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October 11, 2007

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

KG>B 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 BB.

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 BB given

P K G > B \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

P F P Y > P X K G > B \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 FP Y>P X P_F \hookrightarrow P_Y \to \gt P_X

is itself a sequence of groupoids.

B)

P F P Y > P X G > B \array{ P_F &\hookrightarrow& P_Y &\to \gt& P_X \\ &&\downarrow &\Downarrow& \downarrow \\ && G &\to\gt& B }

is the identity transformation

C)

P F P Y K G > B \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

P K G > B \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 nn-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 nn-Transport.

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

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

The differential version (in terms of Lie nn-algebras instead of Lie nn-groupoids) is indicated in section Bundles with Lie nn-algebra connection subsection Bundles with g (n)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.

Posted at October 11, 2007 5:16 PM UTC

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2 Comments & 0 Trackbacks

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?

Posted by: some guy on the street on October 11, 2007 7:11 PM | Permalink | Reply to this

Re: Question on Weak Pullbacks along Sequences

Thanks. Fixed.

Posted by: Urs Schreiber on October 11, 2007 8:19 PM | Permalink | Reply to this

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