# Is the category of toposes cocomplete ?

Hello.

[Edits between brackets.]

Does the [1-]category of [elementary] toposes [with logical morphisms] admit any [1-]colimits ?

[By colimit I mean initial object in the category of outgoing cocones. I'm afraid I don't know what pseudo-colimits, bicolimits or strict 2-limits are, but I can say that I'm not interested by any 2-categorical aspect of my question (at least, for now), just by the 1-categorical one.]

Any reference would be greatly appreciated.

Thanks for any answer and please forgive me for my pityful english.

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you said you mean elementary toposes, but which 1-cells: geometric morphisms? logical? and, which type of colimits: pseudo-colimits? bicolimits? strict 2-limits? – Eduardo Pareja Tobes Jul 20 '11 at 16:23
Logical morphisms preserve logic, so the coproduct of a topos that satisfies some proposition $p$ with another topos that satisfies $\neg p$ would have to satisfy $p\land \neg p$. – Wouter Stekelenburg Jul 21 '11 at 7:48

## 2 Answers

$\mathbf{Log}$, the category of elementary topoi and logical morphisms which preserve everything on the nose is cocomplete; in

Eduardo J Dubuc, GM Kelly - A presentation of topoi as algebraic relative to categories or graphs - Journal of Algebra (website)

this category $\mathbf{Log}$ is shown to be the category of algebras of a finitary monad on $\mathbf{Cat}$; it looks like you can even get this over $\mathbf{Grph}$, like the presentation of cartesian closed categories monadic over graphs in Lambek-Scott intro to higher-order categorical logic. This implies then that it is cocomplete, see the nLab page on colimits in categories of algebras, for example.

For the sake of completeness, a nice account (including explicit constructions) of 2-limits in $\mathbf{Log}$ as a locally groupoidal 2-category (1-cells the standard notion of logical morphism, 2-cells between them restricted to be isomorphisms) is given in chapter III of Steve Awodey's PhD thesis,

S Awodey - Logic in topoi: functorial semantics for higher-order logic - available from his website

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You should consult section B3 Of the Elephant by Johnstone. My understanding is that the 2-category of Grothendieck toposes (over sets) has all small colimits. This holds more generally for bounded toposes over a topos with natural numbers object.

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Just to make things clear: I presume that you are talking about 2-colimits. – André Henriques Jul 20 '11 at 15:30
Yes, I that is what I was trying to get at when I said the 2-category of Grothendieck toposes. – Benjamin Steinberg Jul 20 '11 at 16:11
Thank you. I'm sorry I haven't said I was talking about 1-colimits. – Funny clown Jul 20 '11 at 16:39