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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.
3
votes
Accepted
In what generality does the following statement hold: A fibration is acyclic if and only if ...
Consider sSet×sSet with the induced structure from sSet. The monoidal unit is (1,1) (which is trivially fibrant, as is the unit in any cartesian monoidal model category), and so an object of the form …
- 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
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
6
votes
A model category of spaces where strict commutative monoids are $E_\infty$-spaces
You may also be interested in this paper.
- 66.8k
1
vote
The definition of Reedy category
If I'm not mistaken, here is an "even worse" counterexample than Charles'. Let $R$ be the walking isomorphism $(0\cong 1)$, let $R_+$ be $(0\to 1)$, and let $R_-$ be $(1\to 0)$. The conditions on de …
- 66.8k
7
votes
Accepted
Co/fibrant replacements via coend calculus
Addressing this sort of question was one of the main goals of math/0610194. The best answer I was able to give is that if $B$ has a suitable model structure with respect to which $F$ is objectwise fi …
- 66.8k
4
votes
Basic questions on the homotopy category
A very concrete example of a cospan having no pullback in the homotopy category, which does not require any knowledge of cohomology or Moore spaces, can be found here. It's phrased in terms of the ho …
- 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
7
votes
Accepted
How do various notions of natural transformation relate to various notions of homotopy in $2...
For strict transformations between strict 2-functors, you can just use the cartesian product $\Delta_1\times A$.
For lax transformations between strict 2-functors, this is what the lax version of the …
- 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 …
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
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
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
13
votes
Accepted
Are there types with nontrivial paths in all dimensions? (HoTT)
$\prod_{n\in\mathbb{N}} S^n$ certainly has nontrivial structure at all levels (i.e. "is not a homotopy $n$-type for any finite $n$"). In classical homotopy theory, even $S^2$ by itself has nontrivial …
- 66.8k
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