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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.
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.
- 15.7k
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 $C\to An …
- 15.7k
7
votes
Bar construction in commutative algebras is calculated by pushout
A way to see this which doesn't dive into the specifics of the simplicial diagram "$C\otimes D^{\otimes n}\otimes E$" is to apply 3.2.4.7 to the symmetric monoidal $\infty$-category $Mod_D(\mathcal C) …
- 15.7k
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 …
- 15.7k
5
votes
Accepted
The $E$-(co)homology of $\mathrm{BGL}(R)^+$ and the algebraic $K$-theory of $R$
Let $R$ be a ring. $BGL(R)^+$ is homotopy equivalent to the $0$ component of $\Omega^\infty K(R)$, and it is stably equivalent to $BGL(R)$.
In particular, for a (co)homology theory $E$, understanding …
- 15.7k
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 …
- 15.7k
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 …
- 15.7k
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 …
- 15.7k
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 …
- 15.7k
4
votes
Accepted
Monomorphisms of diagrams in an $\infty$-category
For completeness, and because I cannot figure out the general case (cf. my comment below Daniel's answer), let me prove the following: if $f:\Delta\to C$ is a functor which, when restricted to $\Delta …
- 15.7k
3
votes
Accepted
When is an $\infty$-categorical localization of an additive 1-category enriched in topologic...
The answer is indeed always.
The fact that $\mathcal A$ is an additive 1-category makes it canonically a module over $Proj_\mathbb Z$, the 1-category of finitely generated projective $\mathbb Z$-modul …
- 15.7k
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 $\infty$- …
- 15.7k
5
votes
Accepted
Homotopy coherent localisation of a ring spectrum $E$ at a subset of $\pi_0E$
The answer is yes - more generally, "coherent invertibility" is just invertibility, which is what makes conditions of the form "such and such things are invertible" extremely practical.
A more precise …
- 15.7k
5
votes
Is the concept of an $H$ object still interesting, when we have the $\infty$-version of it?
This is not an answer, but slightly too long for a comment.
The main thing I wanted to say is that there is no "correct" or "incorrect" formalization, they just serve different purposes.
For instance …
- 15.7k
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 …
- 15.7k