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Results tagged with homotopy-theory
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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.
109
votes
What properties make $[0,1]$ a good candidate for defining fundamental groups?
Rather than answering the question, I want to claim that it's the wrong question.
There is a way to define fundamental group(oid)s and other homotopical notions that makes no reference to the unit in …
- 66.8k
32
votes
Homotopy Type Theory: What is it?
That description of the three directions is not too bad, although they are not of course completely separate, nor does everything called HoTT fall under one of them. Also, I'm not sure whether this a …
- 66.8k
29
votes
Defining $SU(n)$ in HoTT
I think Noah's answer is mostly right, but partly misleading, and explaining why will take too much space for a comment, so I'm posting a separate answer.
As Noah says, the main conceptual point is t …
- 66.8k
29
votes
Why is the definition of the higher homotopy groups the "right one"?
I think there's something fundamental missing from all the other answers so far: the modern realization that topological spaces are distinct from $\infty$-groupoids.
Suppose you didn't know about hig …
26
votes
Accepted
Grothendieck derivators vs $\infty$-categories
The short answer is that $(\infty,1)$-categories are the "real" object of interest. Derivators are a tool for working with them that is sometimes (for some people) easier to use, but doesn't remember …
- 66.8k
25
votes
Uniqueness of loop spaces
As Ryan points out, if Y is allowed to be disconnected, then there is no hope, since the loop-space construction sees only the connected component of the basepoint. But even if Y is assumed to be con …
- 66.8k
24
votes
Accepted
Deligne's doubt about Voevodsky's Univalent Foundations
It is a bit difficult to understand what he is asking. The already-linked nForum discussion includes some clarification about his example, which at the meeting took us a while to figure out.
More br …
- 66.8k
22
votes
Accepted
Constructive homological algebra in HoTT
As regards HoTT, my own current opinion is that the best way to do "homological algebra" therein is by working directly with spectra.
With only a working mathematician's knowledge of homological alge …
- 66.8k
21
votes
Role of univalence in homotopy group calculations
"Isomorphic structures are equal" is a cute slogan, but it sometimes gets in the way because it sounds like it's saying that it forces isomorphic structures to be related by the pre-existing notion of …
- 66.8k
17
votes
What determines a model structure?
Mark's answer explains why (1) cofibrations and fibrations do determine the model structure, and Charles' and Tom's examples show that (4) cofibrant objects and fibrant objects do not.
For (2), any w …
- 66.8k
16
votes
Can Homotopy Type Theory or algebraic geometry deal with homotopy fibers in terms of families?
I am answering your "later addon" only, although it seems actually to be a very different question than your original one.
This is perhaps one of the most misunderstood aspects of HoTT and particular …
- 66.8k
16
votes
What are finite homotopy types?
I believe the first two notions coincide, but they definitely don't coincide with the third: as was pointed out in a deleted answer, the nerve of any finite group is a (3) but certainly not a (1) or ( …
- 66.8k
16
votes
Accepted
Definition of homotopy limits
Reid's answer is quite right, but long before "quasicategories" became fashionable, algebraic topologists were doing exactly the same thing using the "simplicial bar construction" and plain old topolo …
- 66.8k
16
votes
Accepted
(∞, 1)-categorical description of equivariant homotopy theory
I think a good reference for the first paragraph is "Equivariant Homotopy and Cohomology Theory" by Peter May and a bunch of other people. Chapter 5 includes "Elmendorf's theorem" that this homotopy …
- 66.8k
16
votes
How should I think about delooping?
I'm not sure whether you'll like this, but my natural response to "how should I think about delooping?" is to invoke (higher) category theory. You may know that a homotopy 1-type, i.e. a space (proba …
- 66.8k