The n-Category Café

I wouldn’t call it particularly “the author’s choice of terminology” in this instance, since that might suggest that the author was doing something idiosyncratic instead of just following a tradition (however irreflectively :-) ).

You are quite right; I realized after I wrote that that it was poor phrasing. Part of the point I wanted to make is exactly what you said: that we can try to be a force for good terminology as well as good mathematics.

Not that one should go around changing terminology willy-nilly. But I think that “Grothendieck topology” is bad enough to be worth trying to change. I like “coverage” for a system of covers on a category, but your suggestion of “local modal operator” for the thing formerly known as a Lawvere-Tierney topology is certainly attractive. It’s a little long though; what about just “local modality”?

Three hits for pseudo\-functor.

I can’t create an entry becasue I don’t know the definitions

Don't let that stop you!

Already many entries have been created, where someone just describes the basic idea, leaving an ellipsis where the definition should go.

Most of these now have definitions, often written by a different person.

Seconded. If you are confused about something, it is likely that other people may also be confused about it, now or in the future, and when the answer is incorporated into the page it will make it that much better of an expository resource.

I agree that questions about a given topic should appear on the page devoted to that topic, rather than “General Discussion”. As long as they’re clearly separated from the main part of page - maybe at the end, in a section called “Discussion” or “Questions” or something - they should be helpful to other readers who have similar questions.

What I’m wondering about is this. After a question has been answered, and some time has passed, would it be okay for me to go back and “polish” the question a bit, or the answers? Often a question will be a mixture of idiosyncratic concerns and some classic “question that everyone has”… and I feel that in the long run, most new readers would prefer to read the answer to a polished version of the question - the sort of thing you see in a FAQ.

For example, in the page monoidal categories, Eric asked if monoidal categories could be defined by internalization. This is a classic question: everyone who learns category theory should eventually ask:

Can we define monoidal categories using internalization? For example, is a monoidal category is a monoid object in $\mathrm{Cat}$?

The answer to this is quite instructive — so this question, and the answer(s), deserve to live permanently on this nLab page.

But Eric didn’t ask the question in its “perfected form”. So, I’m wondering if it’s okay for me to go back and “polish” it.

Obviously this runs the risk of annoying the person who originally asked the question. And this may be even more true when it comes to perfecting people’s answers. So, what should we do?

One possibility is to have a FAQ section that distills the results of discussions, separate from the actual discussion.

I like that idea!

Maybe even with a clear heading

#Note to New Contributors#

Once you edit a page, your name will appear at the bottom, grayed out with a question mark since you don’t have a user page yet. You may take this as an invitation to create your user page and tell us about yourself. But if you don’t want to or don’t have the time right now, you should also feel free to just make a blank user page for yourself, containing only ‘category: people’ (so that you show up in the list of contributors).

I was going to say, “If this is agreeable, I can add it.” But in the wiki spirit, I will just add it and if you’d like to improve upon it or remove it, fell free :)

Yes! I was going to say: “Please go ahead and add whatever you think should be added.” Now Eric already did. Good.

The comments here are quite sensible, except that most of them are based on this misconception.

Bad proofreading; “this misconception” referred to a paragraph that I deleted from my final post, not to Eric's paragraph that I quoted; Eric's paragraph seems to based on that misconception, but it does not actually contain it. (Which is: that category: people is the place to look for a list of contributors.)

Thanks. I will try it.

Hi Tim,

I would love to see anything you have to say, e.g. notes, lectures, whatever. I think I am a good target audience for what you are trying to do (PhD in applied physics/engineering, but have been e-hanging around with these guys for decades).

If you have anything you prefer sending via email, mine is eforgy at yahoo, but providing links works too!

I agree with Tim. And I think that this goes for more than just enrichment (and, obviously, internalization). We naturally like, in a page's first draft, to get down the cool $n$-categorial way of looking at something; but when we get a fleshed-out article that we expect newcomers to learn something from, then we really should start with the familiar and move from there to the new and exciting.

On the other hand, one can certainly define sheaves of anything one likes. It may be a bit harder if the site is also enriched. But I don’t know a whole lot of examples.

The motivating class of exmaples which made me come to post the above comment is the case where $V$ is some category of “higher structures”, such as simplicial sets, $\omega$-categories.

Following our discussion at $\mathrm{\infty }$-stack homotopically we can model a whole lot of $\mathrm{\infty }$-category theory in terms of the $V$-enriched homotopical category of $V$-valued (pre)sheaves on some locally small category $S$.

But thinking about this, the statement becomes most natural if we think of $S$ as being $V$-enriched itself. Mostly implicitly one thinks here of a natural canonical inclusion $\mathrm{Sets}\hookrightarrow V$.

But in most every discussion of covers, this makes people run into a little cheat: in the context of locally small $S$ covers are described by a variety of means, but in most applications a much more elegant description arises after enlarging the perspective suitably: covers in $S$ are just suitable acyclic fibrations in $[S^{\mathrm{op}},V]$ after the Yoneda embedding.

(There is a discussion (which has potential for improvement…) at the end of $n$Lab:sieve which mentions how a sieve is nothing but the presheaf represented by the acyclic fibration in $[S^{\mathrm{op}},V]$ induced from the cover. )

That looking at $S$ as a locally small, i.e. Sets-enriched category here is in some respects suboptimal or at least slightly awkward is rarely mentioned in the litarture. It is mostly considered as obvious, I suppose. I found one remark which mentions the awkwardness explicitly as footnote 3 on p. 10 of Toën’s Higher and derived stacks.

Maybe I am seeing ghosts here, but this made me think that there is reason to comtemplate seriously the possibility that we want to think of $S$ here as $V$-enriched for possibly more general enrichements than those factoring through $\mathrm{Sets}\hookrightarrow V$ and consider $[S^{\mathrm{op}},V]$ in the fully $V$-enriched context. For instance instead of taking $S=\mathrm{Top}$ or $S=\mathrm{Diff}$ as usual, one could start with $S={\mathrm{Top}}^{{\Delta }^{\mathrm{op}}}$ and $S={\mathrm{Diff}}^{{\Delta }^{\mathrm{op}}}$, which would make the covers of spaces be honest objects in $S$.

Whether one does this fully explicitly or not, it seems that a bunch of general questions arising when modelling $\mathrm{\infty }$-categorical structures using “enriched homotopy theory” or “homotopy coherent category theory” should naturally live in a context of “sheaf and topos theory enriched over $V$”. Or that was at least what motivated my remark.

NQ-supermanifolds are equivalent to Lie $\mathrm{\infty }$-algebroids, this is not obvious but is an important fact that must be noted

I doubt it’s even true - super implies $\mathbb{Z}/2$ graded

And the “$N$” in “NQ” says that this ${\mathbb{Z}}_2$-grading is lifted to a $\mathbb{N}$-grading.

This is well known to be true in low degree and has been proposed to be the general definition at least by Pavol Ševera, and is being generally adopted by people thinking about it.

I thought I gave references at the nLab. Maybe these ended up more in the entry on “Lie theory” or somewhere else. I’ll try to go through the entries and make this clearer.

(This gives me a chance to voice a general experience: writing a single thought into a single entry is one thing, producing a coherent wiki quite another which is beginning to require quite a bit of effort…)

Speak of the devil!

I see Urs has just created a page [[Yoneda embedding]] with content that also appears on [[Yoneda lemma]]. As a general practice, I would suggest that once the new page [[Yoneda embedding]] is created, that we replace the Section “Yoneda embedding” with a link to [[Yoneda embedding]] to avoid duplicate content.

I will make the change (and you can revert back if you want to), but suggest this as a way to go moving forward.

But on the topic of “Duplication”, I’ve seen an example or two where content was copied from one page to another. I’m not sure that is a good idea.

When I say that we should duplicate subjects, I don't mean that we should duplicate content. Each page will explain the ideas from the perspective of the terminology and definitions that it uses.

On the other hand, I do think that copying can be of some use, especially when starting a new page. But we shouldn't expect to keep them in sync; again, each page will discuss the material in its own way.

If [[foo]] talks about bars (including, but perhaps not limited to, their relationship to foos), but we realise that we want a separate page [[bar]], then I would initially copy the material on bars from [[foo]] to [[bar]]. Then I would rework that material to start from the notion of bar, which may amount to simply changing the introduction. And then I would abbreviate the discussion on bars in [[foo]], focussing on their relation to foos but removing the general facts about bars. At this point there may (or may not) remain duplicate sentences or even whole paragraphs about the relationship between foos and bars, but I would make no attempt to keep them identical in future edits (unless such an edit amounted to correcting an error or something like that).

I wonder if there is a way to automate this.

But I want everybody to drop a small human-written comment at “latest changes” describing what it is that was changed.

I think we really need this. We can’t just all go around working on the wiki without systematically telling each other what it is we are doing.

A couple questions for the community on this:

I would answer “yes” at least to the second of these questions. I find it very useful if after a day of work on something else, I can just open the $n$Lab, go to the “latest changes” page and get a quick idea for what happened this day. This will help me organize my time and attention for the $n$Lab and make my time spent on it more effectively useful for the general progress.

Logging editorial work at the pages latest changes was just a suggestion on my part, since it seemed to me to be beneficial for everybody involved. But it’s just a suggestion.

Everybody is perfectly free not to follow this suggestion of course. Please, I’d rather have you work on some entry and not log about it than not work on some entry!

Myself, I won’t invest energy in automatizing the “lastest changes” page, for one because for me its value is the human-written comments in it, but if you want to go ahead, don’t let me stop you. Concerning questions about how to change code of the $n$Lab, you’ll have to get in contact with Jacques. Much as I would like to, I don’t see myself finding the time to look into such activity.

But it would generally be good, for other purposes too, if somebody actively involved in the $n$Lab were knowledgable enough to be able to play around with the code.

For what it’s worth, I created this last week:

Causal Graphs

Obviously, I agree with you :)

I created that entry for Brown’s article because it is being referred to several times at several entries. I was thinking that when I next want to refere to it, I wouldn’t have to type the full bibliographical details but just give the link to the article’s own entry (which, on top of the bib information, has now a summary of the article).

Moved it to BrownAHT now.

I don’t think we need a page for every reference, but I can see that it might be useful to have pages for those that are referred to fairly frequently, in addition to those about which we have specific things to say.

The “alpha” convention in BibTeX is what you get when you say \bibliographystyle{alpha}; it produces things like [Bro73]. I would prefer something a little more memorable; BrownAHT is reasonable.

One possible convention (which is more or less what I use when giving tags to my own BibTeX entries) is the following. If the reference has one author and a title that can be summarized or abbreviated in one word, call the reference page “Lastname:Title”. So we could have Johnstone:Elephant. If there are multiple authors, use instead their last initials; so we could have CG:Degeneracy I. (For just two authors, we could be flexible and allow “Cheng-Gurski:Degeneracy I.”) And if the title doesn’t seem amenable to comprehensible abbreviation, just use the first letters; so we could have Brown:AHTGSC. There is still some discretion involved on the part of whoever makes the reference page, but we would end up with names that are fairly short and fairly memorable.

Well, I can’t speak for anyone else’s memory. But I will remember “Elephant” or “Johnstone:Elephant” though I have no chance of remembering “MR1953060.”

When above I wrote:

Yes, I agree. I am in favor of following your suggestion.

before the controversy ensued I was thinking mainly:

yes, let’s adopt the notation $f;g$ Toby suggested when we want to indicate that we are writing composition in the way opposite to the usual one, but natural for diagram compositions.

As in, if in the conventionally labeled diagram

$$\begin{array}{ccccc}x& & \stackrel{gf}{\to }& & y\\ & {}_f\searrow & & {\nearrow }_g\\ & & z\end{array}$$

or equivalently

$$\begin{array}{ccccc}x& & \stackrel{g\circ f}{\to }& & y\\ & {}_f\searrow & & {\nearrow }_g\\ & & z\end{array}$$

I feel that the notation convention for the labels gets in the way of the nature of the diagram, then I can enforce diagrammatically natural notation by saying, hey, I’ll switch to that notation with the semicolon for the present purpose and write (still equivalently)

$$\begin{array}{ccccc}x& & \stackrel{f;g}{\to }& & y\\ & {}_f\searrow & & {\nearrow }_g\\ & & z\end{array}.$$

From the discussion I gather that there is pretty much consensus about this.

The other problem that arose in the discussion is a general one of a wiki which we should talk openly about:

In principle, everything anyone writes into the wiki can be subject to editing by others. This has its advantages (which is why we run the wiki in the first place) and its disadvantages (when the first author does not find the edits of his original material to be an improvement).

I believe the solution is: let’s notify each other as much as possible about our work on the $n$Lab at the “latest changes” place (or even by private email, if urgent/important).If I make a change to something somebidy else wrote and notify him or her shortly afterwards, the edits can be checked and, if there is disagreement, be undone and the issue taken to a discussion section if necessary.

With a complex project as the $n$Lab we are bound to run into disagreement about various details from time to time. But I’m sure we can deal with that.

I should mention that one option for those who from time to time want to contribute material which should not be editable by others (for whatever reason) is to create a separate “web” on the server.

Currently there exist four different “webs”, which you can see here:

$n$Lab web list.

Besides the $n$Lab itself, this currently contains private webs for John Baez, for Eric Forgy and for myself.

John hasn’t done much with his private web so far, so I don’t know what he intends it to be. Myself I am collecting material on projects that I am working on at my private web, stuff which I feel should not be merged into the $n$Lab before it has been further developed/polished/srutinized but which I want to share nevertheless.

Eric Forgy had started to develop some tentative ideas he wants to work on first in an $n$Lab page and then had asked for a private web to pursue that activity there.

I think that we could create such private webs for others who are and have been actively and consistently involved in the development of the $n$Lab (such as everybody participating in this discussion here). If there is any interest, please contact me by private email. These private webs work, technically, precisely as the $n$Lab itself, the only difference being that you have the choice between making it entirely public and editable by everybody (as the $n$Lab itself is) or making it public but editable only using a password (that’s the way I have set up my web currently), or even making it completely password protected, so that viewing and editing is restricted to a chosen circle of people.

I can imagine good use for all three of these options. In general I think it is good to have not too much material separated in different webs, as one of the powers of the $n$Lab will be its connected ness via links and backlinks as a single web, but for some purposes it may be good.

I prefer not being forced to write anything at all between the factors, provided that the context already forces which order is meant. For example, if someone refers to an adjunction F⊣U and then refers to the monad UF, I’ll know what is meant (unless, of course, the adjunction is given as ambidextrous!

I agree that if the context is clear, then you shouldn't feel forced to do anything. (And regardless, you can't actually be forced, of course!) But I or Urs may come by and add in a symbol.

Actually, this example is a good one for showing why such a symbol can be useful. If someone is not familiar with how adjunctions give rise to monads, then they would not know whether $U\circ F$ or $U;F$ is meant (since both exist). Certainly one would want to include an explicit symbol if that (how adjunctions give rise to monads) were the topic!

I agree and would happily go along with that.

On the contrary, I think for people unfamiliar with categories, writing $gf$ seems very strange at first.

(I agree that $g\circ f$ has a well established meaning outside of category theory, and I don't propose to mess with that. But $gf$ for composition of functions is not nearly so well established.)

Especially if one is introduced to categories through diagrams, $fg$ is the natural thing to do. But as it is ambiguous, I wouldn't use it; I'd write $f;g$ instead.

(It is true that $f;g$ is not well known; even with that order, $fg$ is probably more common. But if I started writing $fg$, then I think that people would get confused, even in situations where there was only one possible intepretation.)

Note: All notation above lives in this context:

Your smileys indicate that you're kidding, but I've considered this. However, left to right (and up to down) reinforce the basic intuition of arrows as going from the source to the target.

At least it does if one reads left to right. I wonder what mathematicians who read right to left think. (Maybe we can get David Ben-Zvi to hop over and say something.)

A way to get a good hint as to whether they’re weak enough would be to ask whether Trimble $n$-groupoids model all homotopy $n$-types.

And indeed this was the original motivation. The idea is that they certainly should be sufficient, because the fundamental $n$-groupoid of a space $X$ in my sense is exactly an algebraic structure whose 0-cells are points, whose 1-cells are paths, …, whose $j$-cells are continuous maps $D^j\to X$ for $j$ < $n$, and whose $n$-cells are homotopy classes of maps $D^n\to X$ rel boundary. In other words, the whole point behind the definition I gave was to pin down the algebraic operations that this particular globular set (which was my understanding of what Grothendieck actually meant by ‘fundamental $n$-groupoid’) enjoys.

But the general idea of iterated enrichment, in a setting more general than Trimble’s, is interesting and worth pursuing.

I quite agree (and weak iterated enrichment is certainly in the spirit of what May was after) – Eugenia Cheng and Nick Gurski have done some interesting work in this direction, greatly generalizing the definition I presented and comparing it to other approaches (notably Batanin’s), as well as using this approach to provide a suitable niche for the algebra of $k$-tangles in $n$-space. The original definition I gave may still be of interest, I feel, at least in terms of hoped-for applications to homotopy theory (e.g., classifying $n$-types).

I clearly see Andrew’s point, that a forum software would be best suited for the kind of disucssion in this thread here.

On the other hand, I am hesitant to branch off the n-community into yet another branch. The big advantage of the discussion here is that it is not happening in a separate “room”.

I’d say the best option would be to go along with John’s suggestion:

But perhaps we should have more blog entries about more different aspects of the $n$Lab?

Yes, we should move non-minor new comments to separate entries. For instance discussion of the homotopy hypothesis clearly would deserve an entry of its own.

Just to amplify one aspect of Todd’s nice reply:

the “singular simplicial complex” $S(X)$ associated with any topological space $X$ is its fundamental $\mathrm{\infty }$-groupoid with these thought of as Kan complexes (in the sense: has every right to be addresed as such):

we want any “fundamental $\mathrm{\infty }$-groupoid” of a space $X$ to be a graded combinatorial gadget which in degree $k$ is something like the “collection of all $k$-dimensional cells in $X$” (with some source and target etc. information, which makes them $k$-dimensional paths).

This is precisely how $S(X)$ works: if ${\Delta }^k$ denotes the standard topological $k$-simplex, then the collection of $k$-cells of $S(X)$ is

$S(X{)}^k:={\mathrm{Hom}}_{\mathrm{Top}}({\Delta }^k,X)$,

i.e. the collection of paths in $X$ which have the shape of $k$-simplices, if you wish.

Thanks for your comment Todd :)

Between you and Urs, I’m wracking up so much homework that I’ll never see the end. Now I need to learn about Kan complexes :)

By the way, I didn’t mean to suggest anyone lose faith in the homotopy hypothesis. Rather, I think (feel?) that the homotopy hypothesis might end up being more transparent if looked at via the “directed” homotopy hypothesis. Then the homotopy hypothesis could be seen as something like a special projection of the fundamental category down through $\uparrow I$ to a fundamental groupoid similar to how a space $X$ can be seen as a projection down through its directed cylinder $X\times \uparrow I$.

If directed spaces do begin to take more of a central role (as I think they are), I know of a nice combinatorial gadget, i.e. “diamonds”, that might come in handy :)

Todd’s excellent reply incites me to add my grain of salt.

(i) The Grothendieck Pursuit is not really archived at Bangor. I have a more or less complete copy as does Ronnie. The more important thing to note is that, if you did not know this before, George Maltsiniotis is editing a version which will be available hopefully later this year (typed in Latex).

(ii) Note the important fact that a space $X$ is weakly equivalent to $\mid \mathrm{Sing}(X)\mid$, not homotopy equivalent. Weak equivalence is great for locally nice spaces such as CW-complexes but not for general compact metric spaces which may have singularities. We thus may have some difficulties when trying to evaluate the homotopy type of a spatial topos $\mathrm{Sh}(X)$.

(iii) I raised the question over in the Lab but let me point it out again. What seems to be of interest for directed spaces is the change in topology of the `time-like’ slices, i.e. deadlock points, inaccessible points (so ‘sources’), bifurcation points etc. There are ideas from dynamical system theory and Morse theory, such as the Conley index, that try to look at these homologically. What would be interesting would be to take some of the $\mathrm{\infty }$-category ideas that are around and see if there our extensive categorical toolkit can identify some at least of the sort of structure that can be of interest to the practioners and users of directed spaces.

I feel that since we have ideas such as an initial object and a terminal object, in category theory, we should be able to adapt them to get sources etc. (This may actually apply in cobordism theory as well.)

John,
Thanks for defending my `honor’.

jim

It’s quite up to you whether to take my advice on the matter. But I think automatically numbered and hyperlinked equations are worthwhile having, particularly when they are just as easy to create.

Similarly, I happen to think that the Theorem Environments are underutilized on the nLabs. But, again, tastes may differ.

Thanks for the advice, Jacques – you make some good points, and I do appreciate them.

And I did realize, I hope, that it’s quite up to me whether to take it. ;-)

Will do!

Thanks Urs. That is EXACTLY the kind of thing I was looking for. Can similar pictures be drawn for the all those defining relations, e.g. face maps and degeneracy maps, etc, for cubical sets?

For example, the relations in Proposition 2.8 (page 9) of

Sjoerd Crans, Pasting schemes for the monoidal biclosed structure on $\omega -\mathrm{Cat}$

What does a face map look like? What does a degeneracy map look like? The relations probably follow from simple pictures like the diagrams you gave.

no relation to the parallel transport meaning of “connection”!

Oh! The unfortunate clash of terminology AND notation, i.e. $\Gamma$’s, compounds the likelihood of confusion :)

Can similar pictures be drawn for the all those defining relations, e.g. face maps and degeneracy maps, etc, for cubical sets?

Sure! I started now describing the idea a bit more in the section Idea at $n$Lab: cubical set. Have a look and let me know if this helps.

This is a test to see if I’ve got the basic idea…

It looks like degeneracy maps are very much like identity assigning maps. Each node gets mapped to an “identity loop”, each edge gets mapped to an “identity cylinder”, etc.

The extra degeneracy maps are kind of like degeneracy maps except some of the cells get “pinched”. So instead of an edge getting mapped to an identity cylinder, it might get mapped to an “identity cone” since one node gets mapped to an identity loop and the other remains pinched as a node.

I made an attempt at a picture of an extra degeneracy map. I hope it is in the ballpark of being correct.

My immediate thought was of this painting by Matisse. :D

I don’t see why you say this is `decomposing’; you’re not saying that the natural transformation has to be an isomorphism, are you?

I was just looking for a catchy word in the headline. I chose “decomposition” loosely to hint at the fact that we are not looking, in that bimodule language, at arbitary $S$-bimodules, but just as those that are decomposed into a left and a right one.

Finding a better term for “decomposition” here may go some way towards answering Andrew’s question. Geometrically speaking the idea is that we want to look at maps between test spaces $U\to V$ and all their factorization through a given generalized space $X$ as

$$U\to X\to V.$$

So, yeah, decomposition is not a good term. But rouhgly something like that is going on.

Possibly you are thinking of the last time that I (or somebody else) mentioned strongly inaccessible cardinals to you. Those are easily formulated in structural (ETCS-like) terms, and (as the Wikipedia article explains) are equivalent to Grothendieck universes.

I think I understand the bit about the inaccessible cardinals. This just means that the Grothendieck universe is big enough (not to say “large”) to qualify as a “universe”: we can’t leave it by taking power sets and not by taking unions.

What I am looking for (still, but your and Todd’s latest comments on the nLab helped a lot) is the global story to be told about size issues when introducing functor categories in general and Set-valued presheaf categories in particular from scratch.

So I take it the story (in its nice modern form) goes something as follows:

0) We assume that we understand and agree on the ETCS definition of the category of sets “by grace”. This assumption roots the formalism to come in something not further formally deduceable.

I’ll write $\mathrm{StrucSet}$ for the ETCS category of sets obtained this way, to amplify how we think of it.

1) Given $\mathrm{StrucSet}$ we have a notion of topos objects internal to $\mathrm{StrucSet}$. If everything works as expected, these should be precisely the Grothendieck universes $U$ in their structural incarnation the way you present it, to be thought of in the following in terms of their associated $\mathrm{StrucSet}$-internal categories $U\mathrm{Set}$.

2) Next we consider for each such internal topos object $U\mathrm{Set}$ the categories internal to $\mathrm{StrucSet}$ which happen to come from categories

a) enriched over $U\mathrm{Set}$ : the $U$-categories;

b) internal to $U\mathrm{Set}$: the $U$-small categories.

3) For every given $\mathrm{StrucSet}$-internal category $C$ and every $U\mathrm{Set}$ we consider the presheaf categories $[C^{\mathrm{op}},U\mathrm{Set}]$. For each we are guaranteed to find a topos object $V$ such that $[C^{\mathrm{op}},U\mathrm{Set}]$ is internal to $V\mathrm{Set}$.

4) In this spirit we proceed with our theory, always formally keeping track of to which internal topos object $U$, $V$ we happen to be internal to. We hope/assume/argue that none of our central statements that are derived depend in a crucual way on these choices.

Is this roughly the story of size issues for handling presheaf categories when we place ourselves in the ETCS lore?

Do we want to talk about accessible categories in this context? If so, it will be a relative notion: something like $U$-accessible $V$-categories for all pairs of internal topos objects $U\subset V$.

Whether or not the above is close to being right, it is this kind of story which I would like to see more clearly.

Thanks for the link to algebraic set theory!

Have you already looked into this? Maybe you could write a short summary entry to the $n$Lab.

Otherwise I try to look into this later. Am currently busy with the entries on limits etc…

Urs wrote to me in part:

(Did you not get the email I sent you a few minutes after the server went down?)

No, because (aside from not being able to read my email for an unreleated reason a little while around then) you sent it to toby@math.ucr.edu, which goes straight to the spam folder. I've warned you about this before!

Try toby+*@ugcs.caltech.edu, but replace * with something unique. (In your case, Urs, I usually replace * with urs when sending email to you, so you should have that in your address book; 10 random digits is also a good system.)

Well, it looks like it’s gone down again,

Yup, I noticed, since I was editing, too, again. I have already tried all I can (not much) to get the system back up. But this time the former trick doesn’t seem to work.

Well, we’ll sort this out eventually. For the time being I have only 5 minutes left until I have to run and catch my train. I’ll send you something by email, in case you feel like fiddling around with this…

I was also implicitly asking for someone to explain the lemma. Now they have the opportunity at Coyoneda lemma.

David recalled the statement of the “coyoneda lemma” and then asked

What does it mean?

Let me give that a try:

First recall what the Yoneda lemma “means”, in this way:

a presheaf $X(-):S^{\mathrm{op}}\to \mathrm{Set}$ is an attempt to define a generalized space $X$ by specifying its collections $X(U)$ of probes by test spaces $U\in S$.

But every test space $U$ can also be regarded as a generalized space in this sense, given by the assignment of its probes $Y(U)(-):V\mapsto S(V,U)$.

So there is a portential problem with our thinking of a general presheaf $X(-)$ as being a rule for how to probe a generalized space:

there are now two different sensible definitions for what it might mean to probe the generalized space $X$ by the test space $U$.

On the one hand, there is the set of probes $X(U)$, the “probes by defintion”.

But on the the other hand there are now also the probes in the full sense of generalized spaces $[S^{\mathrm{op}},\mathrm{Set}](Y(U)(-),X(-))$.

If these two sets don’t coincide, out attempt to interpret the presheaf $X(-)$ as a generalized space would be inconsistent.

The Yoneda lemma says: don’t worry, it’s okay, the two sets are naturally isomorphic and the interpretation of a presheaf as a generalized space is consistent in this sense.

I am claiming one can read the coYoneda lemma as stating the analogous, now for generalized quantities: generalized functions.

A copresheaf

$$A:S\to \mathrm{Set}$$

we want to think of as a generalized collection of functions: to every test co-domain $U\in S$ it assigns the set $A(U)$ of co-probes of $A$ by $U$, to be thought of as the set of functions from some unspecified generalized space into $U$.

Again, if that interpretation of co-presheaves as generalized collections of functions to be consistent, one would want that the maps-of-generalized functions from a generalized collection of functions $A$ to an ordinary collection of functions $S(a,-)$ on the test space $a$

$$[S,\mathrm{Set}](A,S(a,-))$$

to be naturally identified with the collection of ways that sends an $A$-functions with values in $d$ to functions on $a$ with values in $d$, such that this is compatible with changing the codomains.

Now take a pen and write out what an element in

$$[(*\downarrow K),D]({\Delta }_a,Q)$$

looks like: it’s for each $d$-valued function in $A$, given by a map of sets

$$*\to A(d)$$

a map

$$a\to d$$

in $S$, hence a $d$-valued function on $a$. And naturality of this assignment for all cotest spaces $d$ means that this respects change of codomains.

So, I’d guess, the coYoneda lemma can be read as stating that the interpretation of co-presheaves as generalized collections of functions (quantities) is consistent, in this sense.

But I’d be grateful for opinions by others.