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Results tagged with homotopy-theory
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user 49
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.
8
votes
How to get by with only functorial cylindrical objects?
You might be interested in looking into enriched model categories. If $\mathcal{V}$ is a monoidal model category (with cofibrant unit, for simplicity) and $\mathcal{C}$ is a $\mathcal{V}$-model categ …
- 66.8k
5
votes
Accepted
Fibrant replacement of an injective model category of enriched diagrams
Section 8 of my paper All (∞,1)-toposes have strict univalent universes shows that under fairly general conditions, injective fibrant replacements can be given by cobar constructions (e.g. the dual of …
- 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 …
- 5,094
10
votes
What is meant by a computational interpretation of univalence?
Roughly speaking, a type theory is computationally adequate if there is an algorithm that evaluates a term belonging to any type into a "normal form" of that type. The simplest form of this is when d …
- 66.8k
6
votes
Is the canonical model structure on strict $\infty$-Cat left proper?
The canonical model structure on 2-Cat is left proper. This is proven in Steve Lack's original paper A Quillen model structure for 2-categories that constructs this model category. The proof involve …
- 66.8k
4
votes
Accepted
Cohomology with local coefficients in homotopy type theory
(I suppose this is actually an answer, so I should post it as one.)
Yes, this generalization is described in the next blog post, since it's needed for the Serre spectral sequence. Make sure you read …
- 66.8k
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
7
votes
Generalized "Homology Whitehead" -- How much does stabilization remember?
Putting together my and Dan's comments deserves to be called an answer. Namely:
If $\mathcal{C}$ is an $(\infty,1)$-topos, then the statement is true when $\mathcal{D}$ is the class of hypercomplete …
- 66.8k
2
votes
Accepted
Categorical Significance of Fibrations
Yes. In roughly your language, the forgetful $(\infty,1)$-functor $\rm Fib\to Top$ is an $\infty$-fibration, where the fiber over a space $X$ is the category of fibrations over $X$, and descent in t …
- 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
7
votes
Accepted
simplicial objects in a model category
No, it is not.
If what you mean by $\rm colim_n$ is the actual colimit of $F$ as a diagram of shape $\Delta$, then this colimit is isomorphic to the coequalizer of the two maps $F([1]) \rightrightarr …
- 66.8k
1
vote
Accepted
Pullback-stable model of fibrewise suspension of fibrations (in simplicial sets, or similar ...
There is an answer to this question in the comments that works for simplicial sets, and more generally for projective model structures on simplicial presheaves. Since 3 years later it hasn't been rec …
- 66.8k
6
votes
Do combinatorial model categories and Quillen adjunctions model presentable $\infty$-categor...
I think it shouldn't be too hard to show using Dugger's technology that $N$ is full, i.e. essentially surjective on 1-morphisms. Suppose $C,D$ are combinatorial model categories and $f : N C \to N D$ …
- 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
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