Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let us denote by $\text{Cat}_c$ the $2$-category, whose objects are cocomplete categories, morphisms are cocontinuous functors and morphisms are natural transformations. Is it then true that $\text{Cat}_c$ is $2$-cocomplete, i.e. has every $2$-functor $C : I \to \text{Cat}_c$, where $I$ is a small $1$-category, a $2$-colimit?

I have the following idea: Let $U :\text{Cat}_c \to \text{Cat}$ be the forgetful $2$-functor. Now $\text{Cat}$ is $2$-cocomplete; a comprehensive reference for this is "The stack of microlocal perverse sheaves" by Ingo Waschkies. Besides, $U$ has a $2$-left adjoint $\widehat{(- )} : \text{Cat} \to \text{Cat}_c$ (see here). So we may consider the cocomplete category $D:=\widehat{\text{colim} UC}$, but of course the functors $C_i \to D$ are not cocontinuous. Thus perhaps we have to define some reflective (and hence cocomplete) subcategory $D'$ of $D$, such that each $C_i \to D$ factors through a cocontinuous functor $C_i \to D'$. But I don't know how to define $D'$.

I'm also interested in colimits in similar $2$-categories, for example $\text{Cat}_{c\otimes}$, which consists of the cocomplete tensor categories. And actually I want something more, namely that these $2$-categories are $2$-locally presentable. Any hints and references are appreciated!

share|improve this question
    
Similar issues are discussed in the papers relating to "totality" in category theory (look e.g. for Max Kelly's articles). Off hand, I do not know the precise answer. –  Zoran Skoda May 14 '11 at 13:28
    
Martin, for your purposes which I am guessing, maybe the work of Durov on vectoids could be also useful. I have sent you some data via personal email. –  Zoran Skoda May 14 '11 at 13:36
    
Thank you. The work is interesting, but it is not useful in this situation. –  Martin Brandenburg May 14 '11 at 17:20
    
Here is an idea. The forgetful functor is not only having a left adjoint but is in fact 2-monadic (see Kelly and Lack emis.dsd.sztaki.hu/journals/TAC/volumes/7/n7 and references therin). Now monadic functors create and preserve limits but not colimits. However, if you have coequalizers (not preserved!) then in 1-categorical situation you can construct colimits on the Eilenberg-Moore category, cf. Borceux, Handbook vol. 2, Prop. 4.3.4. So if you can make coequalizers and the Prop. extends to 2-dim situation you are in business. –  Zoran Skoda May 15 '11 at 21:18
    
I think I can construct $2$-coproducts in $\text{Cat}_c$, but I don't know how to deal with coequalizers. Also, everything becomes more mysterious in the enriched setting, say with $R$-linear categories. In the details, the whole thing becomes even more fiddly. Is the answer to my question unknown, or is it just a "common fact"? –  Martin Brandenburg May 16 '11 at 11:20

1 Answer 1

up vote 7 down vote accepted

In order not to have to worry about size issues, I'm going to answer the following question instead:

For a (small) cardinal number $\kappa$, is the category of small categories with $\kappa$-small 2-colimits 2-cocomplete?

If you take $\kappa$ to be inaccessible, then this will correspond to your question, under a particular choice of foundations. I presume moreover that you mean "2-colimits" in the weak "up-to-equivalence" sense which the nLab uses (which 2-category theorists traditionally call "bicolimits").

The fact that the 2-category Cat of small categories is 2-cocomplete, in this sense, has been well-known to category theorists for decades. It is obvious that Cat is cocomplete as a 1-category (since it is locally finitely presentable), and since it is closed symmetric (cartesian) monoidal, it follows by general enriched category theory that it is cocomplete as a category enriched over itself. In the nLab terminology, it has all strict 2-colimits. We then observe that strict pseudo 2-limits, which are 2-limits that represent cones commuting up to isomorphism but satisfy their universal property up to isomorphism (rather than equivalence), are particular strict 2-limits. Since any strict pseudo 2-limit is also a (weak) 2-limit, Cat is 2-cocomplete.

Now as Zoran pointed out in the comments, there is a 2-monad on Cat whose algebras are categories with $\kappa$-small colimits; let us call this 2-monad $T$. The strict $T$-morphisms are functors which preserve colimits on-the-nose, while the pesudo $T$-morphisms are those which are $\kappa$-cocontinuous in the usual sense (preserve colimits up to isomorphism). Therefore, the question is whether the 2-category $T$-Alg of $T$-algebras and pseudo $T$-morphisms is 2-cocomplete.

The answer is yes: it was proven by Blackwell, Kelly, and Power in the paper "Two-dimensional monad theory" that for any 2-monad with a rank (preserving $\alpha$-filtered colimits for some $\alpha$) on a strictly 2-cocomplete (strict) 2-category, the 2-category $T$-Alg is (weakly) 2-cocomplete. The 2-monad $T$ has a rank (namely, $\kappa$, more or less), so their theorem applies. I believe this all works just as well in the enriched setting.

share|improve this answer
    
The $2$-category of cocomplete categories with chosen colimits (and pseudo morphisms) is $2$-equivalent to the $2$-category of cocomplete categories, right? –  Martin Brandenburg May 26 '11 at 11:47
    
Yes. When your morphisms are pseudo, they "forget" which colimits were "chosen". –  Mike Shulman May 26 '11 at 23:01
    
Thanks. It would be nice if you can actually describe the colimits in my specific exmaple. –  Martin Brandenburg May 29 '11 at 10:32
1  
It would, wouldn't it? (-: Unfortunately, general colimits in categories of algebras for monads are almost always quite nasty. –  Mike Shulman Jun 3 '11 at 4:09

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.