However, this being said, I don’t think Toby would have intended the distinction to mean that morphisms of linear orders strictly preserve the $\lt$ relation.

]]>It’s hard to check now, but my memory is that in the early nLab days it was Toby Bartels, much prompted by constructivist concerns, who introduced total order for the reflexive version.

]]>Yes, the orders classified by cosimplicial sets have to be reflexive, as the category of points is the ind-completion of the simplex category.

But I would say that what’s wrong here is the nlab definition of linear order. According to e.g. wikipedia linear orders are reflexive. Maybe one could call the non-reflexive ones “strict linear orders”?

]]>In a discussion on Zulip this weekend, I realised this is incorrect. The theory of (decidable) linear orders in this sense is classified by the category of presheaves on the opposite of the category of finite orders and *injective* order-preserving functions.

The topos of cosimplicial sets classifies inhabited totally ordered sets, hence having a *reflexive* relation, \({\leq}\). This is what is proved in the Moerdijk textbook cited there. The difference is easy to see if one considers that the unique mapping from the two-element total order to the one-element total order preserves the reflexive order relation but not the antisymmetric one. ]]>

Put a pointer to Diaconescu’s theorem that justifies passing to the presheaf topos in the case of the finite limit theory for groups.

I know it’s mentioned later in the article, but (IMO) the theory of groups seems to be the reference example and the one that’s most fleshed out, so this last detail should be included there.

]]>Factored in the reference to Tierney’s article.

]]>Make the description of Set and 1 as classifying toposes more concise.

]]>@49. In nutshell, my intuitive idea of a model seems to correspond basically to the situation prevalent in classical model theory: a carrier set plus structure. Since it is rather commonplace to blur the distinction between carrier and model, one is then tempted to read (the induced model) $f^*(U)$ as arising from the application of $f^*$ to the object $U$ carrying the generic model (I guess the leading examples as well as the rather casual introduction of the notation in the MacLane-Moerdijk book reinforce this intuition). A nice and useful picture as far as it goes but unfortunately inadequate to handle cases when more than one sort is around as you point out, as well as in the cases that cause my phantom pain here, when there is less than one sort around, creating a strange nostalghia for a concrete carrier object where none is to be found.Thanks for setting me straight!

]]>I think the intent of that paragraph was to illustrate the fact the empty structure isn’t the only thing $Set$ classifies. E.g. it classifies natural number objects, and for any small set $S$, it classifies “objects that are $S$-indexed disjoint unions of the terminal object”.

The concrete example is to show $Set$ classifies initial objects, but this idea is being framed as formulating a theory with a single sort.

]]>the codomain class of this empty function

Well, of course there are multiple empty functions: one from the empty set of sorts to the set of objects of the topos, another from the empty set of function symbols to the set of morphisms of the topos, etc.

the empty collection of maps needs a composition defined in order to give a category of models

Of course there is a unique empty function $\emptyset\times\emptyset\to\emptyset$, defining the empty category.

the idea that any model arises as $f^*(U)$ from the generic model seems to be lost

Not at all: there is exactly one model of the empty theory in any topos, including exactly one (generic) model in $Set$, and $f^*$ indeed maps the latter to the former. There’s no trouble defining $f^*$ on models of the empty theory.

For a signature with sorts one thinks of a model rather in terms of the objects in the image of the assigment

I’m not sure exactly what that means, but if it produces such confusion one should probably stop thinking that way. (-: You certainly can’t in general think of a model as a *subset* of the set of objects of the topos equipped with structure, if that’s what you mean, since some object might have to be used for more than one sort (e.g. a model of any multi-sorted theory in the one-object trivial topos).

Explicitly stated the generic model in the theory of objects.

]]>Well, I don’t doubt that it can be made precise, actually it works pretty well as stated. What I am a bit uneasy about is for one the codomain class of this empty function, then the empty collection of maps needs a composition defined in order to give a category of models and the idea that any model arises as $f^*(U)$ from the generic model seems to be lost. For a signature with sorts one thinks of a model rather in terms of the objects in the image of the assigment and for me this gives the empty assigment the feeling of being a model in name only since the concrete object under it is missing. But feel free to improve or correct the section! I never felt very happy with it and hesitated quite some time before adding it, since it is a bit too long and too syntax leaning in comparison with the rest of the examples, and probably too confusing by bringing in the idea of classification relative to a signature. On the other hand, $Set$ and $1$ are the most basic examples, so they should be mentioned somewhere and then I didn’t want to give impression that $Set$ only classifies the empty theory over the empty signature.

]]>I’m not sure what you mean by “spooky” or “sophistry” there. An empty assignment is of course just an empty function.

]]>Added $Set$ as a classifier for the empty theory. Since the notion of model is a bit spooky in that case it might be helpful to flash out the syntactical sophistry (of course, the notion of empty assigment is still somewhat informal).

]]>What started as typo fixing ended by throwing in some remarks and references on the relation between NNOs and classifying toposes.

]]>I noticed that at *classifying topos* there was no pointer back to *forcing*. So I have now added in the Idea-section right after the mentioning of the universal model the following paragraph:

The fact that a classifying topos is like the ambient set theory but equipped with that universal model is essentially the notion of

forcingin logic: the passage to the internal logic of the classifying toposforcesthe universal model to exist.

Please feel free to improve as need be.

]]>Ringed space is not any more standard than a “space with a structure ring” (or sheaf of rings). BOTH are standard, and many references introduce both terms simultaneously just the full version does not lead to confusion as often as the first (outside of the algebraic geometry community). Also the longer term is more often used when speaking to people of wider community (say in a colloqium talk, as opposed to an expert community). Plus in many languages “ringed space” does not translate, so one has to stick with the literal translation of the long version. And as I said, “locally” is far more confusing, as well as “lined topos”, for which I still tend to forget what is meant. The worst combination is monoided space where one can be mislead to having a space in some monoidal category or something like that (such notions exist). It is like an invitation for a confusion. A similar thing is with “simplicial category” which is ambigous abbreviation for either simplicial object in Cat or a simplicially enriched category. At least when mentioning it the first time within a context/paper/talk one should say the full name and not appeal to the jargon of his microcommunity. (The fact that the first can be viewed in a canonical way as a special case of the second does not make itan excuse, but rather it makes it worse.)

It is so easy to type several more characters and being unambiguous (surely, I myself used the locally ringed spaces in some papers of mine, but I am not happy about that decision, as it was intended to a wider community than alg. geom.). Of course in a context when one repeats often, and uses one and the same version of structure, one can simply make a local convention and say “space” for gadgets within his category of whatevered spaces.

]]>You are replying to #38 from last year, right?

In principle I very much agree with what you say. Only that since the terminology “ringed space” etc. is entirely standard, it is not clear to me if saying “locally algebra-ed space” is more unfriendly to outsiders than changing their standard term and saying “space with structure ring”. But I agree that in principle the latter woule be better.

]]>I prefer to say a “space with a structure sheaf (of rings)” than a strange abbreviation “ringed space”, which I always found misleading and when I was a student it confused me. Even worse with “locally ringed space” as if it were ringed locally, what is not true; it is so much easier to say with a structure local ring and everybody understands. So I like also “topos with a structure ring” and alike phrases which are understandable to people outside the subject unlike the strange abbreviation “ringed topos” (worse with “lined topos” where I still forget what is exactly meant). In fact there is no need to devise words which are unlikely to get acquired by all interested mathematicians Thus monoided and algebred and so on topoi and spaces makes me quite uneasy, it is inventing for no need; it looks also like it is an operation on topoi and not a structure. If it is a structure, one says with such structure. It is friendly, and it is in general a problem with category theory to be unfriendly to outsiders with lots of terminology and conventions. If it were needed than I would agree with the terminology, but rather lets not without need proliferate unclear abbreviations. We should rather save readers from confusion rather than few letters of ink.

]]>I have added to *classifying topos* a section *For inhabited linear orders* with statement and one half of the proof that $Set^{\Delta } = Sh(\Delta^{op})$ classifies those simplicial objects which are nerves of posets that are inhabited linear orders.

have added a section for local algebras to classifying topos, with the 1-categorical analog of structured (infinity,1)-topos.

It is amazing that this 1-categorical analog has not been discussed or at least not discussed prominently before. Looks to me like a major omission. Lurie of course discusses it in the $\infty$-categorical context, but somebody should write it all out just for 1-categories.

I am going to make a puny start. Have been thinking about better terminology, suitably generalizing “locally ringed topos” but being more descriptive than structured topos.

How about

locally algebra-ed topos

??

There are people (Benno vdB, in the context of Bohrification) who speak of “C-stared toposes” to mean ringed toposes whose ring object is in fact a $C^*$-algebra object. So generally “algebra-ed toposes” seems to be a notion to go for. I am just not sure how exactly to spell it. ;-)

]]>I have added to classifying topos a section Geometric morphisms equivalent to morphisms of sites containing the crucial lemma that explains “why classifying toposes work”.

]]>It’s possible that a simplicial approach would work to define higher-lax sorts of limits, although I’m a little skeptical that it would get the directions of everything right; simplicial nerves start to get really weird when the 2- and higher cells are noninvertible. However, I haven’t thought very much about what n-lax things would mean for n any bigger than the smallest possible (wherever the numbering should start), or what they would be good for; maybe you’re right. But my main point remains: lax pullbacks should not be called 2-pullbacks. If you want a number indexing the level of laxness, I would put a prefix on the word “lax” instead.

]]>You said you would have wanted to use “2-pullback” to mean “lax pullback.” I assumed that by “lax pullback” you meant a comma object; did you actually mean what I would call a lax pullback, namely the lax limit of a cospan?

Yes. I am imagining that there is an evident definition of lax $(\infty,n)$-limit for all $n$: imagine you have a simplicial set incarnation of $(\infty,n)$-categories (say along the lines of Verity), then use Joyal’s simplicial formula for the simplicial set of cones over a diagram and find the terminal object in there. In other words, proceed verbatim as for $(\infty,1)$-limits modeled in quasi-categories. Just have more relaxed assumptions on what the simplicial set has to satisfy.

I did not follow the discussion carefully but if you are talking about some sort of classifying object construction for bundles and using steps like Grothendieck construction, then in low dimensions, in my unreliable memory one does deal with the approapriate versions of comma objects. No ?

The Grothendieck construction is a comma object, but the pullback that defines a principal bundle is a homotopy pullback. I am thinking this should remain true if the principal bundle is realized in its topos incarnation.

]]>