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Results tagged with homotopy-theory
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user 7721
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.
17
votes
The most general context of Mather's Cube Theorems
Let $\mathcal{X}$ be an $\infty$-category (i.e., a homotopy theory) which admits small homotopy colimits, a set of small generators, and has the property that homotopy colimits in $\mathcal{X}$ commut …
- 17.8k
17
votes
Strictly commutative elements of $E_\infty$-spaces
Here's something which may be of interest.
Let $E$ be the Lubin-Tate spectrum associated to a formal group of height $n$ over an algebraically closed field of characteristic $p > 0$, and let $X$ be $ …
- 17.8k
106
votes
Accepted
Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?
I don't think I have a compelling answer to this question, but maybe some bits and pieces that will be helpful. One point is that all of the examples that you bring up are related to the first: simpli …
- 17.8k
13
votes
Accepted
Etale cohomology of and forms of algebraic groups
For your first question: the two forms of $GL_2$ differ by modifying the Galois action by an automorphism of $GL_2$. The outer automorphism group of $GL_2$ is cyclic of order $2$: an automorphism is i …
- 17.8k
15
votes
Accepted
Delooping and unreduced operads
The answer to your first question is no: for example, take $X$ to be any connected pointed space, and regard $X$ as a nonunital commutative monoid by saying that the product of
any two points of $X$ i …
- 17.8k
18
votes
Accepted
Where to find the correct result in Higher Algebra, incorrect reference
The correct reference is 6.1.4.14. (And the hypothesis of 6.1.6.27 should refer to countable limits and colimits, rather than finite limits and colimits.)
- 17.8k
29
votes
What are simplicial topological spaces intuitively?
If $\mathcal{A}$ is an abelian category, then the Dold-Kan correspondence supplies an equivalence between the category of simplicial objects of $\mathcal{A}$ and the category of nonnegatively graded c …
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9
votes
Accepted
Homotopy limit-colimit diagrams in stable model categories
This is not true. For example, take $W = X = Y = 0$, and the map $Z \rightarrow V$ to be an isomorphism. Then you've got a homotopy colimit diagram, but it is only a homotopy limit diagram if $Z$ is w …
- 17.8k
15
votes
Accepted
Fibered/cofibered higher categories, relative model structures, slicing, and (∞,2)-category ...
It seems that the first question only makes sense for marked simplicial sets $X$ over $S$
where every edge of $X$ is marked (otherwise, the slice category is not equivalent to marked simplicial sets o …
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15
votes
Accepted
Is an ∞-topos of local homotopy dimension $\leq n$ of homotopy dimension $\leq n$?
Let $\mathcal{X}$ denote the $\infty$-topos $\mathcal{S}_{/S^1}$, whose objects are spaces $X$ with a map $X \rightarrow S^1$. Then $\mathcal{X}$ is generated under colimits by the object given by the …
- 17.8k
10
votes
Accepted
classifying $\infty$-toposes for topological/localic groups?
The projection map $p: \mathbf{R} \rightarrow \ast$ induces a fully faithful embedding of sheaf categories $p^{\ast}: Shv(\ast) \rightarrow Shv( \mathbf{R} )$. This is equally true for sheaves of sets …
- 17.8k
42
votes
What's the current state of the classification of not-fully-extended TQFTs?
When n > 1 the paper that you cite can give you a little bit of traction: the sketch proof of the main result gives a generators-and-relations presentation of (k,k+1,...,k+n)-Bord relative to (k,k+1)- …
- 17.8k
13
votes
Accepted
Which E_∞-spaces are homotopy colimits of k-truncated E_∞-spaces?
If you don't group complete, then free $E_{\infty}$-spaces are $1$-truncated.
Consequently, for $k > 0$, the answer is "all $E_{\infty}$-spaces". When $k=0$, you'll
get those which are homotopy equiva …
- 17.8k