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Stream: learning: questions

Topic: Twisted arrow category and the nerve


view this post on Zulip Arturo De Faveri (Feb 20 2025 at 18:38):

Hi,

almost by chance I bumped into the following construction. Let Tw():CatCat\mathrm{Tw}(-): \mathbf{Cat} \to \mathbf{Cat} be the functor that takes in input a (small) category C\mathcal{C} and returns its twisted arrow category (or, which is the same, the category of elements of the functor C(,)\mathcal{C}(-,-)). Let N:CatsSetN: \mathbf{Cat} \to \mathbf{sSet} be the nerve functor and let el:sSetCat \textrm{el}: \mathbf{sSet} \to \mathbf{Cat} be the functor returning the category of elements of a simplicial set. There is a well-defined natural transformation ε:elNTw\varepsilon: \textrm{el} \circ N \Rightarrow \mathrm{Tw} as follows. For each small category C\mathcal{C}, εC:el(NC)Tw(C)\varepsilon_{\mathcal{C}}: \textrm{el}(N \mathcal{C}) \to \mathrm{Tw}(\mathcal{C}) is the functor acting on the objects as ([n],(f1,,fn))f1fn([n], (f_1, \ldots, f_n)) \mapsto f_1 \circ \cdots \circ f_n.

My question is: I wonder whether this construction is universal in the sense that (el,ε)(\textrm{el}, \varepsilon) is the right Kan extension of Tw\mathrm{Tw} along NN. I don't have a particular reason to believe that, but it type-checks, which is already something. If this is indeed the case I would like to know a bit more what's going on!

Thanks a lot for your patience!