Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with homotopy-theory
Search options questions only
not deleted
user 130058
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
0
answers
83
views
What is the group completion of the underlying multiplicative $\mathbb{E}_\infty$-monoid of ...
I recently noticed the following categorical/universal way to describe the passage from $\mathbb{Z}$ to $\mathbb{Q}$:
We start with the categroy $\mathsf{Sets}^{\mathrm{actv}}_*$ of pointed sets and …
- 11.8k
4
votes
0
answers
97
views
When do the different notions of homotopy inside a general simplicial set agree?
$\newcommand{\defeq}{\overset{\mathrm{def}}{=}}\newcommand{\rmR}{\mathrm{R}}\newcommand{\rmL}{\mathrm{L}}\newcommand{\End}{\mathrm{End}}\newcommand{\Hom}{\mathrm{Hom}}$This question is a sequel to my …
- 11.8k
8
votes
1
answer
350
views
Conservative cocompletion of categories of geometric shapes for homotopy theory
The recent paper
Calin Tataru, Partial orders are the free conservative cocompletion of total orders. arXiv:2404.12924
has shown that the conservative cocompletion of the simplex category $\Delta$ i …
- 11.8k
13
votes
0
answers
232
views
Isbell duality for simplicial sets
$\newcommand{\IsbellSpec}{\mathsf{Spec}}\newcommand{\IsbellO}{\mathsf{O}}\newcommand{\Nat}{\mathrm{Nat}}$Isbell duality sets up an adjunction (see here for a short abstract summary)
$$\mathsf{O}\dashv …
- 11.8k
5
votes
1
answer
458
views
Homotopy groups of categories of elements as higher colimits
Given a diagram of sets $D\colon\mathcal{C}\to\mathsf{Set}$, we have a bijection (Proof)
$$\operatorname{colim}(D) \cong \pi_0 (\textstyle\int_\mathcal{C}D).$$
Is there any known application or signi …
- 11.8k
5
votes
1
answer
222
views
Homotopy coherent localisation of a ring spectrum $E$ at a subset of $\pi_0E$
Homotopy coherent Invertibility.
Similarly to how $\mathbb{E}_k$-commutative spectra are a homotopy-coherent version of homotopy commutative spectra, encoding commutativity with higher homotopies, we …
- 11.8k
4
votes
1
answer
411
views
The “field of fractions” of the sphere spectrum (localization at $\pi_0(\mathbb{S})\setminus...
Perhaps the most common construction of the rational numbers is the one given by taking the field of fractions $\mathrm{Frac}(\mathbb{Z})\cong\mathbb{Q}$ of the ring $\mathbb{Z}$ of integers.
I'm wond …
- 11.8k
4
votes
0
answers
183
views
Valuations and (semi)norms on ring spectra
Valuations and seminorms on rings play a big role in number theory and analytic geometry, with seminorms being heavily used in Berkovich geometry and valuations featuring heavily in adic geometry.
Let …
- 11.8k
6
votes
0
answers
301
views
Looking for analogs of the rational, $p$-adic, and real numbers in homotopy theory via the p...
We can view the construction of the real or $p$-adic numbers as the end result of a series of constructions each of which adds more and more structure to the object at hand:
First, we start with the …
- 11.8k
1
vote
0
answers
80
views
Powersets of simplicial sets vs. powersets of topological spaces
Motivation. Recently I've been trying to understand how well- or ill-behaved are the many different powerset topologies one can put on the powerset of a topological space, and in particular I'm tryin …
- 11.8k
6
votes
1
answer
215
views
Homotopical properties of powersets of simplicial sets
Given a simplicial set $X_\bullet$, define its powerset simplicial set $\mathcal{P}_\bullet(X)$ as the composition
$$\Delta^\mathsf{op}\xrightarrow{X_\bullet}\mathsf{Set}\xrightarrow{\mathcal{P}}\math …
- 11.8k
6
votes
1
answer
253
views
Inexistence of a Kan–Quillen model structure on globular sets
(This is in a sense a follow-up to my earlier question on a geometric definition of globular $\infty$-groupoids)
We know by Scholie 8.4.14 of Cisinski's thesis that the globe category $\mathbb{G}$ is …
- 11.8k
6
votes
0
answers
252
views
A theory of higher limits of (1-)functors, after higher hochschild homology
$\newcommand{\Trans}{\mathrm{Trans}}\newcommand{\H}{\mathrm{H}}\newcommand{\HH}{\mathrm{HH}}\newcommand{\Hom}{\mathrm{Hom}}\newcommand{\Mod}{\mathrm{Mod}}$Recently I noticed that one may regard the co …
- 11.8k
7
votes
1
answer
332
views
Do the various notions of morphism spaces of simplicial sets agree on the underived level?
$\DeclareMathOperator\Ho{Ho}\DeclareMathOperator\Hom{Hom}$There are (at least) seven kinds of morphism spaces for a simplicial set $X$:
The left-pinched morphism space $\Hom^L_X(x,y)$,
The right-pinc …
- 11.8k
7
votes
1
answer
304
views
On morphism spaces of $(\infty,2)$-categories failing to be $\infty$-categories
As noted in Tag 01WZ of Kerodon, the morphism spaces of an $(\infty,2)$-category $C$ can fail to be $\infty$-categories, in contrast to the pinched morphism spaces of $C$.
This feels very counterintui …
- 11.8k