MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Here, I asked wether taking lax pullback preserves continuity, but got no precise answer.

However, I have found this recent article by Riehl and Verity which proves something very similar, but I can't see wether it applies immediately in the context I want.

Can someone familiar with the paper (or perhaps the authors) say if it implies the following:

Suppose $F:C\rightarrow D$ and $G:B\rightarrow D$ are functors of 2-categories which send comma objects to comma objects, is it true that the map $F\downarrow G\rightarrow B$ sends comma objects to comma objects?

share|cite|improve this question
up vote 9 down vote accepted

Our paper specializes to the case of 1-categories with (co)limits and (co)limit-preserving functors, not to 2-categories.

But surely this is true. I'm defining the comma object in the 2-category of (strict) 2-categories, 2-functors, and 2-natural transformations. The defining universal property of $F \downarrow G$ characterizes maps $A \to F\downarrow G$. Considering $A$ to be terminal, the walking arrow, and the walking 2-cell tells us how to define the 2-category $F\downarrow G$. For instance: a 2-cell is a 2-cell in $C$ and a 2-cell in $B$ together with a 2-natural transformation from the image of the former to the image of the latter. The data of this 2-natural transformation consists of a pair of maps in $D$, which is part of the data of the pair of objects serving as domain and codomain of this 2-cell. Note that the underlying category of this 2-category is the comma category of the underlying categories.

Now a cospan in $F \downarrow G$ is just a cospan in the underlying comma category. Form the comma objects in $C$ and $B$. Their images in $D$ are still comma objects and the legs of the cospan together with the 2-cell of the former comma object defines a cone over the latter, inducing a unique map between the comma objects so that everything commutes. This defines the comma object in $F \downarrow G$ and the universal property is proven similarly.

I'd suspect, guided by Blackwell-Kelly-Power's "Two dimensional monad theory" that any PIE-limit (including pseudo-limits and lax-limits) of 2-categories with and 2-functors that preserve certain 2-(co)limits will have those 2-(co)limits and they will be preserved by the legs of the limit cone. But I don't know whether such 2-categories can be characterized as algebras for a 2-monad.

UPDATE: John Bourke has pointed me to a paper On the monadicity of categories with chosen colimits that shows that what you'd hope for is true. For a cosmos $V$ and any class of small weights, the 2-category $V$-CAT has a 2-monad whose algeras are $V$-categories with chosen weighted (co)limits and whose pseudomorphisms are $V$-functors preserving such. The Blackwell-Kelly-Power theorem now implies the result that you want.

share|cite|improve this answer

Your Answer


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.