Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: Twisted arrow category and the nerve

Arturo De Faveri (Feb 20 2025 at 18:38):

Hi,

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

My question is: I wonder whether this construction is universal in the sense that $(\textrm{el}, \varepsilon)$ is the right Kan extension of $\mathrm{Tw}$ along $N$. 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!