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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 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 …
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 …

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