# 2-completeness of stacks

I am looking for a reference which discusses the 2-categorical properties of the 2-category $$St(C,J)$$ of stacks and the stackification $$\dashv$$ inclusion of presheaves 2-adjunction.

My stacks are indexed categories (not necessarily groupoids). I am for example interested in knowing what kind of 2-limits $$St(C,J)$$ has. Is $$St(C,J)$$ 2-(co)complete, what properties do the (co)limits have, etc.? I tried hard to find texts about that, but I wasn't able to find anything useful.

Edit. I know that 2-categories of stacks have all homotopy (co)limits. I am interested in stronger (Cat-enriched) limits like flexible limits. I want to know if categories of stacks have them and how they are formed. I can show that St(C,J) has all the flexible Cat-enriched limits which $$BiCat(C^{op},Cat)$$ admits, so it seems like the only thing missing is a proof that $$BiCat(C^{op},Cat)$$ has flexible limits. The category $$BiCat(C^{op},Cat)$$ itself is the category of pseudoalgebras for a 2-monad on $$\Pi_{C_0}Cat$$. In "Two-dimensional monad theory" the authors claim that any two monad $$T$$ which satisfies some smallness conditions can be replaced by a 2-monad $$T'$$ such that $$T'Alg= PsdTAlg$$ are Cat-enriched equivalent. This would imply that categories of stacks have all flexible limits. But unfortunately the papers which the authors of "Two-dimensional monad theory" promised never appeard. (I can't find them). So I guess what I am really looking for is a reference for the $$T'Alg = PsdTAlg$$ result. It is mentioned very often, so I assume some reference must exist. Where can I find it? :)

• Have you looked at Street's papers on 2-toposes? ("2-dimensional sheaf theory" and "...bicategories of stacks") These 2-categories are reflective in 2-categories of 2-functors into Cat, so they're certainly going to be complete and cocomplete. May 26 at 22:29
• @KevinArlin Not yet, I'll check it out!
– Nico
May 27 at 9:39

I am very interested in this question. I can only write a partial answer, hoping that someone can complete it or suggest other approaches.

As Kevin Arlin pointed out, Street's papers "2-dimensional sheaf theory" and "Characterization of bicategories of stacks" prove that there is a stackification. The former paper considers strict stacks, i.e. strict 2-functors into Cat whose functors of descent are isomorphisms of categories. The latter considers the more standard stacks, i.e. pseudofunctors into Cat whose functors of descent are equivalences of categories.

By Street's former paper, the 2-fully faithful inclusion of strict stacks on C inside [C^op,Cat] has a left exact 2-adjoint, called stackification. From this, looking for example at Kelly's "Basic concepts of enriched category theory" (Section 3.5), it follows that strict stacks have all strict 2-limits and 2-colimits. Strict 2-limits are calculated in [C^op,Cat] and are automatically strict stacks. Strict 2-colimits are the stackification of the strict 2-colimits calculated in [C^op,Cat]. Then strict 2-(co)limits in [C^op,Cat] are calculated pointwise in Cat.

By Street's latter paper, the 2-fully faithful inclusion of stacks on C inside Ps[C^op,Cat] (pseudofunctors and pseudonatural transformations) has a left exact biadjoint, called stackification. From this I believe it follows that stacks have all bilimits and bicolimits, but I couldn't find a reference for this. It should be easy to prove that all bicolimits exist, calculated as the stackification of the bicolimit in Ps[C^op,Cat], since the stackification preserves bicolimits (being a left biadjoint) and the counit of the stackification-inclusion biadjunction should be an equivalence (since the right biadjoint is 2-fully faithful). But bilimits are more delicate. It would be then great to say that bi(co)limits in Ps[C^op,Cat] are calculated pointwise in Cat, which I believe is true, but again I couldn't find any reference on this.

We also have that Ps[C^op,Cat] has all flexible limits, by Remark 7.4 of Bird, Kelly and Power's "Flexible limits for 2-categories". So it would be nice to say that also stacks have all flexible limits (or at least PIE limits). This would allow for example to consider comma objects instead of just bicommas. A way to show that Ps[C^op,Cat] has all flexible limits could go through proving that the inclusion of stacks inside Ps[C^op,Cat] needs to be 2-monadic, for example by the 2-dimensional Beck's monadicity theorem of Le Creurer, Marmolejo and Vitale's "Beck's theorem for pseudo-monads". But I am not sure if this is true. Then, again, are flexible limits in Ps[C^op,Cat] calculated pointwise?

• Oh lol, I was always under the impression that comma objects are homotopical meaningful limits.
– Nico
May 31 at 17:05
• I just checked comma objects, and they seem to be okay. $f/g$ is a stack (respectively prestack) when domains and codomain of $f$ and $g$ are (pre)stacks. But I think that it works because comma objects are homotopical okay, there are no equalities appearing in the universal property. What is a bicomma object?
– Nico
May 31 at 17:27
• After inspecting my proof a little I think I would have gotten in trouble with 2-universal properties which involve $=$-2-cells. Some of the PIE-limits are not pseudo, right? It might be a good idea to check them by hand.
– Nico
May 31 at 17:29
• Flexible limits are homotopically meaningful strict 2-limits. I think the only difference between a bicomma and a comma is that a bicomma might only induce an equivalence of categories between maps in and cones, whereas a comma induces an isomorphism of categories. It's almost a category error to ask whether a PIE limit is pseudo, because PIE limits come up in the context of strict weighted limits in a category-enriched category. It just so happens that pseudo limits, with an isomorphism of categories in the defining universal property, can be defined as certain (strict!) weighted limits. Jun 1 at 16:56
• I have just had an idea for commas in stacks. Start from a diagram for a comma in stacks. If you embed the diagram in Ps[C^op,Cat], you can find the comma there. Assume that it's true that stacks have all bilimits calculated in Ps[C^op,Cat]. Then since the comma in Ps[C^op,Cat] that we have found also satisfies the universal property of a bicomma, it needs to be a stack. So it is a comma in stacks for the original diagram. Can this be extended to general flexible limits? Maybe saying that products are also biproducts, inserters are also biinserters and so on? Jun 2 at 16:30