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    • CommentRowNumber1.
    • CommentAuthordanlior2
    • CommentTimeMay 17th 2010


    Let CC be a category and let PCPC be the category of subcategories of CC . Then (C):C opPC(- \downarrow C) : C^{op} \rightarrow PC is a functor and it’s colimit is CPCC \in PC.

    Is colimit(nerve(C))=nerve(C)colimit(nerve(- \downarrow C)) = nerve(C) ?

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 17th 2010

    Sorry, I’m having some trouble parsing this. If I have an object cc of CC, which subcategory is (cC)(c \downarrow C) supposed to be? Ordinarily I would interpret the notation as denoting the comma category whose objects are morphisms cdc \to d where dd is an object of CC, and whose morphisms are commutative triangles with vertex cc, but that’s not a subcategory of CC.

    As for the next part, am I to interpret the nerve we’re taking the colimit of as this composite:

    C op(C)PCCatnerveSet Δ op,C^{op} \stackrel{(- \downarrow C)}{\to} P C \hookrightarrow Cat \stackrel{nerve}{\to} Set^{\Delta^{op}},

    assuming that the first arrow makes sense?

    • CommentRowNumber3.
    • CommentAuthordanlior2
    • CommentTimeMay 18th 2010

    Hi Todd,

    Thanks for you quick reply.

    Your interpretation of (cC)(c \downarrow C) is the same as mine. So is your interpretation of the the the nerve we’re taking colimits of.

    I think of (cC)(c \downarrow C) as a subcategory of CC by sending the object cc 0c \rightarrow c_0 to c 0c_0 and the morphism cc 0c 1c \rightarrow c_0 \rightarrow c_1 to the morphism c 0c 1c_0 \rightarrow c_1.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 18th 2010

    Well, it’s not a subcategory by any definition I know of (certainly not injective on objects; it’s faithful but not full), but it seems not to matter since we can just bypass PCP C and go straight to (C):C opCat(- \downarrow C): C^{op} \to Cat. Someone around here may know right away, but I’ll try to give it a think when I get a chance.

    • CommentRowNumber5.
    • CommentAuthordanlior2
    • CommentTimeMay 18th 2010

    Yes of course, you’re right. (cC)(c \downarrow C) is not a subcategory or CC for general categories CC. However, I neglected to mention that in the particular situation that I’m considering, CC is a partially ordered set. In particular, it’s hom-sets have cardinality at most 1. I think that for such categories CC, (cC)(c \downarrow C) is a subcategory of CC.

    I also agree with you that this point doesn’t really matter anyway since we can bypass PC and go straight to CatCat.

    I would have saved confusion if I simply stated things that way to start with. Thanks for helping me clarify my question. I just hope that someone can give me an answer.


    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 18th 2010

    This is related to questions about test categories or the like. I recall seeing some result like this in the book on the homotopy theory of Grothendieck by Maltsiniotis, but my memory may be faulty.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 19th 2010
    • (edited May 19th 2010)

    Dan, I’ve thought a little about your question, and I think the answer is ’yes’, and the answer is not hard to see. Let’s see if I have this right:

    Your statement about the colimit in CatCat of the (cC)(c \downarrow C) being isomorphic to CC intrigued me – I had never seen that before – but on reflection it was something fairly obvious, in fact basically the Yoneda lemma in disguise. The objects of colim c:C op(cC)colim_{c: C^{op}} (c \downarrow C) are equivalence classes of arrows cdc \to d where the equivalence \sim is generated by

    (cfd)(cgcfd)(c \stackrel{f}{\to} d) \sim (c' \stackrel{g}{\to} c \stackrel{f}{\to} d)

    and it is immediate that every g:cdg: c \to d is equivalent to 1 d:dd1_d: d \to d; this of course is just a form of the Yoneda lemma.

    Now let’s look at your problem, which compares the nerve(C)nerve(C) to the colimit of

    C op(cC)CatnerveSet Δ opC^{op} \stackrel{(c \downarrow C)}{\to} Cat \stackrel{nerve}{\to} Set^{\Delta^{op}}

    Since colimits in Set Δ opSet^{\Delta^{op}} are computed pointwise, we just have to show the colimit of

    C op(cC)CatnerveSet Δ opev nSet,C^{op} \stackrel{(c \downarrow C)}{\to} Cat \stackrel{nerve}{\to} Set^{\Delta^{op}} \stackrel{ev_n}{\to} Set,

    where ev nev_n is evaluation at an object nn, agrees with nerve(C) nnerve(C)_n. This is

    C op(cC)Cathom([n],)SetC^{op} \stackrel{(c \downarrow C)}{\to} Cat \stackrel{\hom([n], -)}{\to} Set

    Now an nn-simplex in the comma category (cC)(c \downarrow C), which is an element of this composite, is the same as an (n+1)(n+1)-simplex beginning with the vertex cc, and the colimit (in SetSet) consists of equivalence classes of (n+1)(n+1)-simplices where a simplex beginning with cc is deemed equivalent to a simplex beginning with cc' obtained by pulling back along any g:ccg: c' \to c. And again, it is a triviality that each (n+1)(n+1)-simplex

    c(d 0d n)c \to (d_0 \to \ldots \to d_n)

    is equivalent to

    d 01 d 0(d 0d n)d_0 \stackrel{1_{d_0}}{\to} (d_0 \to \ldots \to d_n)

    but the collection of such d 0d nd_0 \to \ldots \to d_n is the same as nerve(C) nnerve(C)_n. This proves your conjecture.

    Edit: By the way, this reminds me of the tangent category stuff that originated at the n-Café in a discussion that included Urs, David Roberts, and me, and which was developed further by Schreiber-Roberts. I think I recall now remarking on the Yoneda lemma in this connection.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeMay 20th 2010
    • (edited May 20th 2010)

    Now that I’ve had a chance to think about this properly, Todd is exactly right. It is also a special case of a general fact about (2-sided) bar constructions. Specifically, the nerve of C is the bar construction B(*,C,*)B(*,C,*) (where ** denotes the functor constant at a terminal object), while the nerve of cCc\downarrow C is the bar construction B(C(c,),C,*)B(C(c,-),C,*). Since colimits of a functor F:C opDF:C^{op}\to D are given by tensor products of functors * CF* \otimes_C F, and such tensor products come inside a bar construction (since colimits commute with colimits), we have

    colim cN(cC)=* cCB(C(c,),C,*)=B(* cCC(c,),C,*)=B(*,C,*)=NC \colim^c N(c\downarrow C) = * \otimes_{c\in C} B(C(c,-),C,*) = B(* \otimes_{c\in C} C(c,-), C, *) = B(*,C,*) = N C

    where * cCC(c,)=**\otimes_{c\in C} C(c,-) = * by the co-Yoneda lemma.

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 20th 2010

    Ah, ah, ah – excellent point, Mike. Thanks.