/sci/ - Science & Math

>>9371562
define "the homotopy type of topological spaces"

>>9371575
∞-groupoids should have the same homotopy theory of topological spaces.

So an ∞-groupoid should not be exactly equivalent to a topological space, but equivalent to a class of topological spaces up to weak equivalence(i.e. all their homotopy groups are isomorphic).

Strictly speaking, ∞-groupoids should form an (∞,1)-category and this (∞,1)-category should be equivalent to an (∞,1)-category obtained by "localizing" the category of topological spaces at weak homotopy equivalence.

>>9371562
Wrong

>>9371562
>should be
>should have
Absolute state of algebraic wank.

>>9372841
>homotopy hypothesis
>hypothesis

>>9372869
https://arxiv.org/abs/1705.02240
The cobordism hypothesis can be proved, so where are the proofs of this homotopy "hypothesis"?
Answer: there are none, because cobordism hypothesis is the cornerstone of something concrete (i.e. TQFT) while this homotopy hypothesis is the cornerstone of absolute algebraic wank.

>>9372889
It can be proven once you actually define an ∞-groupoid. It was first proposed by Grothendieck, before ∞-groupoids were even defined, as a sort of guiding principle.

The common definition of an ∞-groupoid is a Kan Complex. And Kan Complexes, considered up to weak homotopy equivalence of simplicial sets, are equivalent topological spaces considered up weak homotopy equivalence.

>>9372898
>it can be proven once you assume that it can be proven
Amazing.

>>9372905
Only if you use the homotopy hypothesis as motivation for a definition.

Conceptually an ∞-category should be a category with higher morphisms.

i.e. Instead of just objects and morphisms between objects, we have 2-morphisms between 1-morphisms, 3-morphisms between 2-morphisms, etc.

And then an (∞,n)-category should be an ∞-category where all k-morphisms, for k>n, are invertible.

Since a groupoid is a category where all morphisms are invertible, an ∞-groupoid should be an (∞,0)-category in the sense described above.

Under this motivation, there isn't any apriori reason to model ∞-groupoids as topological spaces.

However if you are familiar with the definition of a Kan complex, they make perfect sense for a model of ∞-groupoids.

And then from there, you can prove Kan complexes (up to weak equivalence) are equivalent to topological spaces (up to weak equivalence).

>>9372963
What do higher morphisms and this specific way of defining (infty,n)-categories model? What uses this kind of specific structure?

>>9373589
The idea is ∞-categories "remember" higher homotopical information.

There is definitely motivation from stable homotopy theory.

A more geometric motivation has to do with intersection theory. A pullback (i.e. intersection) of schemes/varieties, in the category of schemes, does not remember intersection multiplicities. However an (∞-)pullback in the ∞-category of derived schemes does remember this information.

>>9373743
>mfw intersection cohomology is useful in equivariant integration
What the fuck did you just change my mind???