of sequences.

The axioms above are all true of the set of convergent sequences which arise from a topology on a set. In fact, this process gives a full and faithful embedding of sequential spaces into subsequential spaces. Thus sequential spaces live inside both topological and subsequential spaces. They are coreflective in the former and reflective in the latter: given a topological space $X$, its sequentially open sets constitute a new (finer) topology; given a subsequential space $Y$, we can consider the “sequentially open” sets with respect to its convergent sequences, and then take all convergent sequences in the resulting (sequential) topological space. Observe that the sense of the adjunction in each case comes from the fact that we either throw in more open sets - so there is a natural map $(X)_\text{seq} \to X$, or throw in more convergent sequences - so there is a natural map $Y \to (Y)_\text{seq}$.

In the following I shall denote the category of sequential spaces by $\mathcal{F}$ and that of subsequential spaces by $\mathcal{F}'$.

### Johnstone’s topos

Let $\Sigma$ be the full subcategory of $\mathcal{T}$ on the objects $1$ (the singleton space) and $\mathbb{N}^+$ (the one-point compactification of the discrete space of natural numbers). The arrows in this category can be described without topology as well: as functions, the maps $\mathbb{N}^+ \to \mathbb{N}^+$ are the eventually constant ones and the ones that “tend to infinity”.

Given an infinite subset $T \subseteq \mathbb{N}$, let $f_T$ denote the unique order-preserving injection $\mathbb{N}^+ \to \mathbb{N}^+$ whose image is $T \cup \{\infty\}$. One can check that there is a Grothendieck coverage $J$ on $\Sigma$ where $1$ is covered by only the maximal sieve, and where $\mathbb{N}^+$ is covered by any sieve $R$ such that:

1. $R$ contains all of the points $n \colon 1 \to \mathbb{N}^+$, $n \in \mathbb{N}^+$.
2. For any infinite subset $T \subseteq \mathbb{N}$ there exists an infinite subset $T' \subseteq T$ such that $f_{T'} \in R$.

The topos $\mathcal{E}$ is then defined to be $\mathrm{Sh}(\Sigma,J)$.

The objects in our topos are a slight generalization of subsequential space. If $X \in \mathcal{E}$, then $X(1)$ is its set of points, and $X(\mathbb{N}^+)$ is its set of convergent sequences. Each point $n \colon 1 \to \mathbb{N}^+$ induces a ‘projection map’ $X(n)\colon X(\mathbb{N}^+) \to X(1)$, giving you the point of the sequence at time $n$. The unique map $\mathbb{N}^+ \to 1$ induces a map $X(1) \to X(\mathbb{N}^+)$, which sends each point to a canonical choice of constant sequence. Note that there may be more than one convergent sequence with the same points, thus it may be helpful to think of $X(\mathbb{N}^+)$ as the set of proofs of convergence for sequences.

Clearly we can embed $\mathcal{F}'$, the subsequential spaces, into $\mathcal{E}$: the points are the same, and the convergence proofs are just the convergent sequences. The first two axioms are satisfied because of the equations that hold in $\Sigma$. The third axiom is encoded into the coverage. Conversely, any object $X$ of $\mathcal{E}$ for which the projection maps $X(n)\colon X(\mathbb{N}^+) \to X(1)$, $n \in \mathbb{N}^+$ are jointly injective is isomorphic to one coming from a subsequential space. There is a functor $H \colon \mathcal{T} \to \mathcal{E}$ sending $X \mapsto \hom_\mathcal{T}(-,X)$, and it is indeed sheaf-valued since it is equal to the composite of the coreflection $\mathcal{T} \to \mathcal{F}$ with the inclusions $\mathcal{F} \to \mathcal{F}' \to \mathcal{E}$. In fact, the Grothendieck coverage defining $\mathcal{E}$ is canonical, so it is the largest for which this functor is well-defined.

We can use observation (B) to think of $\mathcal{E}$ as all spaces constructed from gluing sequences together. It is just about possible that we could have motivated the construction of $\mathcal{E}$ this way: classically, any sequential space $X$ is the quotient in $\mathcal{T}$ of a metrizable space, which may be taken to be a disjoint union of copies of $\mathbb{N}^+$ - one for every convergent sequence in $X$. Compare this with the canonical representation of a presheaf as a colimit of representables (one for each of its elements).

### Colimits

It turns out that $\mathcal{F}'$ is the subcategory of $\neg\neg$-separated objects in $\mathcal{E}$, hence it is a reflective subcategory. $\mathcal{F}$ is reflective in $\mathcal{F}'$, hence it is also reflective in $\mathcal{E}$. In particular, all limits in $\mathcal{F}$ are preserved by the inclusion into $\mathcal{E}$. Take some caution, however, since products do not agree with those in $\mathcal{T}$: one has to take the sequential coreflection of the topological product. This is only a minor issue; having to modify the product arises in other “convenient categories” such as compactly-generated spaces.

The colimits in $\mathcal{F}$ do agree with those in $\mathcal{T}$ because it is a coreflective subcategory. Surprisingly, the inclusion $\mathcal{F} \to \mathcal{E}$ preserves many of these colimits.

Theorem Let $X$ be a sequential space, and $\{U_\alpha \mid \alpha \in A\}$ an open cover of $X$. Then the obvious colimit diagram in $\mathcal{F}$: $\begin{matrix} U_\alpha\cap U_\beta & \rightarrow & U_\alpha & & \\ & \searrow & & \searrow & \\ U_\beta \cap U_\gamma & \rightarrow & U_\beta & \rightarrow & X \\ \vdots & \searrow & & \nearrow & \\ & & U_\gamma & & \\ & & \vdots & & \end{matrix}$ is preserved by the embedding $\mathcal{F} \to \mathcal{E}$.

Proof The recipe for this sort of theorem is: take the colimit in presheaves, show that the comparison map is monic, then show that it is $J$-dense, for then it will exhibit $X$ as the colimit upon reflecting into the topos $\mathcal{E}$. The colimit $L$ in presheaves is calculated “objectwise”, so $L$ has the same points of $X$, but only those convergent sequences which are entirely within some $U_\alpha$ (hence the comparison map $L \to X$ is monic). To sheafify, we need to add in all those sequences $x \in X(\mathbb{N}^+)$ which are locally in $L$, i.e. for which the sieve $\{f \colon ? \to \mathbb{N}^+ \mid X(f)(x) \in L(?) \}$ in $\Sigma$ is $J$-covering. For any $x \in X(\mathbb{N}^+)$, this sieve clearly contains all the points $1 \to \mathbb{N}^+$. But $x$ must also be eventually within one of the $U_\alpha$, so the second condition for the covering sieves is also satisfied. $\square$

There are several other colimit preservation results one can talk about (with similar proofs to the above). The amazing consequence of these is that the colimits used to construct CW-complexes are all preserved by the embedding $\mathcal{F} \to \mathcal{E}$. Thus classical homotopy theory embeds into $\mathcal{E}$ and we have successfully found a topos of spaces which agrees with the classical theory.

### Geometric realization

Let $\Delta$ be the category of non-zero finite ordinals and order-preserving maps. Then objects of the presheaf category $[\Delta^\mathrm{op},\mathrm{Set}]$ are known as simplicial sets.

Theorem $[\Delta^\mathrm{op},\mathrm{Set}]$ is the classifying topos for intervals in $\mathrm{Set}$-toposes.

The closed unit interval $[0,1]$ is sequential and is in fact an interval (a totally ordered object with distinct top and bottom elements). Thus it corresponds to a geometric morphism $\mathcal{E} \to [\Delta^\mathrm{op},\mathrm{Set}]$ (an adjunction $(f^\star \dashv f_\star)$ with $f^\star$ left-exact).

Theorem If $S \in [\Delta^\mathrm{op},\mathrm{Set}]$ is a simplicial set, then $f^\star(S)$ is its geometric realization, considered as a sequential space and hence as an object of $\mathcal{E}$. If $X \in \mathcal{E}$ is a sequential space, then $f_\star(E)$ is its singular complex.

The usual geometric realization is not left-exact if considered to take values in $\mathcal{T}$, one must choose a “convenient subcategory” first, and then there is some work to do in proving it. Here the left-exactness just arises out of the general theory of geometric morphisms. Should we wish to do so, the above method allows us to replace $[0,1]$ with any other object that the internal logic of $\mathcal{E}$ sees as an interval to get a different realization of simplicial sets.

The above is far from a complete survey of “On a Topological Topos”, which contains several more results of interest relating to $\mathcal{E}$ and captures the elegance of using the site $\Sigma$ for calculation - I thoroughly recommend taking a look if you know some topos theory. We have seen enough though to understand that for many spaces the sequential properties align with the topological properties. Unfortunately, $\mathcal{E}$ is yet to receive the attention it deserves.

Posted at April 7, 2014 5:38 AM UTC

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### Re: On a Topological Topos

In our discussions we could hardly ignore Lawvere’s notion of cohesion as a characteristic property of categories of spaces. So it was interesting to notice that this topological topos fails to be cohesive over Set, since the inclusion of discrete spaces does not have a left adjoint (as Tim pointed out, this functor does not preserve infinite products). This could be taken as a first indication that this topos fails to capture the homotopical properties of its objects.

Posted by: Alexander Campbell on April 7, 2014 9:36 AM | Permalink | Reply to this

### Re: On a Topological Topos

I wonder if we can salvage some semblance of cohesion? The “discrete spaces” functor doesn’t have a left adjoint, but certain objects do have reflections along it – in particular, any sequential space has a reflection given by taking its connected components. As Sean discussed, Johnstone characterizes the subsequential spaces $\mathcal{F}' \subset \mathcal{E}$ as the separated objects in the double-negation topology, but I think not all of these have a discrete reflection. I wonder if the sequential spaces $\mathcal{F}$ admit some kind of topos-theoretic characterization within $\mathcal{E}$?

Posted by: Tim Campion on April 7, 2014 1:52 PM | Permalink | Reply to this

### Re: On a Topological Topos

Is cohesion really characteristic, or is it an idealisation?

1. It is not clear to me whether $\mathbf{Top}$ has a $\pi_0$ functor. I am sure we get $\pi_0$ if we restrict to locally connected spaces, however.
2. $\mathbf{Haus}$ does not have a “codiscrete” functor, of course. Nor does $\mathbf{Mfd}$.
3. $\mathbf{sSet}$ has all the desired adjoints, but $\pi_0$ does not preserve powers.

The only category of spaces that I know satisfies all of Lawvere’s axioms is the category of Kan complexes – but then we don’t have finite limits!

Posted by: Zhen Lin on April 7, 2014 2:41 PM | Permalink | Reply to this

### Re: On a Topological Topos

I attended a talk today where the speaker cited a topos of topological spaces. For a fun second I thought he might mean Johnstone’s consequential spaces, but he explained that he wanted Kan complexes (if I heard correctly). I was surprised to hear the claim that Kan complexes are a topos. If such an obvious example is present, why the need for Johnstone’s elaborate construction?

But Zhen, you point out that Kan fibrations do not contain finite limits. Is that obvious? Maybe if you take the equalizer of the inclusion of two $(n-1)$-hemispheres of a sphere, you get a simplicial set with no $n$-simplex filler? But surely that’s only if you take the equalizer in simplicial sets instead of Kan complexes. Can you clarify?

Posted by: Joe Hannon on April 8, 2014 2:57 AM | Permalink | Reply to this

### Re: On a Topological Topos

Perhaps the speaker was using the implicit infinity-category convention.

The (full sub)category of Kan complexes is not closed under equalisers, as you say. But I am sure that it simply doesn’t have equalisers. Indeed, $\Delta^0$ is a Kan complex, so equalisers must have the expected vertex set, and the full subcategory of Kan complexes is an exponential ideal of the category of simplicial sets, so exponentiating by $\Delta^n$ lets us deduce the same about the sets of $n$-simplices.

Posted by: Zhen Lin on April 8, 2014 8:28 AM | Permalink | Reply to this

### Re: On a Topological Topos

The existence of $\pi_0$ means that a cohesive category should be thought of not as an arbitrary category of spaces, but as a category of locally connected spaces. In particular, according to observation (B), a cohesive topos should be a category of spaces obtained by gluing together (or “locally modeled on”) locally connected spaces. From this perspective, it is unsurprising that Johnstone’s topos is not cohesive, since $\mathbb{N}^+$ is not locally connected.

Posted by: Mike Shulman on April 7, 2014 7:10 PM | Permalink | Reply to this

### Re: On a Topological Topos

Oh – it only just struck me that even the discrete space functor fails to preserve infinite products even when taking values in $\mathbf{Top}$. So it doesn’t have a left adjoint of cohesion even in that case (even though it’s not a topos anyway). This despite the fact that the “connected components” of an aribtrary space are well-defined in point-set topology. The map from a totally disconnected space, for instance, to its connected components is not continuous, so it doesn’t form a reflection.

But the discrete space functor from $\mathbf{Set}$ to locally connected spaces does have a left adjoint, given by taking connected components. (This implies that products of locally connected spaces are not taken as in $\mathbf{Top}$.) If we take an arbitrary small, full subcategory of the category of locally connected spaces, and look at the category of sheaves in the canonical topology, do we expect to get a cohesive topos?

Is cohesion something that only really applies to gros toposes? Or is it also meaningful for petit toposes? Is the category of sheaves over a locally connected space cohesive?

Posted by: Tim Campion on April 10, 2014 1:46 AM | Permalink | Reply to this

### Re: On a Topological Topos

These questions are addressed in Lawvere’s short article Categories of spaces may not be generalized spaces as exemplified by directed graphs. Proposition 1 shows that localic toposes (such as sheaves on a topological space) cannot be “sufficiently cohesive” over Set - meaning they are not cohesive over Set with connected subobject classifier (see Axiomatic cohesion for this definition).

Posted by: Alexander Campbell on April 10, 2014 4:30 AM | Permalink | Reply to this

### Re: On a Topological Topos

Is the category of sheaves over a locally connected space cohesive?

Cohesivity is (strong) local connectedness plus locality. Locality says that the topos, considered “petitely”, must be a sort of “fat point”.

products of locally connected spaces are not taken as in Top

the “connected components” of an aribtrary space are well-defined in point-set topology. The map from a totally disconnected space, for instance, to its connected components is not continuous

Indeed: so in general, one should consider the connected components of a non-locally-connected space to be not a set but a space, with the induced quotient topology. Another way to say this is that we can generalize from a left adjoint to a left pro-adjoint, leading to shape theory.

Posted by: Mike Shulman on April 10, 2014 7:01 PM | Permalink | Reply to this

### Re: On a Topological Topos

Yes, the category of locally connected spaces is coreflective in $Top$. The coreflector retopologizes a space $X$ by letting the connected components of open sets of $X$ generate a new topology. (So, for anyone reading this: to take an infinite product of locally connected spaces, one takes the product in $Top$ and then retopologizes that according to this recipe.)

Posted by: Todd Trimble on April 10, 2014 8:18 PM | Permalink | Reply to this

### Re: On a Topological Topos

Ah, very nice, thanks. So the category of locally connected spaces is cohesive over Set (though it is not of course a topos).

Posted by: Mike Shulman on April 10, 2014 11:25 PM | Permalink | Reply to this

### Re: On a Topological Topos

Is there a “pure logic” way to see that $[\Delta^{\mathrm{op}}, \mathbf{Set}]$ is the classifying topos for intervals (= total orders with distinct top and bottom elements, as Sean says)? Does it somehow follow directly from the fact that $\Delta^{\mathrm{op}}$ is equivalent to the category of finitely presentable intervals in $\mathbf{Set}$? And how is this related to the fact that $\Delta$ is the category of finite inhabited total orders (NOT intervals!) in $\mathbf{Set}$?

On the flip side, what kind of geometric insight is there to be gained by viewing the singular complex / geometric realization adjunction as being generated by an “interval object”?

Posted by: Tim Campion on April 7, 2014 5:01 PM | Permalink | Reply to this

### Re: On a Topological Topos

Is there a “pure logic” way to see that $[\Delta^{\text{op}}, Set]$ is the classifying topos for intervals (= total orders with distinct top and bottom elements, as Sean says)?

You mean something other than the proof in Moerdijk-Mac Lane p. 457 (Chapter VIII, Section 8)? I’m not sure it’s “pure logic” but it’s not really “pure geometric” either.

Does it somehow follow directly from the fact that $\Delta^{\text{op}}$ is equivalent to the category of finitely presentable intervals in $Set$?

Unless I’m confused, I think it does in the sense that if for some theory $T$, $Mod(T)=Hom(Set,[\mathcal{C},Set])$ (i.e. if $T$ is of presheaf type) then we can take $\mathcal{C}$ to be the full subcategory of finitely presentable objects in $Mod(T)$. So in this case since $Mod(T)$ is the category of intervals, the fact that $[\Delta^{\text{op}},Set]$ is the classifying topos for intervals would follow directly from the fact that $\Delta^{\text{op}}$ is equivalent to the category of finitely presentable intervals. But this assumes a lot of stuff on accessible/presentable categories, so I’m not sure you’d like to say that it follows “directly”. Or perhaps you had something different in mind.

Posted by: Dimitris on April 7, 2014 10:54 PM | Permalink | Reply to this

### Re: On a Topological Topos

The proof that in Mac Lane and Moerdijk that $\Delat^{\mathrm{op}}, \mathbf{Set}]$ classifies intervals is similar to Johnstone’s proof: both are rather “fiddly”, I’d say: they require really unpacking the terms of Diaconescu’s theorem and proving them bit-by-bit.

The nlab entry at Theories of Presheaf Type claims that the classifying topos $\mathbf{B}[\mathbb{T}]$ of a geometric theory $\mathbb{T}$ is a presheaf category (the theory is “of presheaf type”) if and only if the category $\mathrm{Mod}(\mathbb{T})$ of models in $\mathbf{Set}$ is finitely accessible. The linked paper by Beke refers to Mac Lane and Moerdijk pp. 381-386 for a proof, which is confusing because Mac Lane and Moerdijk don’t discuss accessible categories at all… Beke does prove, following Joyal and Wraith, under slightly stronger hypotheses, that the classifying topos is the presheaf topos on the finitely presentable objects of $\mathrm{Mod}(\mathbb{T})$ (apparently in general it might be presheaves on some other category, and this can happen for the theory of fields, at least under certain axiomatizations). The proof is short, but I don’t really follow it. It takes the locally finitely presentable case for granted, without citation.

This is exactly the sort of thing I had in mind: some systematic framework into which the fiddly calculations fall. I wish I could follow the details.

Posted by: Tim Campion on April 8, 2014 3:57 AM | Permalink | Reply to this

### Re: On a Topological Topos

You won’t find the proof stated in topos-theoretic language, but it is in e.g. [Adámek and Rosický, Proposition 2.26].

But there are subtleties. The category of fields is $\aleph_0$-accessible, hence is the category of models of a theory of presheaf type. This theory is not the coherent theory of fields.

Posted by: Zhen Lin on April 8, 2014 8:35 AM | Permalink | Reply to this

### Re: On a Topological Topos

Yes, it was the Beke paper that I had in mind in the argument above.

It takes the locally finitely presentable case for granted, without citation.

The fact that $Mod(T)$ is locally finitely presentable can be seen to follow from e.g. Theorem D.2.3.6 in the Elephant, since $T$ in this case is a finite limit theory. So you are asking about the fact that $Set^{\mathcal{C}_T}$ is the classifying topos for $T$ where $\mathcal{C}_T$ are the fp objects in $Mod(T)$? I don’t think I’ve ever seen that worked out explicitly but I think it follows from the discussion leading up to Corollary 1.2.5 in Makkai-Pare’s Accessible Categories, if that’s at all helpful.

Also you’re right that there is no mention whatsoever of accessible categories in Moerdijk Mac-Lane. I guess you need both Adamek-Rosicky and Mac Lane-Moerdijk to get Beke’s proposition 0.1. $(ii) \Rightarrow (iii) \Rightarrow (iv) \Rightarrow (ii)$ is provided by Mac Lane-Moerdijk Theorem 2 [bis] on p. 382 since $Flat(\mathcal{C})$ is the free filtered cocompletion of $\C$ and also obviously a category of models for the geometric theory of flat functors $\C \rightarrow Set$. And then $(i) \Leftrightarrow (ii)$ follows from the Representation Theorem 2.26 quoted by Zhen.

By the way in their remark after Theorem 2.26, Adamek-Rosicky write that the fact that $\mathcal{C}$ can be chosen to be the finitely presentable objects of $Mod(T)$ “generalizes [well not really in this case since we are talking about finite presentability] the well-known fact that each Scott domain is a free completion of its finite elements.” I don’t really understand this, but it sounds like a “pure logic” fact.

Posted by: Dimitris on April 8, 2014 4:40 PM | Permalink | Reply to this

### Re: On a Topological Topos

It’s very simple: if $\mathcal{C}$ is a $\kappa$-accessible category, then $\mathcal{C}$ is the free completion of the full subcategory of $\kappa$-presentable objects under $\kappa$-filtered colimits. Conversely, the free completion of any small category under $\kappa$-filtered colimits is $\kappa$-accessible. The order-theoretic analogue of $\aleph_0$-accessible categories are the so-called “algebraic posets”.

I suppose you can phrase it in logic as a downward Löwenheim–Skolem theorem of some kind. It is quite telling that the proof that the category of models for a small geometric sketch is accessible requires such a theorem.

Posted by: Zhen Lin on April 8, 2014 8:30 PM | Permalink | Reply to this

### Re: On a Topological Topos

I understand that finitely accessible categories are the Ind-completions of small categories, which is $(i) \Leftrightarrow (iii)$, so in theory I should be satisfied with Mac Lane and Moerdijk for establishing Beke’s Proposition 0.1. Having digested this a bit more, I think I’d be inclined to say that there’s very little “topos-theoretic content” here – it’s mostly just playing with accessible categories.

Anyway, as Zhen hinted at, we can conclude from the finite accessibility of the category of $\mathbf{Set}$-intervals that $[\Delta^{\mathrm{op}}, \mathbf{Set}]$ classifies a geometric theory whose $\mathbf{Set}$-models are intervals. But in order to identify which geometric theory, we need to do more work, like Johnstone does, or using a more technical theorem like Beke’s Theorem 1.1 and Example 3.3. The way Beke’s theorem works is we let $T$ be the theory of a partial order with bottom and top elements. This is a Cartesian theory. We let $T^+$ be $T$ augmented by the axiom $\forall x, y . x \leq y \vee y \leq x$, a theory of intervals. Then because in $\mathbf{Set}$, every interval is a filtered colimit of intervals which are finitely presentable when considered as models of $T$, and also there are “enough” models of $T^+$, it follows that $T^+$ is classified by copresheaves on its objects which are finitely presentable when considered as models of $T$.

I’m a little concerned about the condition that $T^+$ has “enough $\mathbf{Set}$-models”. Does this just mean that the inverse image functors of the $\mathbf{Set}$-points of its classifying topos $\mathbf{B}[T^+]$ are jointly conservative? If so, how do we verify this without constructing $\mathbf{B}[T^+]$ (which is, after all, what the the theorem is supposed to do for us)?

Posted by: Tim Campion on April 9, 2014 3:26 AM | Permalink | Reply to this

### Re: On a Topological Topos

There are “classical completeness theorems” that assert that the classifying topos of, say, theories in coherent logic automatically have enough points. (This particular one is also known as Deligne’s theorem on coherent toposes.) The theory of intervals is a coherent theory, so its classifying topos, whatever it may be, must have enough points.

Posted by: Zhen Lin on April 9, 2014 8:49 AM | Permalink | Reply to this

### Re: On a Topological Topos

As Sean pointed out, an interesting consequence of $\mathcal{E}$’s being a topos is that it is balanced: any monic epic is an isomorphism. This is kind of interesting. In $\mathbf{Top}$, a map which is the identity on points but goes from a finer topology to a coarser one violates balancedness. How are things different in $\mathcal{E}$?

The monics $X \to Y$ in $\mathcal{E}$, like in any sheaf topos, are the maps that are pointwise injective as maps of presheaves. So they are the functions which are injective on points ($X(1) \to Y(1)$ is injective) and injective on “proofs of convergence” ($X(N^+) \to Y(N^+)$ is injective). In particular, if $X$ is subsequential, so that a “proof of convergence” is determined by its underlying sequence, then it’s necessary and sufficient that $X \to Y$ be injective on points.

The epics are a little more subtle. In a topos, a map $f: X \to Y$ is epic iff the usual logical formula $\forall y \exists x . f(x)=y$ is valid in the internal logic. I’m a little hazy on how this interpretation works exactly, but I think it should mean that $f$ is surjective on points, and that any convergent sequence in $Y$ has a subsequence which is image of a convergent sequence in $X$. Or rather, this should be said of “proofs of convergence”, but it comes to the same thing if $Y$ is subsequential.

So being an epimorphism in $\mathcal{E}$ is a stronger condition than being an epimorphism in $\mathbf{Top}$, or in $\mathcal{F}$ or $\mathcal{F}'$. This is because there are more maps out of a given object to exhibit a map as a non-epimorphism. Any surjection is an epimorphism in the category $\mathcal{F}'$ of subsequential spaces, so we need to use all the flexibility that $\mathcal{E}$ provides to get the strengthened epimorphism condition: exhibiting a surjection $X \to Y$ as a non-epimorphism in $\mathcal{E}$ by two maps $Y \overset{\to}{\to} Z$ requires sending some convergence proof in $Y(N^+)$ to two different proofs in $Z(N^+)$; in particular $Z$ can’t be subsequential.

As a cogenerator, this means that the subobject classifier $\Omega$ must be very far from subsequential. This is borne out in Johnstone’s description of it: $\Omega$ has two points $\{t, f\}$, and every sequence converges, but for sequences converging to $t$ there are many, many different proofs.

Does this make it easier or harder to do topology? I’m not sure.

Posted by: Tim Campion on April 9, 2014 3:36 AM | Permalink | Reply to this

### Re: On a Topological Topos

Something which hasn’t been mentioned yet (at least, not on this blog post) is that the category of subsequential spaces is, while not a topos, at least a quasitopos. Intuitively, a quasitopos is “a topos that may not be balanced”. Moreover, subsequential spaces are:

• a Grothendieck quasitopos: the category of presheaves on a small category (here Johnstone’s category $\Sigma$) that are sheaves for one topology (here, Johnstone’s topology $J$ described above) and separated for another topology (here, the topology where $\mathbb{N}^+$ is covered by the collection of all points $1\to \mathbb{N}^+$).

• the category of concrete sheaves on a concrete site. This is a special sort of Grothendieck quasitopos, where the second topology consists of the “surjections on global points” in the site.

• the category of separated objects for the double negation Lawvere-Tierney topology on Johnstone’s topos $\mathcal{E}$.

A surprising amount of topos theory can also be done for quasitoposes, so if one is bothered by the non-balancedness of Johnstone’s topos, one might consider using subsequential spaces instead.

Posted by: Mike Shulman on April 9, 2014 6:03 AM | Permalink | Reply to this

### Re: On a Topological Topos

One answer to the question of why you’d want a topos of spaces is so that you get a cartesian closed category. One thing I was surprised to learn a few years ago is that there is in fact a (unique) closed monoidal structure on the category of all spaces.

This is all worked out very clearly in the obvious chapter of Borceux’s Handbook of categorical algebra volume II. I don’t know whether the results are due to him.

The first theorem is that any closed monoidal structure on all spaces must have the underlying set of the monoidal product the cartesian product of the underlying sets, and it must have the underlying set of the internal hom equal to the set of all continuous functions between the given spaces.

The next result is that there is a unique such structure, with the function spaces given the topology of pointwise convergence. So the monoidal product is not the cartesian product: “asterisks” are open sets.

The “pointwise convergence” appearing here suggests a connection to the Johnstone topos. Can anyone illuminate this?

Posted by: Emily Riehl on April 11, 2014 10:45 PM | Permalink | Reply to this

### Re: On a Topological Topos

Pardon me for being dense, but what’s an asterisk, in this context?

Posted by: David Roberts on April 12, 2014 1:54 AM | Permalink | Reply to this

### Re: On a Topological Topos

Yeah, I should have spelled this out.

The product is topologized so that $f \colon X \times Y \to Z$ is continuous just when $f$ is separately continuous in each variable, i.e., when $f(x,-) \colon Y \to Z$ and $f(-,y) \colon X \to Z$ is continuous for each point $x \in X$ and $y \in Y$. So this is the largest topology so that the maps $(x,-) \colon Y \to X\times Y$ and $(-,y) \colon X \to X \times Y$ are continuous.

Borceux points out that an “asterisk” is then an open neighborhood of the origin in $\mathbb{R} \times \mathbb{R}$; the figure is at the top of page 359. What it looks like is a four-leaf clover whose leaves align with the axes and meet at the origin. The boundary points are not in the open set, with the exception of the origin.

If you look at horizontal or vertical cross sections of the asterisk, you see open intervals in $\mathbb{R}$, so this is open in the asterisk topology. But of course it’s not open in the usual product topology because this set contains no open ball around the origin.

Posted by: Emily Riehl on April 12, 2014 6:21 PM | Permalink | Reply to this

### Re: On a Topological Topos

That’s very interesting! I had never run across that result.

For what it’s worth, the function spaces in Johnstone’s topos are not pointwise convergence; they’re “continuous convergence”. A sequence of functions $f_n$ converges continuously to a limit $f$ if for any sequence $x_n$ in the domain converging to a point $x$, the images $f_n(x_n)$ converge to $f(x)$.

Posted by: Mike Shulman on April 12, 2014 6:29 AM | Permalink | Reply to this

### Re: On a Topological Topos

Oh, hmm, nevermind then.

Posted by: Emily Riehl on April 12, 2014 6:22 PM | Permalink | Reply to this

### Re: On a Topological Topos

This is an interesting notion of convergence: I wonder if it has a standard name?

Sean mentioned in the seminar that the subsequential spaces form an exponential ideal in Johnstone’s topos: if $Y$ is subsequential and $X$ is arbitrary, then $[X,Y]$ is subsequential. I’ve just managed to convince myself of this by unwinding the definitions; apparently it follows directly from the fact that the subsequential spaces are the separated objects for some topology on Johnstone’s topos, i.e. the fact that they form a Grothendieck quasitopos, as Mike mentioned above.

In particular, this means that subsequential spaces form a cartesian closed subcategory of Johnstone’s topos. I’m curious now: do the sequential spaces have similar nice properties? Are they cartesian closed? If not, then are the subsequential spaces somehow a “minimal” cartesian closed supercategory of the sequential spaces?

Posted by: Tim Campion on April 13, 2014 3:23 PM | Permalink | Reply to this

### Re: On a Topological Topos

It is the smallest topology with respect to which the evaluation map is (sequentially) continuous, and is closely related to the topology of uniform convergence on compacts. I would hypothesise it is equivalent to uniform convergence on (sequential) compacts when the range is a uniform space.

Posted by: Tom Ellis on April 13, 2014 4:35 PM | Permalink | Reply to this

### Re: On a Topological Topos

I think “continuous convergence” is the standard name.

do the sequential spaces have similar nice properties? Are they cartesian closed?

According to the nlab, yes.

Posted by: Mike Shulman on April 14, 2014 3:37 AM | Permalink | Reply to this

### Re: On a Topological Topos

As for the attribution: Borceux refers to a paper of Pedicchio and Solomini. In his book, Borceux only proves the existence of a symmetric monoidal closed structure on $\mathbf{Top}$, but not its uniqueness. Pedicchio and Solomini do prove the uniqueness, and in fact they show that $\mathbf{Top}$ admits a unique monoidal biclosed structure, symmetric or not. In contrast, they mention that there is a proper class of monoidal closed structures on $\mathbf{Top}$ which are not biclosed.

Skimming their introduction, they apparently use a notion of Isbell called a topological topology, which is a topology on the lattice of open sets of a topological space $X$ which makes arbitrary union and finite intersection into continuous maps. Weird!

Posted by: Tim Campion on April 13, 2014 1:49 AM | Permalink | Reply to this

### Re: On a Topological Topos

It seems like Peter Scholze’s work on condensed mathematics is much like top. Compactly generated spaces embed fully faithfully into condensed sets (sheaves over the etale site on a point, covers are jointly surjective collections of maps), not minding certain size constraints they put on the category. I think this is equivalent to sheaves on the category of compact hausdorff spaces.

Posted by: Dean Young on August 9, 2019 4:17 PM | Permalink | Reply to this

### Re: On a Topological Topos

Yes, condensed sets and the related pyknotic sets are certainly another approach to this sort of thing. Do they have good behavior relative to colimits like Johnstone’s topos?

Posted by: Mike Shulman on August 10, 2019 10:10 AM | Permalink | Reply to this

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View Single Post
 2003-12-03, 14:04 #3 GP2     Sep 2003 5·11·47 Posts One of the scripts I run regularly checks for exponents that are assigned but have already been factored. The most recent run showed: Code: Exponents assigned (LL testing) that already have a factor? ------- factors ------- 10548491,114813042292255308481 20861773,53368898818455899608663 21236561,1830770648128687616906281 21250771,231495979763797238268689074541177 21397807,125957393078551400623 21398227,88156696745048623921 ------- STATUS_L.TXT ------- 10548491,D ,64,,13.9,16.7,60.7,02-Dec-03 19:54,19-Nov-03 05:05,Team_Prime_Rib,Reboot_It180 20861773, ,66,,45.3,3.6,45.6,21-Oct-03 17:52,18-Oct-03 20:38,bsman,p41800s 21236561, ,66,,14.2,2.8,62.8,,18-Nov-03 22:27,CDavis,P2530 21250771, ,66,,13.3,29.7,74.7,,19-Nov-03 18:34,DL1ELY,DL1ELY 21397807, ,66,,2.9,11.1,71.1,,30-Nov-03 04:15,.,C19214038 21398227, ,66,,2.9,28.1,85.1,,30-Nov-03 04:47,curtisc,ccc54 When I find cases like this, I re-report the factors to PrimeNet using the manual report page, under Dummy_Results account created by garo. This hopefully prevents an entirely useless LL test from being done... hopefully, the next time Prime95 updates its info with the server, it sees "this exponent is not assigned to you" and deletes it from the worktodo.ini file of whoever it was assigned to. The factor did make it into the "weekly" data files (Dec 2 release), which is how I detected it, so presumably you got credit for it. I haven't figured out how this situation occurs, but every week there are at least one or two cases of assigned exponents that already have a factor.


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Now showing items 1-2 of 2

• #### A functional limit theorem for stochastic integrals driven by a time-changed symmetric sigma-stable Levy process

(2014-12-31)
Under proper scaling and distributional assumptions, we prove the convergence in the Skorokhod space endowed with the $M_1$-topology of a sequence of stochastic integrals of a deterministic function driven by a time-changed ...
• #### On the convergence of quadratic variation for compound fractional poisson processes

(2012-12-31)
The relationship between quadratic variation for compound renewal processes and M-Wright functions is discussed. The convergence of quadratic variation is investigated both as a random variable (for given t) and as a ...


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# boltzmann constant

Q B p a additional driving pressure.   beschrieben werden kann: Bei Raumtemperatur (T = 293 K) beträgt der Wert der Temperaturspannung ungefähr 25 mV. [6][7] and at the standard state temperature of 298.15 K (25.00 °C; 77.00 °F), it is approximately 25.69 mV. {\displaystyle \Omega } und tritt beispielsweise im Gasgesetz für ideale Gase als eine der möglichen Proportionalitätskonstanten auf: Bezogen auf Normalbedingungen (Temperatur . Diese lautet: In Halbleitern besteht eine Abhängigkeit der Spannung über einen p-n-Übergang von der Temperatur, die mit Hilfe der Temperaturspannung 1. The Boltzmann constant is a very important constant in physics and chemistry. [3] "The amount of thermal radiation emitted increases rapidly and the principal frequency of the radiation becomes higher with increasing temperatures". Die Boltzmann-Konstante (Formelzeichen oder ) ist eine Naturkonstante, die in der statistischen Mechanik eine zentrale Rolle spielt.   mit der Maßeinheit 1/mol die Avogadro-Konstante ist.   oder T Dazu kommen bei ausreichend hohen Temperaturen noch Schwingungen der Bindungen. Dies ist in einem abgeschlossenen (isolierten) System stets der Fall (Zweiter Hauptsatz der Thermodynamik). Monatomic ideal gases (the six noble gases) possess three degrees of freedom per atom, corresponding to the three spatial directions, which means a thermal energy of 3/2kT per atom. See the linked section for details. He made significant contribution to the field of statistical mechanics. {\displaystyle Q} [2] The equation can also be derived from Planck's law, by integrating over all wavelengths at a given temperature, which will represent a small flat black body box. In semiconductors, the Shockley diode equation—the relationship between the flow of electric current and the electrostatic potential across a p–n junction—depends on a characteristic voltage called the thermal voltage, denoted VT. So hat Wasser eine extrem hohe Wärmekapazität durch eine große Zahl solcher Schwingungsfreiheitsgrade. k   stellt dabei jene Energie dar, um die Entropie Ω B The Boltzmann constant has the dimension energy divided by temperature, the same as entropy. Updates? In thermodynamics, Boltzmann constant is the physical constant relating average kinetic energy of the gas particles and temperature of the gas represented by k or k B.The value of Boltzmann constant is measured using J/K or m 2 Kgs-2 K-1.Boltzmann constant is one of the seven defining constants that have been given exact definitions. In classical statistical mechanics, this average is predicted to hold exactly for homogeneous ideal gases. The first and third values are exact; the second is exactly equal to 1380649/16021766340. Aus Wikipedia, der freien Enzyklopädie. What is Stefan Boltzmann Law? Pro, Vedantu   kann die Gasgleichung umformuliert werden zu: Allgemein ergibt sich für die mittlere kinetische Energie eines klassischen punktförmigen Teilchens im thermischen Gleichgewicht mit S h The Boltzmann constant (k B or k) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. We also know for n  =  1 mole, we get Na =  no of particles in one mole (Avogadro’s number) which has no unit. 1, 1895, S. 40, siehe Ingo Müller, Zuletzt bearbeitet am 31. B. in den Sitzungsberichten der Wiener Akademie 1877 oder den Vorlesungen über Gastheorie, Bd. It plays a major role in the statistical definition of entropy. Slovenian physicist Josef Stefan formulated the constant in 1879, and it was later derived in 1884 by Austrian physicist Ludwig Boltzmann. Where  N  is the number of molecules   and  Na  is the Avagadro's number or Avogadro Constant, Value of  Na=  6.022 x 10 ^ 23 molecules per mole. A k mittels der Avogadro-Konstanten NA. L It is used to express Boltzmann factor. The Stefan–Boltzmann constant (also Stefan's constant), a physical constant denoted by the Greek letter σ (sigma), is the constant of proportionality in the Stefan–Boltzmann law: "the total intensity radiated over all wavelengths increases as the temperature increases", of a black body which is proportional to the fourth power of the thermodynamic temperature. Kb is a Bridge between Macroscopic and Microscopic Physics. At lower temperatures, not all these degrees of freedom may fully participate in the gas heat capacity, due to quantum mechanical limits on the availability of excited states at the relevant thermal energy per molecule. Corrections? k Boltzmann constant (= 1.38065812 e −23 J/K) κ curvature. Das statistische Gewicht For a Classical system at equilibrium at temperature (E), the average energy per Degree of freedom is, In the elementary illustration of the gas comprising D non-interacting atoms, each atom has  three transitional degrees of freedom (X, Y, & Z directions). {\displaystyle k} Diese Gleichung, in leicht abgewandelter Nomenklatur, ist im Grabstein von Ludwig Boltzmann am Wiener Zentralfriedhof eingraviert. There was considerable disagreement in the second half of the nineteenth century as to whether atoms and molecules were real or whether they were simply a heuristic tool for solving problems. Die Boltzmann-Konstante hat die Dimension Energie/Temperatur. Value of Boltzmann’s constant in erg/kelvin, Relation Between the Length of a Given Wire and Tension for Constant Frequency Using Sonometer, NCERT Solutions for Class 12 Biology Chapter 12, NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry, NCERT Solutions for Class 11 Physics Chapter 2, NCERT Solutions for Class 12 Physics Chapter 5, NCERT Solutions for Class 12 Physics Chapter 4, NCERT Solutions for Class 12 Physics Chapter 2, NCERT Solutions for Class 11 Physics Chapter 6, NCERT Solutions for Class 12 Physics Chapter 1, NCERT Solutions for Class 12 Physics Chapter 9, Important Questions for CBSE Class 12 Physics Chapter 5 - Magnetism And Matter, Important Questions for CBSE Class 12 Physics, Important Questions for CBSE Class 11 Physics Chapter 2 - Units and Measurement, Important Questions for CBSE Class 11 Physics, Important Questions for CBSE Class 12 Physics Chapter 4 - Moving Charges and Magnetism, Important Questions for CBSE Class 11 Physics Chapter 6 - Work, Energy and Power, Important Questions for CBSE Class 12 Physics Chapter 2 - Electrostatic Potential and Capacitance, Important Questions for CBSE Class 12 Physics Chapter 1 - Electric Charges and Fields, Important Questions for CBSE Class 12 Biology Chapter 12 - Biotechnology and its Applications, Important Questions for CBSE Class 11 Physics Chapter 7 - Systems of Particles and Rotational Motion, CBSE Class 12 Physics Magnetism And Matter Formula, CBSE Class 11 Physics Work, Energy and Power Formulas, CBSE Class 12 Physics Electrostatic Potential and Capacitance Formula, CBSE Class 12 Physics Moving Charges and Magnetism Formula, CBSE Class 12 Physics Electric Charges and Fields Formula, CBSE Class 12 Physics Ray Optics and Optical Instruments Formula, CBSE Class 11 Physics Systems of Particles and Rotational Motion Formulas, CBSE Class 12 Physics Dual Nature of Radiation and Matter Formula, CBSE Class 12 Physics Question Paper 2020, Previous Year Question Paper for CBSE Class 12 Physics, Previous Year Question Paper for CBSE Class 12 Physics - 2015, Previous Year Question Paper for CBSE Class 12 Physics - 2018, Previous Year Question Paper for CBSE Class 12 Physics - 2014, Previous Year Question Paper for CBSE Class 12 Physics - 2013, Physics Question Paper for CBSE Class 12 - 2016 Set 1 C, Previous Year Question Paper for CBSE Class 12 Physics - 2019, Previous Year Physics Question Paper for CBSE Class 12 - 2017, Vedantu {\displaystyle S} Pro, Vedantu Diese Gleichung verknüpft – über die Boltzmann-Konstante als Proportionalitätsfaktor – die Mikrozustände des abgeschlossenen Systems mit der makroskopischen Größe der Entropie und bildet die zentrale Grundlage der statistischen Physik. How Boltzmann’s constant links the macroscopic and microscopic worlds may perhaps be easiest seen like this: k is the gas constant R (remember the … The thermal voltage depends on absolute temperature T as. Allgemeiner tritt die Boltzmann-Konstante in der Wahrscheinlichkeitsdichte beliebiger Systeme der Statistischen Mechanik im thermischen Gleichgewicht auf: Die thermische Wahrscheinlichkeitsdichte solcher Systeme bei der Thermodynamischen Temperatur $T$ lautet $e^{-\frac{E}{k T}}/Z$ mit einer Normierungskonstanten $Z$, wobei $E$ die Energie ist. For a classical system at equilibrium at temperature T, the average energy per degree of freedom is kT/2. In a p -n junction diode, the thermal voltage (U) is related to the absolute temperature (E) with this equation: U = Kb x E /  n,  n is the charge on an electron which is equal to 1.6 x 10 ^ - 19 C. Vedantu academic counsellor will be calling you shortly for your Online Counselling session. Ein Molekül ohne eine solche Symmetrie hat 3 Rotationsfreiheitsgrade, also insgesamt 6. The constant of proportionality k serves to make the statistical mechanical entropy equal to the classical thermodynamic entropy of Clausius: One could choose instead a rescaled dimensionless entropy in microscopic terms such that. of each H2O (water) molecule.   (Ergebnisraum) der entsprechend möglichen Mikrozustände (bzw. the Nernst equation); in both cases it provides a measure of how much the spatial distribution of electrons or ions is affected by a boundary held at a fixed voltage. We can see that Boltzmann's Constant (Kb) has the same dimensional formula as of Entropy (S). 0 Planck's 1920 lecture continued:[12]. ρ Boltzmann-Konstante - Boltzmann constant. Sorry!, This page is not available for now to bookmark. In the simplest example of a gas consisting of N noninteracting atoms, each atom has three translational degrees of freedom (it can move in the x-, y-, or z-directions), and so the total thermal energy of the gas is 3NkT/2. Any vessel filled with water, cover it with a lid, and put it on fire, now before it starts boiling, just remove the lid, put your hand over the vessel of the steam, and it feels hot to your hand. The constant relates the average kinetic energy of molecules of a gas with thermodynamic temperature. Yunus A. Çengel, McGraw Hill, 2007, Learn how and when to remove this template message, "2018 CODATA Value: Stefan–Boltzmann constant", Scientists whose names are used in physical constants, List of scientists whose names are used as SI units, https://en.wikipedia.org/w/index.php?title=Stefan–Boltzmann_constant&oldid=967639468, Wikipedia articles with style issues from March 2020, Creative Commons Attribution-ShareAlike License, This page was last edited on 14 July 2020, at 12:10. {\displaystyle N_{\mathrm {L} }={\tfrac {N}{V_{0}}}} Zuvor war das Kelvin anders definiert, und


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# Excel VBA macro loops through 20-28 sheets taken from filenames in cells and pastes them to the main workbook

My apologies for a "repost", my original post was only partial and confusing. Here is a better snippet. This runs fairly slow (30 seconds or so) cycling through 20 workbooks. Any suggestions for improvement would be appreciated; I have left out my dimming of variables at the beginning, as well as my error handling portions, as those all work just fine.

For even MORE details, the user enters file paths into a column of up to 28 filepaths, and then the adjacent column derives the file name from the path. This macro, once initiated via a command button, begins cycling through all those filepaths, opening the document, copying and pasting it to the main workbook, until it reaches an empty cell in the column, to which then it stops.

There are also a few other small parts in there, like updating the last modified date when the button is clicked, as well as taking information from the first opened sheet to let the user know (on the main page) what kind of information is being pulled or has been pulled. Additionally, I often include frequent status bar updates for my impatient users who don't like staring at a spinning cursor without being told what's going on.

Private Sub CommandButton19_Click()

Application.DisplayStatusBar = True
Application.ScreenUpdating = False 'run faster
Application.DisplayAlerts = False 'prevents clipboard prompt
Application.Cursor = xlWait

On Error GoTo HandleIt

Dim RangeDifferentiator As Integer
Dim i As Long
Dim SheettoCopyLoop
Dim FileToCopyLoop
Dim FileNameLoop

For i = 1 To 28 'start loop (28 because the last row being used is 34. If we need more rows this needs to change)

RangeDifferentiator = i + 5
SheettoCopyLoop = Range("D" & RangeDifferentiator)
FileToCopyLoop = Range("E" & RangeDifferentiator)
FileNameLoop = Range("F" & RangeDifferentiator)

If FileToCopyLoop <> "" Then

Application.StatusBar = "Pulling data for " & SheettoCopyLoop & "..." 'let's us know which one it's pulling
Sheets(SheettoCopyLoop).Cells.ClearContents
Workbooks.Open Filename:= _
FileToCopyLoop
'Needs optimizing
Sheets("Summary").Cells.Copy

'Establishes if this is a MTD or YTD file and up to what month it's pulling
If i = 1 Then
Workbooks("YTD File.xlsm").Sheets("SM Summaries").Range("C2").Value = ActiveWorkbook.Sheets("Summary").Range("C6").Value
Workbooks("YTD File.xlsm").Sheets("SM Summaries").Range("C4").Value = ActiveWorkbook.Sheets("Summary").Range("C5").Value
End If

'Needs optimizing
Windows("YTD File.xlsm").Activate
Sheets(SheettoCopyLoop).Select
Sheets(SheettoCopyLoop).Range("A1").Select
ActiveSheet.Paste
ActiveSheet.Range("A1").Select
Windows(FileNameLoop).Close
Range("A1").Select

Else
Application.ScreenUpdating = True
Application.Cursor = xlDefault
MsgBox "Data pull complete!", vbInformation, "Success!"
Exit Sub
End If

Next i
Exit Sub

'The following is all error handling to help show what went wrong and where
HandleIt:
If SheettoCopyLoop <> "" Then
errorhandler = MsgBox("Oops! An error has occured on page " & SheettoCopyLoop & ". Check to make sure file exists/file path is correct.", vbExclamation, "Error!")
Application.StatusBar = "Error occured on page " & SheettoCopyLoop 'Show where error occured on status bar if applicable
Else
errorhandler = MsgBox("Oops! An unknown error has occured.", vbCritical, "Error!")
Application.StatusBar = "Unknown error"
End If
Range("H7").Select
Application.ScreenUpdating = True
Application.Cursor = xlDefault
Exit Sub
End Sub

I'm not sure why it takes 30 seconds to process the 20 files, because with my almost empty test files, your code only takes 2.202 seconds (21 files). I'm guessing those files are quite large, contain a lot of objects, charts, data (?)

If they have a lot of rows I'd suggest changing them to the xlsb format, which will compress them quite a bit (size could go down to 40%), and they will be a lot faster to open and close

XLSB Format

XLSB is the Excel binary format. It is structured as a compressed folder that contains a large number of binary files. It is much more compact than the XLS format, but the amount of compression very much depends on the contents of the workbook. For example, ten workbooks show a size reduction factor ranging from two to eight with an average reduction factor of four...

Another consideration is that interacting with files over the network is considerably slower than the local drive

Related to the code, there are a few basic issues:

• To properly replace the select statements you need to refer to all objects explicitly:

Windows("YTD File.xlsm").Activate
Sheets(SheettoCopyLoop).Select
Sheets(SheettoCopyLoop).Range("A1").Select
ActiveSheet.Paste

use:

Windows("YTD File.xlsm").Sheets(SheettoCopyLoop).Range("A1").Paste

• Indent your code to make it easier to read, and distinguish hierarchies of logic and nesting
• It's better to prevent all errors than to rely on the "On Error GoTo" statements
• There are few errors that cannot be avoided
• The GoTo statement breaks the flow and structure of the code
• Turn off Application.Calculation (to manual) if possible
• Don't force the user to dismiss dialog boxes after every iteration
• They only need to be aware if their task cannot be completed

I made some changes, such as copying the entire sheets to the main Workbook, instead of copying only the data, so here is how I would change your code:

Option Explicit

Private Sub CommandButton19_Click2()
Dim wb As Workbook, ws As Worksheet, newWB As Workbook
Dim wsSMS As Worksheet, wsS As Worksheet, wsNew As Worksheet
Dim fileInfo As Variant, sheetToCopy As String, r As Long
Dim fileToCopy As String, fileName As String, t As Double

t = Timer

OptimizeXl True, xlWait

Set wb = ThisWorkbook
Set wsSMS = wb.Worksheets("SM Summaries")

fileInfo = ws.Range("D6:F34").Value2

For r = 1 To UBound(fileInfo)               'max xl sheet name length = 31 chars
sheetToCopy = Left(fileInfo(r, 1), 31)  'column 1 of Range("D6:F34") (Col D)
fileToCopy = fileInfo(r, 2)             'column 2 of Range("D6:F34") (Col E)
fileName = fileInfo(r, 3)               'column 3 of Range("D6:F34") (Col F)

'Continue only if all necessary file details are not empty
If Len(fileToCopy) > 0 And Len(sheetToCopy) > 0 And Len(fileName) > 0 Then
If Len(Dir(fileToCopy)) > 0 Then    'continue only if file exists
Application.StatusBar = "Pulling data for " & fileName & "..."

If WsExists(newWB, "Summary") Then
'Delete existing sheet named "sheetToCopy"
If WsExists(wb, sheetToCopy) Then wb.Worksheets(sheetToCopy).Delete
Set wsS = newWB.Worksheets("Summary")
If r = 1 Then
wsSMS.Range("C2").Value = wsS.Range("C6").Value
wsSMS.Range("C4").Value = wsS.Range("C5").Value
End If
'Copy new sheet into this workbook, and rename it to "sheetToCopy"
wsS.Copy After:=wb.Worksheets(wb.Worksheets.Count)
Set wsNew = wb.Worksheets(wb.Worksheets.Count)
wsNew.Name = sheetToCopy
Else
Debug.Print "Invalid sheet name for file on row " & r
End If
newWB.Close False
Else
Debug.Print "Invalid fileToCopy on row " & r & " (" & fileToCopy & ")"
End If
Else
Debug.Print "Empty fileToCopy, sheetToCopy, or fileName on row " & r
End If
Next
ws.Activate
OptimizeXl False
Debug.Print "V2 Time: " & Format(Timer - t, "0.000") & " sec"
MsgBox "Data pull complete!", vbInformation, "Success!"
End Sub

Private Sub OptimizeXl(ByVal stat As Boolean, Optional ByVal curs As Long = xlDefault)
Application.Calculation = IIf(stat, xlCalculationManual, xlCalculationAutomatic)
Application.ScreenUpdating = Not stat
Application.Cursor = curs
End Sub

Private Function WsExists(ByRef wb As Workbook, ByVal wsName As String) As Boolean
Dim ws As Worksheet
If Not wb Is Nothing Then
If Len(wsName) > 0 Then
For Each ws In wb.Worksheets
If ws.Name = wsName Then
WsExists = True
Exit Function
End If
Next
End If
End If
End Function

Test results:

V1 Time: 2.202 sec (your code)
V2 Time: 1.012 sec

• Was just able to view this now, very helpful, thank you very much! I had actually started using Option Explicit about a month ago, I didn't include it here... Wish I had known more about that when I was writing hundreds of pages in Word VBA and running into errors all the time... One thing I must say, though, is I'm very adamant about keeping in dialog boxes for after all interactions. Why? Because my users aren't always tech savvy, and if they click a button and aren't told what happened, they might think it just doesn't work and try clicking it again. I keep those there for guidance. – dwirony Jul 26 '17 at 15:01


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# In a Chess Tournament Each of the 6 Players Will Play Every Other Player Only Once. How Many Matches Will Be Played During the Tournament? - Mathematics

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MCQ

In a chess tournament, each of the 6 players will play every other player only once. How many matches will be played during the tournament?

#### Options

• 12

• 15

• 30

• 36

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#### Solution

15
Explanation:
Required number of matches = "^6C_2

= (6!)/(2!xx4!)=15

Concept: Permutation and Combination (Entrance Exam)
Is there an error in this question or solution?
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# Triangles in triangles

Geometry Level 4

Suppose there is an equilateral triangle with an area of 2016 square units. $$AB, BC, CA$$ was divided by the points $$D, E, F$$ respectively such that $$AD:DB=BE:EC=CF:FA=1:2$$. The intersection of $$AE, BF$$ and $$CD$$forms a triangle $$XYZ$$. Find the area of triangle $$XYZ$$.

×