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Results tagged with homotopy-theory
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user 1306
Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
16
votes
What are Homotopy rings good for?
The structure is just that of a graded Lie ring (homologically graded - to create the correct Koszul signs), once you shift degrees by 1. This structure is not at all exotic, you see it in Gerstenhabe …
- 16.9k
9
votes
Accepted
How to define the equivalence of Maurer-Cartan elements in an $L_{\infty}$-algebra?
To add a bit to what Damien says, addressing your question on how to generalise the gauge approach (which is equivalent to the approach outlined by Damien, as proved by several people):
You can view …
- 16.9k
6
votes
Who introduced the notion of 2-categories?
Following the comment of varkor, I re-opened Catégories structurées of Charles Ehresmann (published 1963), and I believe that Section 4 "Catégories doubles" and Section 5 "Catégories $n$-uples" give t …
- 16.9k
3
votes
Accepted
Homotopy transfer in the opposite direction
Let us denote by $p\colon Y\to X$ and $i\colon X\to Y$ the maps of your SDR. Since $pi=\mathop{\mathrm{id}}\nolimits_X$, the map $i$ is injective, and is an isomorphism with its image. The map $\pi=i\ …
- 16.9k
2
votes
What is a Homotopy between $L_\infty$-algebra morphisms II
First of all, you might want to look at the MO question How to define the equivalence of Maurer-Cartan elements in an $L_\infty$-algebra?, since of course this is effectively what you need (morphisms …
- 16.9k
1
vote
Is super-vector spaces a "universal central extension" of vector spaces?
I am not sure if that's what you want, but let me try though. When I hear of extensions, one of the pictures I have in my mind is something like crossed products, and I want to push that analogy. Ther …
- 16.9k