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I am starting an entry internal (infinity,1)-category about complete Segal-like things.
This is prompted by me needing a place to state and prove the following assertion: a cohesive $\infty$-topos is an “absolute distributor” in the sense of Lurie, hence a suitable context for internalizing $(\infty,1)$-categories.
But first I want a better infrastructure. In the course of this I also created a “floating table of contents”
and added it to the relevant entries.
I have now added in the section Suitable ambient contexts Jacob Lurie’s definition of “absolute distributor” for the case of $\infty$-toposes.
But I have reformulated it, observing that for $\infty$-toposes $\mathbf{H}$ the condition is equivalently simply this:
$\mathbf{H}$ is $\infty$-connected;
the codomain fibration over the discrete objects is a canonical $(\infty,2)$-sheaf.
Can you say anything about what sort of “well-behavedness” of the theory of internal $(\infty,1)$-categories this definition guarantees? I would have expected intenal $(\infty,1)$-categories in any $(\infty,1)$-topos to be well-behaved.
Also, I didn’t realize that the nLab was using “$\infty$-connected” to include “locally $\infty$-connected”. I don’t really like that, as it doesn’t faithfully extend the case of 1-toposes and even topological spaces, which can be connected without being locally connected. Can we change it?
Can you say anything about what sort of “well-behavedness” of the theory of internal (∞,1)-categories this definition guarantees?
I still need to understand this better myself. I think a key point is that this assumption is needed for the model category structure for complete Segal objects in a given model structure of the ambient thing to present the correct $\infty$-category. So: to get prop. 1.5.4 in Lurie’s article.
But I really haven’t absorbed this satisfactorily yet. I am working on it…
Also, I didn’t realize that the nLab was using “∞-connected” to include “locally ∞-connected”. I don’t really like that, as it doesn’t faithfully extend the case of 1-toposes and even topological spaces, which can be connected without being locally connected. Can we change it?
Sure, my fault. We had this discussion before. I guess I was being careless. I’ll change it.
Mike,
the “well-suitedness for internalization”-assumption (Lurie: “distributor”) is about formalizing the “completeness” condition on a complete Segal space. One needs (def. 1.2.7) a way to say internally that for an $n$-fold complete Segal object $X_\bullet$ in $\mathcal{C}$ the object $X_0$ is “an $\infty$-groupoid”.
The strategy to do so is pretty much the “internalization of discrete objects” that we had long discussion about: declare there to be a subcategory of discrete objects, to be thought of as the inclusion of a base $\infty$-topos of $\infty$-groupoids. Then say that $X_0$ is in that sub-category.
…continuing the above thought:
In consequence, this also means that, given the way that a cohesive $\infty$-topos over the base topos $\infty Grpd$ is “well-suited” (“absolute distributor”), a complete Segal object in there is not what I expected it to be – namely a general $(\infty,2)$-sheaf over a given site of definition – instead its underlying $(\infty,1)$-sheaf is constant.
Hm…
Attempt at clarification:
Since the codomain fibration of any $(\infty,1)$-topos is always a canonical $(\infty,2)$-sheaf, every $(\infty,1)$-topos is automatically a “distributor”, hence admits a good theory of complete Segal objects.
But it is an “absolute distributor” only precisely if it is also locally and globally $\infty$-connected. Only in this case does, apparently, the proof apply which says that we have a good model category presentation for the complete Segal objects in the $\infty$-topos.
I need to call it quits for today. Will spell this out more nicely tomorrow…
What exactly is the goal here? I mean, what sort of internal “complete Segal objects” are we trying to talk about? To me, a complete Segal object in an $(\infty,1)$-topos should not require any conditions on $X_0$ to be “a groupoid”, because all objects of that topos “are groupoids” in the internal sense. In fact from the internal point of view of an $(\infty,1)$-topos, the objects of that topos are precisely the “$\infty$-groupoids.”
Right, that’s what I said in my last message. All $\infty$-toposes are “distributors”. Sorry for causing intermediate confusion.
Jacob Lurie’s definition of “absolute distributor
this link is broken.
Okay, fixed. But there are about a dozen links here and in the entry to this article! :-)
Well then, what is the goal of being able to add the adjective “absolute”?
The answer is still as in #4. I promise to report back when I know more! :-)
That’s ok - just being pedantic.
So looking at it again this morning, it seems that the notion of “absolute distributor” is indeed not really essential. It’s mostly a convenience for the definition of $(\infty,n)$-categories by iteration. In particular the proof of the bulk of prop. 1.5.4 does not need the “absolute”.
I am further working on that entry, I think now it’s slowly taking shape.
I have
renamed it to category object in an (infinity,1)-category
added a reminder on groupoid objects;
added the definition of category objects internal to $(\infty,1)$-topos;
“played Bourbaki” by renaming Jacob Lurie’s “category object” to “pre-category object” and his “complete Segal object” to just “category object”. That seems only reasonable.
added a bit more here and there.
While, on the one hand, I agree that it’s the “complete” ones that deserve to be thought of as “internal categories” when doing internal category theory, I would find it kind of unfortunate if not every “groupoid object” were a “category object”. I don’t really have an alternative proposal at the moment, though.
I’ve also never been entirely thrilled with the phrase “category object”, since an internal category consists (in the classical case) of two objects, and (in the $\infty$-case) of an infinite number of objects (together with some morphisms between them). How about “internal category in an (infinity,1)-category”?
I was starting to have similar worries about my decision.
But now before I go and change all the entries again, let’s sort it out once and for all. What would be the canonical good choice of terminology, improving on
or
that are currently in the literature?
Do we want to settle for
?
The pre/complete issue here is a subtle one, which maybe deserves better discussion in itself anyway. Currently I just have a half-sentence in the entry saying that “the notion of homotopy in the internal category should be compatible with that in the ambient category”. Which goes in the right direction, but should be improved on.
If you want it to, a category object may consist of only one object: the object of morphisms. Some of these morphisms happen to be (identity morphisms of) objects, but that’s no matter. (Of course, there are a bunch of structure morphisms too, but that’s normal.)
An $\infty$-category object also consists of only one object, although you can’t blithely call its elements $\infty$-morphisms; in fact, each one is a $k$-morphism for some finite $k$ (and hence for every smaller $k$). But you can still describe it with this one object (and its structure morphisms).
@Toby: That’s certainly true for 1-categories. And I believe it for an internal strict $\infty$-category in a well-behaved category like a topos, where we can take the “union” of all the objects of $k$-morphisms in a well-behaved way. I don’t see any way to do it for an internal strict $\infty$-category in a general category with finite limits, and it’s not clear to me how to go about it for weak $\infty$-categories even in a topos, let alone weak $\infty$-categories internal to an $(\infty,1)$-category.
What would be the canonical good choice of terminology
I don’t know! I’ve been wondering that for some time. With all due respect to Charles Rezk, I don’t really like the adjective “complete” here; it doesn’t convey any of the right intuition to me and it has other conflicting meanings.
Part of the problem is that the “right” notion of internal category depends on the ambient category level. If you take the notion of internal complete Segal object and interpret it literally in an $(\infty,1)$-category that happens to be a 1-category, then you don’t get the correct notion of “internal category” in a 1-category: the completeness becomes an extra “rigidity” condition.
One possible name for an “internal pre-category” is an “$(\infty,2)$-congruence”. I’m not sure that that’s better.
Chris Schommer-Pries has told me that the point of an “absolute distributor” is to enable us to define enriched categories as particular internal categories. In the 1-categorical case, if $C$ is an extensive category with finite products, then we can sort of identify $C$-enriched categories with $C$-internal categories whose object-of-objects is a coproduct of copies of the terminal object. “Absolute distributors” (which really need a better name) are supposed to let us identify a class of internal complete Segal objects that act like “enriched” rather than “internal” categories.
Thanks, Mike!
Okay, I have further worked on the entry, filling in the previously missing further definitions and statements, and including a paragraph on “enrichement”.
(I have also made the change of terminology from “category object” to “internal category” suggested by Mike above, and made some other terminology choices. Feel free to disagree with these choices, I am just trying out what might work.)
So what I need to better understand is this: in a locally and globally $\infty$-connected $(\infty,1)$-topos $(\Pi \dashv Disc \dashv \Gamma) : \mathbf{H} \to \infty Grpd$, there are two canonical choices of “distributors” / “choices of internal groupoids”:
consider the identity $id : \mathbf{H} \to \mathbf{H}$;
consider the discrete object inclusion $Disc : \infty Grpd \hookrightarrow \mathbf{H}$.
Accordingly there are the “$\mathbf{H}$-internal categories” in $Cat(\mathbf{H})$ and the “$\mathbf{H}$-enriched categories” in $Cat_{\infty Grpd}(\mathbf{H})$.
Intuitively, judging from the 1-category theory, one would expect that the latter are included in the former. Here this inclusion cannot be given by the naive inclusion of the underlying simplicial objects, so if there is an inclusion, it’s not right away obvious. But I am wondering if there actually is one. Or else, what one can say about the relation between the two.
Need to think about this.
How attached are we to the name “groupoid object”? If we instead called those “$(\infty,1)$-congruences” or “internal pregroupoids” then the terminology would all be consistent.
I was wondering about this, following your previous suggestion, but couldn’t make up my mind.
But “internal pregroupoid” might work well. (For some reason I don’t develop strong emotions for the “congruence”-terminology, not sure why).
I was thinking: probably what makes it hard to better see the big pattern at work is that $(\infty,1)$-category theory is a degenerate context for internal groupoid theory. If we looked at groupoid objects and category objects in an $(\infty,2)$-category, we might get a better feeling for what things should be called, because then internal groupoids would no longer be simply constant simplicial objects, in general! (I guess).
Wouldn’t internal groupoids in an $(\infty,2)$-category naturally be simply $(\infty,0)$-truncated objects?
Right, I was walking in the wrong direction. As we increase $(n,r)$, the internal $(n' \lt n, r' \lt r)$-categories become simpler, namely are constant simplicial objects on $(n',r')$-truncated objects.
I presume that by “don’t develop strong emotions for” you mean “don’t like”? (-:
I really just mean that I don’t feel a particular pull towards the using the term. But maybe I just need to get used to it. I certainly do appreciate the fact that it is a good candidate.
Maybe what bothers me is that it seems too degenerate a case to name the general concept by. This becomes more pronounced if we say equivalently “internal equivalence relation” for “congruence”. Do we really want to say that an (pre-)$\infty$-groupoid is an $\infty$-equivalence relation? We could, but it seems more natural to go the other way and highlight that an equivalence relation is just a degenerate case of a groupoid.
Maybe it would help me if you amplified arguments for why “$(n,r)$-congruence” is a great choice. I am very much undecided, so this might easily convince me.
Well, the point of saying “congruence” rather than “equivalence relation” is that then we (hopefully) don’t get the baggage of the name seeming degenerate. A set is a 0-groupoid, but we don’t call $n$-groupoids “$n$-sets”; similarly an equivalence relation is a 0-congruence but we don’t need to feel like we have to call $n$-congruences “$n$-equivalence relations”. The idea of “congruence” is that it is the sort of thing you take a quotient of; it’s property/structure/stuff on $X_0$ which (in good categories) determines an effective epi out of $X_0$.
Of course, the original literal meaning of “congruence” was an equivalence relation, so your objection does have some force. I’m not saying it’s a great choice; I’m undecided too.
How about if we amplify the descent aspect of the situation more?
Because, why is it is that the “groupoid objects” in an $(\infty,1)$-topos determine the actual internal groupoids, but are different from them? Of course it’s because these groupoid objects are really an internal groupoid but equipped with extra data, namely with a “covering”, from which the actual internal groupoid is obtained by gluing. And there are many inequivalent ways to get one and the same internal groupoid from such coverings.
From this perspective also the terminology “internal pre-groupoid” is not accurate. They are more like “internal post groupoids”, namely an internal groupoid and more.
However, what stops me from carrying this way of thinking further is that I don’t feel I have a good intuition for the analogous story now as it comes to “internal pre-categories”.
That’s a good point. Wouldn’t an “internal pre-category” just be an internal category equipped with a covering of its core?
I am also still bothered by the fact that in a 1-category regarded degenerately as an $(\infty,1)$-category, it’s the “internal pre-categories” that are the right notion of internal category.
Oh! I just realized that internal categories in a 1-category, in the usual sense, do also have extra data, namely a covering of their (internal) groupoid of objects by an (internal) set. Hence why we have to remove that extra data by passing to stacks or using anafunctors. And that’s true even in Set, except that we usually obscure it by using AC.
So maybe “internal pre-categories” should be called something like “covered internal categories”?
internal categories in a 1-category, in the usual sense, do also have extra data, namely a covering of their (internal) groupoid of objects by an (internal) set.
[…]
So maybe “internal pre-categories” should be called something like “covered internal categories”?
I am still not sure if I understand the step from the first to the second sentence in the above quote. So given a simplicial object satisfying the Segal condition, but not being a groupoid object: how do you think of it as covering what?
I think of the groupoid $X_0$ as covering the core of its CSS-reflection.
I think of the groupoid $X_0$ as covering the core of its CSS-reflection.
Here you are thinking internal to an $(n \gt 1,1)$-category, don’t you. But in a 1-category? There $X_0$ is the set of objects covering the set of isomorphism classes of objects.
Maybe we are talking past each other. I thought one trouble was that CSS yoga does not seem to make much sense internal to a 1-category. But then in #35 you seemed to say that you have a way of looking at the situation so that the two pictures do unify after all. This I don’t see yet.
Oh, I misunderstood what you were saying. Internal to a 1-category, I want to think of the object $X_0$ as covering the core of the stackification of the internal category. (I’m thinking of my 1-category as being a topos here, or at least a site.) Does that help?
Here’s another suggestion of a name for incomplete Segal objects: $\infty$-strict internal $(\infty,1)$-categories. In general, a $k$-strict $n$-category should be an $n$-category equipped with an essentially surjective functor from a $k$-groupoid. A 0-strict 1-category then reduces to Toby’s notion of strict category.
But what’s the relation to Segal-completeness?
Segal-completeness forces the essentially surjective functor to be an equivalence onto the core, so that it adds no additional structure.
What I don’t quite understand yet is why you say that this is a good name to use for incomplete Segal objects.
Maybe we should say: a category object is $n$-strict if it receives an (n-1)-connected morphism from an $n$-groupoid?
Then an incomplete Segal object would be 0-strict, and a complete Segal object would be $\infty$-strict, I suppose.
It seems to me that receiving an essentially surjective morphism from an $n$-groupoid happens more often in practice. For instance, what I called 0-strict $(\infty,1)$-categories (an essentially surjective morphism from a set) include many models for $(\infty,1)$-categories other than CSS, such as “Segal categories” and simplicial categories and $A_\infty$-categories. Also, the usual notion of “internal category” in a 1-category is the same as a 0-strict one as well.
Okay. But what do you think about that idea about connectedness?:
I haven’t tried to check it formally, but inuitively it should be true that an incomplete Segal space $X$ is complete Segal precisely if in the $(\infty,1)$-category of Segal spaces the inclusion $Core(X) \to X$ of the core is $\infty$-connected.
Because a morphism should be $\infty$-connected if all its homotopy fibers are contractible, but the homotopy fibers in the $(\infty,1)$-category of Segal objects only see the core of $X$ inside $X$.
If this is correct, and given that essentially surjective = (-1)-connected, I would feel it pretty natural to consider the interpolation between “strict” and “complete” by n-connected covers by a $k$-groupoid.
Mike, I like this concept of a $k$-strict $n$-category. I’ve been meaning to figure this out: how a strict category and a strict 2-category are both strict but at different levels. So a strict category is $0$-strict, while a strict $2$-category is $1$-strict.
Except that by your definition, it’s different. An essentially surjective functor from a $1$-groupoid to a $2$-category has extra morphisms that don’t associate strictly. I conclude that Urs is right; this functor should be full. And not just full on isomorphisms but on all morphisms, so we need to use a category rather than a groupoid as the domain.
How does this fit in with the Segal objects?
Toby, thank you for bringing that in! Maybe we are groping our way towards a good definition. I agree that a strict 2-category, as the term is usually used, is a 2-category equipped with an essentially surjective and full functor from a (non-strict) 1-category. There seem to be two dimensions at play: we are looking in general at an $n$-category equipped with a functor out of a $k$-category which is $m$-connected (by which I mean locally surjective on $j$-morphisms for $j\le m+1$). The examples we have so far include:
Maybe a fully general terminology would include both $k$ and $m$? What sort of $(\infty,n)$-categories do we get from (for instance) incomplete iterated Segal spaces?
@Urs 45: how are you defining the $(\infty,1)$-category of Segal spaces? As a full subcategory of $\infty Gpd^{\Delta^{op}}$?
how are you defining the (∞,1)-category of Segal spaces? As a full subcategory of $\infty Grpd^{\Delta^{op}}$?
Yes, I am thinking of the usual definition. By localization of $PSh_\infty(\Delta)$ at the spine inclusions.
Mike, could you indicate why these characterizations
will indeed give equivalent characterizations? I see that an incomplete Segal space naturally comes with a (-1)-connected functor out of an $(\infty,0)$-groupoid, but I am not sure I see why $(\infty,1)$-categories equipped with such a functor would be equivalent to incomplete Segal spaces. Similarly for the second item. I am not saying that I have reason to doubt it, but I don’t see the argument. Do you find this obvious?
Sorry for taking a month to reply to this. I don’t have a proof of either statement in the $\infty$-case. Both seem “intuitive” to me but that is not the same as “obvious”; I admit that my intution may be wrong. However, analogous truncated statements are true, which is part of where my intution comes from.
For instance, consider Reedy fibrant internal categories in Gpd. These are an obvious sort of “truncated” incomplete Segal space. For internal categories in Gpd, Reedy fibrancy is, I’m pretty sure, equivalent to being a framed bicategory. Thus, by Appendix C of my paper (made groupoidal), to give such a thing is equivalent to giving a (2,1)-category equipped with a bijective-on-objects functor out of a groupoid.
For a truncated version of the second statement, consider internal categories in Gpd whose groupoid of objects is discrete. These are just (2,1)-categories as usually defined, but the fact that we’ve specified a discrete groupoid of objects gives us a (2,1)-categorical version of a strict category.
Does that make sense?
Sorry for taking, in turn, six months to reply to this. :-o
Yes, that makes sense, thanks. So I suppose the upshot is that the incomplete Segal category associated with an $\infty$-category $\mathcal{C}$ and an essentially surjective functor $i \colon\mathcal{K} \to \mathcal{C}$ out of an $\infty$-groupoid $\mathcal{K}$ is something like the the lax Cech nerve of that functor. (?) And that it is complete Segal precisely if $i$ is the core inclusion.
Yes, exactly.
Thanks, Mike. This seems more than obvious now that we say it, but I would like this to be made explicit. I gave it a start in the Examples section at Segal space, but discussion of that should go in the corresponding thread.
I have been working through all of category object in an (infinity,1)-category, but not quite done yet with what I am aiming at.
While I am at it: there is a certain lack of written-out examples of “distributors”, choice of groupod objects, does anyone have further insights to offer on this?
So we have for every $n$ that $\infty Grpd \hookrightarrow (\infty,n)Cat$ is a “distributor”. It seems natural to expect that for $\mathcal{D}$ any $\infty$-site, it follows from this that also
$Sh_\infty(\mathcal{C}, \infty Grpd) \hookrightarrow Sh_\infty(\mathcal{C}, (\infty,n)Cat)$is a “distributor”. Possibly that’s tautological from staring at the definition a bit and using something like $\mathcal{C} \simeq Func_{limpres}(\mathbf{H}^{op}, (\infty,n)Cat)$ . But also, maybe it’s too late for me tonight. Did anyone think about this?
Hi Urs, you know that distributor is another name for profunctor? Perhaps they should be called ’Lurie distributors’… Jean Benabou would have a fit with the appropriation of his terminology :-S
Spot on, mate! Best to heed that warning…
When you look at the entry you’ll see that I am calling them “choice of groupoids”, not “distributors”.
re #55: never mind, now that I am awake again, I see that this is the statement of “variant 1.3.8” of Lurie’s article.
I think even calling them Lurie distributors is not great. There’s nothing like a good descriptive name like ‘choice of groupoid objects’.
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