Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
What is there to say in general about fracture theorems, fracture squares, etc.?
Mike had a post on The Propositional Fracture Theorem, then observed that a construction in tangent cohesive -toposes was a case of fracture.
It crops up as ’chromatic fracture squares’ here and here. On this theme, there’s a blog post Topologized objects in algebraic topology
Chromatic fracture is a big deal in algebraic topology; its noble goal is to reconstruct higher chromatic spectra from chromatic layers and gluing data.
If you find an answer, I’d love to hear it.
Are there always a pair of modalities involved + some glue, allowing you to recapture the whole ?
Sometimes there seems to be more than one modality, e.g. the fracture theorem of spaces into localizations at all primes and rationalization.
And in the case of the blog post, you might have divided your space up into a collection of open sets and the closed complement of their union.
Oh, does that relate to what Urs is doing with those filtrations, maybe here filtration by support?
Yeah, I thought briefly about whether such a decomposition of a space would give rise to a fracture theorem that looks like the one for localization, but I didn’t succeed. Perhaps I just didn’t think hard enough, though.
1 to 6 of 6