In Lurie's Higher Topos Theory Proposition A.3.2.4, the author used Proposition A.2.6.15 to prove that for any combinatorial monoidal model category $\mathbf{S}$ with all objects cofibrant and weak equivalences stable under filtered colimits, the category $\mathbf{Cat_S}$ is a left proper combinatorial model ctegory, where he implicitly used the statement:

$\mathbf{Cat_S}$ is locally presentable.

Why is this statement true? Can anybody give me some help? Thanks in advance!


1 Answer 1


This is proved in the paper

Kelly, G. M.; Lack, Stephen. $\scr V$-Cat is locally presentable or locally bounded if $\scr V$ is so. Theory Appl. Categ. 8 (2001), 555--575. http://tac.mta.ca/tac/volumes/8/n23/8-23abs.html

as an instance of the fact (proved by Gabriel and Ulmer) that the category of algebras for a finitary monad on a locally $\lambda$-presentable category is locally $\lambda$-presentable (for $\lambda$ a regular cardinal).

To wit, Kelly and Lack show that if $\scr V$ is a monoidally cocomplete category, then the category ${\scr V}\textbf{-Cat}$ of $\scr V$-enriched categories is finitarily monadic over the category ${\scr V}\textbf{-Gph}$ of $\scr V$-enriched graphs, which they show to be locally $\lambda$-presentable if $\scr V$ is so. (Note that, by definition, a combinatorial monoidal model category is in particular monoidally cocomplete and locally presentable.)

  • 1
    $\begingroup$ Thanks for your help! $\endgroup$
    – Frank Kong
    Apr 20, 2019 at 3:40

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