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Chad Nester (Jul 07 2020 at 13:39):I don't know if this counts as a basic question, but I can't find anything in the literature and I don't see an appropriate existing thread. Perhaps someone here knows!
Q: Is the category of elementary toposes and logical morphisms known to be locally finitely presentable?
Chad Nester (Jul 07 2020 at 13:44):(realizing this is probably one of those "this depends..." questions, let's start with toposes internal to Set, or similar)
Nathanael Arkor (Jul 07 2020 at 14:04):I think it is. The category of elementary toposes and logical morphisms is the category of algebras for a finitary monad on Cat (this is shown in Dubac–Kelly's A Presentation of Topoi as Algebraic Relative to Categories of Graphs), and the category of algebras for a finitary monad on a locally finitely presentable category is also locally finitely presentable by the final remark of Chapter 2 of Locally Presentable and Accessible Categories (page 124).
Chad Nester (Jul 07 2020 at 14:06):Thanks! I think that answers my question :)
Reid Barton (Jul 07 2020 at 15:08):It depends :upside_down:
Let's consider a simpler question: is the category of (categories with a terminal object) and (functors preserving the terminal object) lfp?
If one interprets the question as being about categories with a chosen terminal object and functors which strictly (i.e., up to equality) preserve the chosen terminal object, then the answer is yes, by the above sort of argument. In fact, Freyd's "Aspects of topoi" more or less explicitly describes a finitary essentially algebraic theory of which elementary topoi in this sense (i.e., with chosen terminal object / pullbacks / subobject classifier or whatever the exact list) are the models.
If one interprets the question in the more usual way as about categories with at least one terminal object and functors which preserve the property of being a terminal object, and if one is interested in the 1-category of such categories and functors, then the answer is no. In fact there is not even an initial category-with-terminal-object, because if we pick a category with at least two terminal objects, there cannot be exactly one terminal-object-preserving functor from any category-with-terminal-object to it.
If however one considers the question to be about 2-categories or (2,1)-categories, then presumably the answer is yes again but one has to first understand what is meant by "locally finitely presentable 2-category".