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Part of higher category theory that for instance in Algebraic Topology enables us to capture finer homotopic distinctions. As in say Eilenberg-Maclane spaces.
3
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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 …
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 …
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 …
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)
$$\mathsf{O}\dashv …
5
votes
1
answer
150
views
Co/limits and 2-co/limits of categories in the $\infty$- and $(\infty,2)$-category of $\inft...
Recently, in a conversation with Gabriel, the following question came up:
Question. Do co/limits of categories taken in the $\infty$-category of $\infty$-categories agree with the usual co/limits tak …
5
votes
1
answer
458
views
Homotopy groups of categories of elements as higher colimits
Given a diagram of sets $D\colon\mathcal{C}\to\mathsf{Set}$, we have a bijection (Proof)
$$\operatorname{colim}(D) \cong \pi_0 (\textstyle\int_\mathcal{C}D).$$
Is there any known application or signi …
5
votes
1
answer
222
views
Homotopy coherent localisation of a ring spectrum $E$ at a subset of $\pi_0E$
Homotopy coherent Invertibility.
Similarly to how $\mathbb{E}_k$-commutative spectra are a homotopy-coherent version of homotopy commutative spectra, encoding commutativity with higher homotopies, we …
4
votes
1
answer
411
views
The “field of fractions” of the sphere spectrum (localization at $\pi_0(\mathbb{S})\setminus...
Perhaps the most common construction of the rational numbers is the one given by taking the field of fractions $\mathrm{Frac}(\mathbb{Z})\cong\mathbb{Q}$ of the ring $\mathbb{Z}$ of integers.
I'm wond …
6
votes
1
answer
215
views
Homotopical properties of powersets of simplicial sets
Given a simplicial set $X_\bullet$, define its powerset simplicial set $\mathcal{P}_\bullet(X)$ as the composition
$$\Delta^\mathsf{op}\xrightarrow{X_\bullet}\mathsf{Set}\xrightarrow{\mathcal{P}}\math …
9
votes
0
answers
246
views
Applications of the simplex $2$-category and its higher dimensional cousins
The simplex category $\Delta$ has a $2$-category refinement $\Delta_2$ given by the full sub-$2$-category of the $2$-category $\mathsf{Cat}$ spanned by the ordinal categories $𝟘$, $𝟙$, $𝟚:=𝟙\star� …
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 …
9
votes
1
answer
331
views
Is there a "geometric definition" of globular $\infty$-groupoids/categories?
The nLab page on $\infty$-categories splits the known definitions of $\infty$-categories into two types:
Algebraic $\infty$-categories, in which composition is expressed "externally", e.g. as a some …
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 …
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 …
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 …