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I am thinking about a theory of approximations to the category of sets. To that end, I want to say that the category of sets is a colimit over compact categories. The categories that I am thinking are things like $1, 2, 3 \ldots$ which are the single object, the two object single arrow and then, 3, which has three objects and two arrows (understood as the composition category).

I asked the general question for this some time ago and got an answer. So Set must be a colimit over a diagram with just $1,2,3$. What is the theory of this?

There must be an ambient category in which this all lives and that category might be locally finitely presentable. If this is true, what is the theory of this?

So, I am wondering if there is a category of topoi that contains Set and is locally finitely presentable. Also, does $1,2,3$ form the basic categories such that all objects are colimits over diagrams of these categories? Is there another basis, other than $1,2,3$?

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    $\begingroup$ What do you mean by "composition of categories"? The way you phrase it now, all the categories you will consider are isomorphic (they are all the one-point category). That is not going to give you enough building blocks. $\endgroup$ – Mark Kamsma Jun 12 at 14:14

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