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Hello,
Let C be a category and let PC be the category of subcategories of C . Then (−↓C):Cop→PC is a functor and it’s colimit is C∈PC.
Is colimit(nerve(−↓C))=nerve(C) ?
Sorry, I’m having some trouble parsing this. If I have an object c of C, which subcategory is (c↓C) supposed to be? Ordinarily I would interpret the notation as denoting the comma category whose objects are morphisms c→d where d is an object of C, and whose morphisms are commutative triangles with vertex c, but that’s not a subcategory of C.
As for the next part, am I to interpret the nerve we’re taking the colimit of as this composite:
Cop(−↓C)→PC↪Catnerve→SetΔop,assuming that the first arrow makes sense?
Hi Todd,
Thanks for you quick reply.
Your interpretation of (c↓C) is the same as mine. So is your interpretation of the the the nerve we’re taking colimits of.
I think of (c↓C) as a subcategory of C by sending the object c→c0 to c0 and the morphism c→c0→c1 to the morphism c0→c1.
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 PC and go straight to (−↓C):Cop→Cat. Someone around here may know right away, but I’ll try to give it a think when I get a chance.
Yes of course, you’re right. (c↓C) is not a subcategory or C for general categories C. However, I neglected to mention that in the particular situation that I’m considering, C is a partially ordered set. In particular, it’s hom-sets have cardinality at most 1. I think that for such categories C, (c↓C) is a subcategory of C.
I also agree with you that this point doesn’t really matter anyway since we can bypass PC and go straight to Cat.
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.
dan
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.
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 Cat of the (c↓C) being isomorphic to C 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 colimc:Cop(c↓C) are equivalence classes of arrows c→d where the equivalence ∼ is generated by
(cf→d)∼(c′g→cf→d)and it is immediate that every g:c→d is equivalent to 1d:d→d; this of course is just a form of the Yoneda lemma.
Now let’s look at your problem, which compares the nerve(C) to the colimit of
Cop(c↓C)→Catnerve→SetΔopSince colimits in SetΔop are computed pointwise, we just have to show the colimit of
Cop(c↓C)→Catnerve→SetΔopevn→Set,where evn is evaluation at an object n, agrees with nerve(C)n. This is
Cop(c↓C)→Cathom([n],−)→SetNow an n-simplex in the comma category (c↓C), which is an element of this composite, is the same as an (n+1)-simplex beginning with the vertex c, and the colimit (in Set) consists of equivalence classes of (n+1)-simplices where a simplex beginning with c is deemed equivalent to a simplex beginning with c′ obtained by pulling back along any g:c′→c. And again, it is a triviality that each (n+1)-simplex
c→(d0→…→dn)is equivalent to
d01d0→(d0→…→dn)but the collection of such d0→…→dn is the same as nerve(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.
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,*) (where * denotes the functor constant at a terminal object), while the nerve of c↓C is the bar construction B(C(c,−),C,*). Since colimits of a functor F:Cop→D are given by tensor products of functors *⊗CF, and such tensor products come inside a bar construction (since colimits commute with colimits), we have
colimcN(c↓C)=*⊗c∈CB(C(c,−),C,*)=B(*⊗c∈CC(c,−),C,*)=B(*,C,*)=NCwhere *⊗c∈CC(c,−)=* by the co-Yoneda lemma.
Ah, ah, ah – excellent point, Mike. Thanks.
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