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Results tagged with homotopy-theory
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user 5587
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.
44
votes
Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?
There are several people here much more qualified to speak about that, so I shall just give you some pointers now. One of the questions Grothendieck tried to answer when writing "Pursuing Stacks" was …
9
votes
1
answer
791
views
How do various notions of natural transformation relate to various notions of homotopy in $2...
In what follows, $2$-categories will be strict, and "$2$-functor" will mean "strict $2$-functor". (Please mention which terminological conventions you are using when answering.) I guess that the answe …
- 2,413
4
votes
1
answer
531
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Is there a standard name for a 2-category which has an object z such that, for every object ...
Motivation
In Pursuing Stacks, Grothendieck defines what he calls a basic localizer, which is, to put it roughly, a class of functors between small categories with which one can make homotopy in $Ca …
- 2,413
19
votes
Accepted
What is the homotopy theory of categories?
I am not knowledgeable enough to have much to say I have not writen in my answer to a previous question of yours, and I think that David Roberts's answer (or, rather immodestly, my previous one) provi …
- 2,413
4
votes
Is there a topos theoretic interpretation/proof of Quillen's Theorem A?
I have no compelling answer to this question myself, but you may find relevant results and ideas in the work of Grothendieck, Maltsiniotis and Cisinski in homotopical algebra. Have you looked at Pursu …
- 2,413
19
votes
1
answer
874
views
Does the category of strict $2$-categories together with Dwyer-Kan equivalences provide a mo...
The question is the title.
In what follows, all $2$-categories and $2$-functors will be strict. Let $2-Cat$ denote the categories whose objects are $2$-categories and whose morphisms are $2$-functors …
- 2,413
16
votes
Conjectures in Grothendieck's "Pursuing stacks"
This is more a comment than an answer, but its length makes me post it as an answer. I want to react to what I have just read, for the first time, about "Pursuing Stacks" at the nLab, and the words us …
- 2,413
39
votes
3
answers
4k
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In which situations can one see that topological spaces are ill-behaved from the homotopical...
In the eighties, Grothendieck devoted a great amount of time to work on the foundations of homotopical algebra.
He wrote in "Esquisse d'un programme": "[D]epuis près d'un an, la plus grande partie d …