### From Simplicial Sets to Categories

#### Posted by John Baez

There’s a well-known nerve of a category, which is a simplicial set. This defines a functor

$N \colon Cat \to sSet$

from the category of categories to the category of simplicial sets. This has a left adjoint

$F \colon sSet \to Cat$

and *this left adjoint preserves finite products*.

Do you know a published reference to a proof of the last fact? A textbook explanation would be best, but a published paper would be fine too. I don’t want you to explain the proof, because I think I understand the proof. I just need a reference.

Posted at August 31, 2019 6:49 AM UTC
## Re: From Simplicial Sets to Categories

According to 1.11 of Joyal’s notes here - https://www.math.uchicago.edu/~may/IMA/Joyal.pdf , the functor $\tau_1: S \to Cat$ preserves finite products by a result of Gabriel and Zisman, but I don’t know what the exact reference is. Maybe it’s really old?