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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 …
39 votes
3 answers
4k views

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 …
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 …
Jonathan Chiche's user avatar
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 …
Jonathan Chiche's user avatar
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 …
Jonathan Chiche's user avatar
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 …
Jonathan Chiche's user avatar
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 …
Jonathan Chiche's user avatar
4 votes
1 answer
531 views

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 …
Jonathan Chiche's user avatar