2
$\begingroup$

When I say "$\infty$-category", I mean a simplicial set such that all inner horns have not necessarily unique fillers. In Lurie's Higher Topos Theory there's the following

Definition. Given an $\infty$-category $\mathcal{C}$, a simplicial set $L$, a map of simplicial sets $p : L \to \mathcal{C}$, there's a unique $\infty$-category $\mathcal{C}_{/p}$, the over-category, such that, for every simplicial set $K$, we have $$\operatorname{Hom}_{\mathbf{sSet}}(K,\mathcal{C}_{/p}) \cong \operatorname{Hom}_p(K \star L,\mathcal{C}),$$ where $K \star L$ means the join of the simplicial sets $K$ and $L$, and where the right-hand side means all morphisms $K \star L \to \mathcal{C}$ commuting with the inclusion morphism $L \to K \star L$ and $p : L \to \mathcal{C}$.

It's then said that this "can be dualized", leading to the notion of an under-category; no further details are given. I've tried working them out myself, but it does not make sense to me.

Attempt 1. Given an $\infty$-category $\mathcal{C}$, a simplicial set $L$, a map of simplicial sets $p : \mathcal{C} \to L$, there's a unique $\infty$-category $\mathcal{C}_{p/}$ such that, for every simplicial set $K$, we have $$\operatorname{Hom}_{\mathbf{sSet}}(K,\mathcal{C}_{p/}) \cong \operatorname{Hom}_p(L \star K,\mathcal{C}),$$ where the right-hand side refers to morphisms $L \star K \to \mathcal{C}$ commuting with the maps $L \star K \to L$ and $p : \mathcal{C} \to L$. This can't be it, because there's no such thing as a map $L \star K \to L$.

Attempt 2. Given an $\infty$-category $\mathcal{C}$, a simplicial set $L$, a map of simplicial sets $p : \mathcal{C} \to L$, there's a (unique) $\infty$-category $\mathcal{C}_{p/}$ such that, for every simplicial set $K$, we have $$\operatorname{Hom}_{\mathbf{sSet}}(C_{p/},K) \cong \operatorname{Hom}_p(\mathcal{C},L \star K),$$ where the right-hand side refers to morphisms $\mathcal{C} \to L \star K$ commuting with the maps $p : \mathcal{C} \to L$ and the inclusion map $L \to L \star K$. This can't be it, bcause there's only one such map in the right-hand side, namely the composition.

It's probably something really obvious I'm missing, but for now I have the following

Question. What is an under-category?

$\endgroup$
0
$\begingroup$

Attempt 1 is the right isomorphism, but you dualized too much. $Hom_p(L\star K, \mathcal C)$ still means the maps commuting with the inclusion of $L$ and the map $p:L\to \mathcal C$. To help see this, consider the case that $L$ is terminal and $\mathcal C$ is a category-you should recover the slice category of the objects under $p$.

$\endgroup$
  • $\begingroup$ Thank you. Things always make sense in hindsight; you need a diagram in $\mathcal{C}$ to define a category under this diagram, so it wouldn't even make sense to start with a map $p : \mathcal{C} \to L$ rather than a map $p : L \to \mathcal{C}$. $\endgroup$ – guest May 30 '18 at 15:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.