Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality education elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 8th 2012

    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 (,1)(\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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 8th 2012
    • (edited Mar 9th 2012)

    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 H\mathbf{H} the condition is equivalently simply this:

    1. H\mathbf{H} is \infty-connected;

    2. the codomain fibration over the discrete objects is a canonical (,2)(\infty,2)-sheaf.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeMar 8th 2012

    Can you say anything about what sort of “well-behavedness” of the theory of internal (,1)(\infty,1)-categories this definition guarantees? I would have expected intenal (,1)(\infty,1)-categories in any (,1)(\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?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 8th 2012
    • (edited Mar 8th 2012)

    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…

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 8th 2012

    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.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 8th 2012
    • (edited Mar 8th 2012)

    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 nn-fold complete Segal object X X_\bullet in 𝒞\mathcal{C} the object X 0X_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 0X_0 is in that sub-category.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMar 8th 2012
    • (edited Mar 8th 2012)

    …continuing the above thought:

    In consequence, this also means that, given the way that a cohesive \infty-topos over the base topos Grpd\infty Grpd is “well-suited” (“absolute distributor”), a complete Segal object in there is not what I expected it to be – namely a general (,2)(\infty,2)-sheaf over a given site of definition – instead its underlying (,1)(\infty,1)-sheaf is constant.

    Hm…

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 8th 2012
    • (edited Mar 8th 2012)

    Attempt at clarification:

    Since the codomain fibration of any (,1)(\infty,1)-topos is always a canonical (,2)(\infty,2)-sheaf, every (,1)(\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…

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeMar 8th 2012

    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 (,1)(\infty,1)-topos should not require any conditions on X 0X_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 (,1)(\infty,1)-topos, the objects of that topos are precisely the “\infty-groupoids.”

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMar 8th 2012
    • (edited Mar 8th 2012)

    Right, that’s what I said in my last message. All \infty-toposes are “distributors”. Sorry for causing intermediate confusion.

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 9th 2012
    • (edited Mar 9th 2012)

    Jacob Lurie’s definition of “absolute distributor

    this link is broken.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2012
    • (edited Mar 9th 2012)

    Okay, fixed. But there are about a dozen links here and in the entry to this article! :-)

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeMar 9th 2012

    Well then, what is the goal of being able to add the adjective “absolute”?

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2012
    • (edited Mar 9th 2012)

    The answer is still as in #4. I promise to report back when I know more! :-)

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 9th 2012

    That’s ok - just being pedantic.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2012
    • (edited Mar 9th 2012)

    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 (,n)(\infty,n)-categories by iteration. In particular the proof of the bulk of prop. 1.5.4 does not need the “absolute”.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2012
    • (edited Mar 9th 2012)

    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 (,1)(\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.

    • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeMar 9th 2012

    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”?

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2012
    • (edited Mar 9th 2012)

    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

    • Segal object / complete Segal object

    or

    • category object / complete Segal object

    that are currently in the literature?

    Do we want to settle for

    • internal pre-category / internal category

    ?

    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.

    • CommentRowNumber20.
    • CommentAuthorTobyBartels
    • CommentTimeMar 10th 2012

    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 kk-morphism for some finite kk (and hence for every smaller kk). But you can still describe it with this one object (and its structure morphisms).

    • CommentRowNumber21.
    • CommentAuthorMike Shulman
    • CommentTimeMar 12th 2012

    @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 kk-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 (,1)(\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 (,1)(\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.

    • CommentRowNumber22.
    • CommentAuthorMike Shulman
    • CommentTimeMar 12th 2012

    One possible name for an “internal pre-category” is an “(,2)(\infty,2)-congruence”. I’m not sure that that’s better.

    • CommentRowNumber23.
    • CommentAuthorMike Shulman
    • CommentTimeMar 15th 2012

    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 CC is an extensive category with finite products, then we can sort of identify CC-enriched categories with CC-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.

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2012

    Thanks, Mike!

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2012
    • (edited Mar 15th 2012)

    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 (,1)(\infty,1)-topos (ΠDiscΓ):HGrpd(\Pi \dashv Disc \dashv \Gamma) : \mathbf{H} \to \infty Grpd, there are two canonical choices of “distributors” / “choices of internal groupoids”:

    1. consider the identity id:HHid : \mathbf{H} \to \mathbf{H};

    2. consider the discrete object inclusion Disc:GrpdHDisc : \infty Grpd \hookrightarrow \mathbf{H}.

    Accordingly there are the “H\mathbf{H}-internal categories” in Cat(H)Cat(\mathbf{H}) and the “H\mathbf{H}-enriched categories” in Cat Grpd(H)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.

    • CommentRowNumber26.
    • CommentAuthorMike Shulman
    • CommentTimeMar 15th 2012

    How attached are we to the name “groupoid object”? If we instead called those “(,1)(\infty,1)-congruences” or “internal pregroupoids” then the terminology would all be consistent.

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2012
    • (edited Mar 15th 2012)

    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 (,1)(\infty,1)-category theory is a degenerate context for internal groupoid theory. If we looked at groupoid objects and category objects in an (,2)(\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).

    • CommentRowNumber28.
    • CommentAuthorMike Shulman
    • CommentTimeMar 15th 2012

    Wouldn’t internal groupoids in an (,2)(\infty,2)-category naturally be simply (,0)(\infty,0)-truncated objects?

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2012
    • (edited Mar 15th 2012)

    Right, I was walking in the wrong direction. As we increase (n,r)(n,r), the internal (n<n,r<r)(n' \lt n, r' \lt r)-categories become simpler, namely are constant simplicial objects on (n,r)(n',r')-truncated objects.

    • CommentRowNumber30.
    • CommentAuthorMike Shulman
    • CommentTimeMar 15th 2012

    I presume that by “don’t develop strong emotions for” you mean “don’t like”? (-:

    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2012
    • (edited Mar 15th 2012)

    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)(n,r)-congruence” is a great choice. I am very much undecided, so this might easily convince me.

    • CommentRowNumber32.
    • CommentAuthorMike Shulman
    • CommentTimeMar 15th 2012

    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 nn-groupoids “nn-sets”; similarly an equivalence relation is a 0-congruence but we don’t need to feel like we have to call nn-congruences “nn-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 0X_0 which (in good categories) determines an effective epi out of X 0X_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.

    • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2012
    • (edited Mar 15th 2012)

    How about if we amplify the descent aspect of the situation more?

    Because, why is it is that the “groupoid objects” in an (,1)(\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”.

    • CommentRowNumber34.
    • CommentAuthorMike Shulman
    • CommentTimeMar 15th 2012

    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 (,1)(\infty,1)-category, it’s the “internal pre-categories” that are the right notion of internal category.

    • CommentRowNumber35.
    • CommentAuthorMike Shulman
    • CommentTimeMar 15th 2012

    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”?

    • CommentRowNumber36.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2012

    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?

    • CommentRowNumber37.
    • CommentAuthorMike Shulman
    • CommentTimeMar 16th 2012

    I think of the groupoid X 0X_0 as covering the core of its CSS-reflection.

    • CommentRowNumber38.
    • CommentAuthorUrs
    • CommentTimeMar 16th 2012

    I think of the groupoid X 0X_0 as covering the core of its CSS-reflection.

    Here you are thinking internal to an (n>1,1)(n \gt 1,1)-category, don’t you. But in a 1-category? There X 0X_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.

    • CommentRowNumber39.
    • CommentAuthorMike Shulman
    • CommentTimeMar 16th 2012

    Oh, I misunderstood what you were saying. Internal to a 1-category, I want to think of the object X 0X_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?

    • CommentRowNumber40.
    • CommentAuthorMike Shulman
    • CommentTimeMay 18th 2012

    Here’s another suggestion of a name for incomplete Segal objects: \infty-strict internal (,1)(\infty,1)-categories. In general, a kk-strict nn-category should be an nn-category equipped with an essentially surjective functor from a kk-groupoid. A 0-strict 1-category then reduces to Toby’s notion of strict category.

    • CommentRowNumber41.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2012

    But what’s the relation to Segal-completeness?

    • CommentRowNumber42.
    • CommentAuthorMike Shulman
    • CommentTimeMay 19th 2012

    Segal-completeness forces the essentially surjective functor to be an equivalence onto the core, so that it adds no additional structure.

    • CommentRowNumber43.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2012
    • (edited May 19th 2012)

    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 nn-strict if it receives an (n-1)-connected morphism from an nn-groupoid?

    Then an incomplete Segal object would be 0-strict, and a complete Segal object would be \infty-strict, I suppose.

    • CommentRowNumber44.
    • CommentAuthorMike Shulman
    • CommentTimeMay 20th 2012

    It seems to me that receiving an essentially surjective morphism from an nn-groupoid happens more often in practice. For instance, what I called 0-strict (,1)(\infty,1)-categories (an essentially surjective morphism from a set) include many models for (,1)(\infty,1)-categories other than CSS, such as “Segal categories” and simplicial categories and A A_\infty-categories. Also, the usual notion of “internal category” in a 1-category is the same as a 0-strict one as well.

    • CommentRowNumber45.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2012

    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 XX is complete Segal precisely if in the (,1)(\infty,1)-category of Segal spaces the inclusion Core(X)XCore(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 (,1)(\infty,1)-category of Segal objects only see the core of XX inside XX.

    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 kk-groupoid.

    • CommentRowNumber46.
    • CommentAuthorTobyBartels
    • CommentTimeMay 21st 2012

    Mike, I like this concept of a kk-strict nn-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 00-strict, while a strict 22-category is 11-strict.

    Except that by your definition, it’s different. An essentially surjective functor from a 11-groupoid to a 22-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?

    • CommentRowNumber47.
    • CommentAuthorMike Shulman
    • CommentTimeMay 21st 2012

    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 nn-category equipped with a functor out of a kk-category which is mm-connected (by which I mean locally surjective on jj-morphisms for jm+1j\le m+1). The examples we have so far include:

    • A strict category is a 1-category equipped with a (-1)-connected functor out of a 0-category.
    • A strict 2-category is a 2-category equipped with a 0-connected functor out of a 1-category.
    • An incomplete Segal space is an (,1)(\infty,1)-category equipped with a (-1)-connected functor out of an (,0)(\infty,0)-category.
    • A Segal category is an (,1)(\infty,1)-category equipped with a (-1)-connected functor out of a 0-category.

    Maybe a fully general terminology would include both kk and mm? What sort of (,n)(\infty,n)-categories do we get from (for instance) incomplete iterated Segal spaces?

    • CommentRowNumber48.
    • CommentAuthorMike Shulman
    • CommentTimeMay 21st 2012

    @Urs 45: how are you defining the (,1)(\infty,1)-category of Segal spaces? As a full subcategory of Gpd Δ op\infty Gpd^{\Delta^{op}}?

    • CommentRowNumber49.
    • CommentAuthorUrs
    • CommentTimeMay 21st 2012
    • (edited May 21st 2012)

    how are you defining the (∞,1)-category of Segal spaces? As a full subcategory of Grpd Δ op\infty Grpd^{\Delta^{op}}?

    Yes, I am thinking of the usual definition. By localization of PSh (Δ)PSh_\infty(\Delta) at the spine inclusions.

    • CommentRowNumber50.
    • CommentAuthorUrs
    • CommentTimeMay 21st 2012

    Mike, could you indicate why these characterizations

    • An incomplete Segal space is an (∞,1)-category equipped with a (-1)-connected functor out of an (∞,0)-category.
    • A Segal category is an (∞,1)-category equipped with a (-1)-connected functor out of a 0-category.

    will indeed give equivalent characterizations? I see that an incomplete Segal space naturally comes with a (-1)-connected functor out of an (,0)(\infty,0)-groupoid, but I am not sure I see why (,1)(\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?

    • CommentRowNumber51.
    • CommentAuthorMike Shulman
    • CommentTimeJun 18th 2012

    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?

    • CommentRowNumber52.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2012

    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 𝒞\mathcal{C} and an essentially surjective functor i:𝒦𝒞i \colon\mathcal{K} \to \mathcal{C} out of an \infty-groupoid 𝒦\mathcal{K} is something like the the lax Cech nerve of that functor. (?) And that it is complete Segal precisely if ii is the core inclusion.

    • CommentRowNumber53.
    • CommentAuthorMike Shulman
    • CommentTimeNov 26th 2012

    Yes, exactly.

    • CommentRowNumber54.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2012

    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.

    • CommentRowNumber55.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2012
    • (edited Nov 27th 2012)

    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 nn that Grpd(,n)Cat\infty Grpd \hookrightarrow (\infty,n)Cat is a “distributor”. It seems natural to expect that for 𝒟\mathcal{D} any \infty-site, it follows from this that also

    Sh (𝒞,Grpd)Sh (𝒞,(,n)Cat) 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 𝒞Func limpres(H op,(,n)Cat)\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?

    • CommentRowNumber56.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 27th 2012

    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

    • CommentRowNumber57.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 27th 2012

    Spot on, mate! Best to heed that warning…

    • CommentRowNumber58.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2012

    When you look at the entry you’ll see that I am calling them “choice of groupoids”, not “distributors”.

    • CommentRowNumber59.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2012

    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.

    • CommentRowNumber60.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 28th 2012

    I think even calling them Lurie distributors is not great. There’s nothing like a good descriptive name like ‘choice of groupoid objects’.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)