The n-Category Café

Question on Weak Pullbacks along Sequences

Posted by Urs Schreiber

Lately I have been running into the following issue, whose full meaning I am trying to better understand.

It’s about a situation where we have two composable morphisms of groupoids

$$K \hookrightarrow G \to\gt B$$

with the property that the image of the left monomorphism is entirely included in the preimage under the right epimorphism of all identity morphisms in $B$.

I’ll address this situation as a short sequence of groupoids. (Here I am not concerned with whether and how this sequence might be “exact”.)

Then with any morphism into $B$ given

$$\array{ &&&& P \\ &&&& \downarrow \\ K &\hookrightarrow& G &\to\gt& B }$$

I want to have a sensible notion of what it means to pull this back weakly along the sequence.

It seems that I know (motivated by my application where this arises in) what the right answer is. What I am looking for is the right question that yields this answer.

Here is what the answer is supposed to be, for which I am looking for the right question:

Answer: given the above setup, we want to complete to a diagram

$$\array{ P_F &\hookrightarrow& P_Y &\to \gt& P_X \\ \downarrow &\Downarrow&\downarrow &\Downarrow& \downarrow \\ K &\hookrightarrow& G &\to\gt& B }$$

with the special property that

A)

$$P_F \hookrightarrow P_Y \to \gt P_X$$

is itself a sequence of groupoids.

B)

$$\array{ P_F &\hookrightarrow& P_Y &\to \gt& P_X \\ &&\downarrow &\Downarrow& \downarrow \\ && G &\to\gt& B }$$

is the identity transformation

C)

$$\array{ P_F &\hookrightarrow& P_Y && \\ \downarrow &\Downarrow&\downarrow && \\ K &\hookrightarrow& G &\to\gt& B }$$

is the identity transformation.

Question: What exactly is the question that yields this answer?

It must be something about forming a weak limit of the diagram

$$\array{ &&&& P \\ &&&& \downarrow \\ K &\hookrightarrow& G &\to\gt& B }$$

But there are a couple of flavors of weak limits. I have some ideas about how the conditions A), B) and C) should arise by using these. But don’t feel fully comfortable yet. If you can help, please do.

My application

This arises when forming non fake-flat $n$-transport by pulling back classifying maps along suspended universal bundles in their groupoid incarnation. More details can be found in the slide show String- and Chern-Simons $n$-Transport.

The big picture is indicated in section $n$-Categorical Background subsection $G_{(n)}$-bundles with connection.

The definition with conditions A), B) and C) as above is mentioned in section Parallel $n$-Transport subsection Non fake-flat $n$-transport.

The differential version (in terms of Lie $n$-algebras instead of Lie $n$-groupoids) is indicated in section Bundles with Lie $n$-algebra connection subsection Bundles with $g_{(n)}$-connection.

To navigate these slides, make use of the hypertext tools of your pdf-reader: use your arrow keys to browse forward, click on the underlined links to pass to special sections, use the pdf-reader’s back button to get back.