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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.
5
votes
Base change of schemes preserves $\mathbb{A}^1$-contractibility
Yes.
(EDIT : Brian Shin pointed out to me that I absolutely did not need smoothness for that argument - I got confused in thinking it was needed to define $Sm/k \to Sm/L$ but of course a pullback of a …
- 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
Left Kan extension and finite product preserving
Yes, but this is a completely general phenomenon unrelated to animated rings and sheaves. The general (surprising!) phenomenon is that the left Kan extension of any product preserving functor $C\to An …
- 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
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
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
1
vote
Function space and contractibility
If there exists a homotopy equivalence $f: X\to Y$ and every other $h$ is homotopic to $f$, then $f$ is homotopic to a constant map, so that $X,Y$ are contractible, and therefore so is $map(X,Y)$.
Exc …
- 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
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
3
votes
Accepted
Linearity of topological periodic cyclic homology
If you want a full module structure (rather than just an "action map" $TP(A)\otimes TP(B)\to TP(B)$, which is enough for some arguments), the reasonable notion would be for $B$ to be an $A$-module in …
- 15.9k
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
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
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
4
votes
Accepted
Computing homotopy colimit of a space with free $S^1$-action
Note that in their context, $C$ has an action of $B\mathbb Z$, not of $\mathbb Z$ ! (Otherwise $C/B\mathbb Z$ wouldn't make sense)
This amounts essentially to a self natural transformation of the iden …
- 15.9k
4
votes
Accepted
Equivariant colimit and equivariant functors
Q1: For any $C,D\in Fun(BG,Cat_\infty)$, $Fun(C,D)$ acquires a $G$-action too. Informally, this is described as $F\mapsto gF(g^{-1}-)$, and this is in fact an accurate description if $G$ is a discrete …
- 15.9k