### 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.

## Re: Question on Weak Pullbacks along Sequences

This may be a silly question, but in the phrase “non-fake-flat”, what is the default associativity? Because (non-fake)-flat and non-(fake-flat) look different to me. Ought I to know by now that the right sense of “flat” is “fake-flat” in this context?