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Results tagged with homotopy-theory
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user 102343
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.
22
votes
Accepted
On the connections between condensed mathematics and homotopy theory
The way in which "condensed sets are similar to topological spaces" is very different from the way in which "$\infty$-groupoids are similar to topological spaces". In fact, condensed mathematics is, i …
- 15.9k
13
votes
Accepted
Homotopy groups of categories of elements as higher colimits
To answer these questions, the best is to note that $|\int_C D|$, the geometric realization of this total category, is equivalently the colimit of $D$, viewed as a functor with values in the $\infty$- …
- 15.9k
13
votes
Accepted
Homotopy coherent colimits in chain complexes
The result is not only true for simplicial model categories, but for plain combinatorial model categories too - this is Higher Algebra 1.3.4.25..
In fact, for this you can reduce to the case of simpli …
- 15.9k
12
votes
Multiplicative Structures on Moore Spectra
I randomly ran into this old question - the state of the art is now the following paper of Robert Burklund where he proves that many Moore spectra do, in fact, have $A_\infty$ -structures. For example …
- 15.9k
9
votes
Accepted
Group completion of $\mathbb{E}_{\infty}$-monoids via tensor products
Yes, for the same reason. Let me sketch a proof.
1- $QS^0\otimes X$ is group-complete. Indeed, its $\pi_0$ is $\mathbb Z\otimes \pi_0(X)$, and that's a group for the usual reasons. Another way to prov …
- 15.9k
9
votes
Accepted
Is there a Dold-Kan theorem for circle actions?
No, they are not equivalent, even for $C = Sp$.
Indeed, the category of spectra with $S^1$-action is also the category of $\mathbb S[S^1]$-modules, and is compactly generated by a single object.
On th …
- 15.9k
8
votes
Homotopy coherent generalization of classifying space theory
I think it is worth expliciting skd's answer.
There is a chain of equivalences $$\mathcal S_{/BG} \simeq Fun(BG, \mathcal S) \simeq Mod_G(\mathcal S)$$ each of which is, at an informal level, easy to …
- 15.9k
8
votes
Accepted
Two definitions of a monad on an ∞-category
This paper (and specifically Section 8 thereof) by Rune Haugseng essentially fully answers the question, proving that the two notions are indeed equivalent.
- 15.9k
7
votes
How to construct $X \oplus \Sigma X$ from $X \oplus \Sigma X \oplus \Sigma X \oplus \Sigma^2...
Whenever $P$ is a summand of $Q$, you can construct $P\oplus\Sigma P$ in one step from $Q$: if $e$ is the idempotent that projects onto $P$, then the cofiber of $1-e$ is $P\oplus\Sigma P$.
You can app …
- 15.9k
7
votes
Accepted
Are morphisms in a stable $\infty$-category generated by split injections?
Any map $f:A\to B$ fits in a cofiber sequence of arrows $(0\to A)\to (A\to A\oplus B)\to (A\to B)$
In other words, any map is a cofiber of split inclusions. But now cofibers (as any colimit, by the Bo …
- 15.9k
7
votes
Bar construction in commutative algebras is calculated by pushout
A way to see this which doesn't dive into the specifics of the simplicial diagram "$C\otimes D^{\otimes n}\otimes E$" is to apply 3.2.4.7 to the symmetric monoidal $\infty$-category $Mod_D(\mathcal C) …
- 15.9k
6
votes
Accepted
Groupoidification of infinity categories and geometric realization
Yes, they are equivalent, and this is why people sometimes use $|C|$ to denote $Str(C)$.
Consider the following composite $Fun(\Delta^{op},\mathrm{Grpd}) \to Fun^{cpl, Segal}(\Delta^{op}, \mathrm{Grpd …
- 15.9k
6
votes
Accepted
Mapping spaces in complete Segal spaces and quasi-categories
This might not be what you want, but you can go the other way around: to a quasicategory $C$ you can associate a Segal space via $NC: [n]\mapsto Fun(\Delta^n, C)^\simeq$, by which I mean the largest s …
- 15.9k
6
votes
Accepted
A space $X$ is $k$-truncated iff $\text{Fun}(S,X)$ is $k$-truncated
Let $S^n := \Delta^n/\partial \Delta^n$; and let me assume for simplicity that $X$ is connected.
We have a (homotopy) fiber sequence $\Omega^n X \to X^{S^n} \to X$.
In particular, for $n>k$, $\Omega^n …
- 15.9k
6
votes
Is the Grothendieck construction a homotopy pullback?
The analogy is of course a correct/useful analogy, but I think any model structure for which the literal statement is correct must have $Set_* \simeq *$, so it would be a bit too coarse to do anything …
- 15.9k