Created descent morphism.

In adding links, I discovered that Euclidean-topological infinity-groupoid and separated (infinity,1)-presheaf use the phrase “descent morphism” to refer to the *comparison functor* mapping into the category of descent data. If no one has any objections, I would like to change this to avoid confusion, but I’m not sure what to change it to: would “comparison functor” be good enough?

I a searching a proof for the fact(s) claimed in this post: https://nforum.ncatlab.org/discussion/560/codomain-stacks/

Is there a proof on the condition for a self-indexing to be a stack? I would like a reference. If there is not, some particular cases or proofs with similar ideas would also be appreciated.

Thanks!

]]>New entries descent along a torsor and Schneider’s descent theorem. Some changes and literature additions to a number of related entries.

]]>I have a very strange question. When people tell me that stacks are abstract nonsense that only algebraic geometers care about (or something similar) I try to point out that stacks occur in most branches of math even if you don’t think of them as stacks (most prominently the stack of vector bundles for differential geometers).

A joke I often make is that there is probably even a stack of measures on a (sufficiently nice) topological space that might interest measure theorists. This question is for people who know some measure theory. Today I was talking about this with an analyst, but the language barrier made it hard to actually verify.

Suppose $X$ is nice enough so that nothing fishy goes on with Borel measures. Then take $Op(X)$ to be the standard site of open sets. Form the category of pairs $(U, \mu)$ where $U$ an open and $\mu$ a Borel measure on $U$. We’ll do something kind of dumb for the arrows and say $(U,\mu)\to (V, \nu)$ is an inclusion $V\subset U$ together with an actual equality of measures $\mu|_V=\nu$.

Just take the forgetful functor and since everything is so rigid this forms a category fibered in sets over $Op(X)$ and is actually seems to be a stack. Now it probably isn’t interesting at all considering you can’t ever have a non-trivial automorphism of $(U,\mu)$. I was wondering if anyone has ever thought of this, or checked this, or come up with something more interesting in a similar vein that I can start using as an example. In theory you could try to use cohomology or something to study measures in this way or talk about “deformations of measures” or something.

The thing that the analyst did say is that measure theorists might care if you could somehow do this up to “bi-Lipschitz equivalence”. We thought about that. The arrows $(U,\mu)\to (V,\nu)$ would then be a bi-Lipshcitz homeomorphism $f:U\to U$ so that $(f_* \mu)|_V=\nu$. It also seems to form a stack (in sets) and moreover you could get lots of automorphisms (for instance take the disk in $\mathbb{R}^2$ and area measure, then any rigid automorphism gives the same measure back) so the stackiness is actually useful. Can anyone else quickly see if this is obviously true or not true for some reason?

]]>Remake of Street’s Gummersbach paper: Characterization of Bicategories of Stacks (zoranskoda).

]]>