Higher categorical analogue of concreteness

2012-10-18T22:15:21Z

I am going to make this question deliberately vague.

A category is concrete if its objects can be realized as sets with extra structure (in particular, it admits a faithful functor to Set).

The examples I know of non-concretizable categories (the homotopy category, Cat with naturally isomorphic functors identified) are obviously derived from forgetting structure of higher categories. Is there some sense in which, given any locally small category, the objects of that category can be realized themselves as (small) higher categories with extra structure, analogous to considering faithful functors to Set?

EDIT: I think my question is as Jeremy suggested below: Given a locally small category, is it always the homotopy category of a concrete $(\infty,1)$-category, for some reasonable definition of concrete?

Answer by Nate Ackerman for Higher categorical analogue of concreteness

2012-10-19T02:36:51Z

If you require you category to be actually small (and not just locally small) then the answer is yes.

Suppose $C$ is a small category. We can then define the category $C'$ as follows:

Objects($C'$) = ${ C/t: t \in$ Object(C)$}$
Morphisms($C'$) are functors between the corresponding categories

There is then a functor $F:C \rightarrow C'$ where

$F(t) = C/t$ for any object $t$ of $C$
For any map $\alpha:s \rightarrow t$ in $C$, $F(\alpha):C/s \rightarrow C/t$ is the functor where:
- When $\beta:p \rightarrow s$ is an object of $C/s$ then $F(\alpha)(\beta) = \alpha \circ \beta:p \rightarrow t$ is the corresponding object of $C/t$.
- Suppose $P:p\rightarrow s$ and $Q:q\rightarrow s$ are objects of $C/s$ and $g: P \rightarrow Q$ is a morphism in $C/s$ (i.e. a map $g:p \rightarrow q$ such that $q \circ g = p$).
  
  Then $F(\alpha)(\gamma) = g$ (as a map from $F(\alpha)(P) \rightarrow F(\alpha)(Q)$)

When dealing with locally small (but not necessarily small) categories however you have to be careful about set theoretic size issues.

Higher categorical analogue of concreteness - MathOverflow

Higher categorical analogue of concreteness

Answer by Nate Ackerman for Higher categorical analogue of concreteness