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It's possible to view nonconnectivity for spectra as arising from enlarging Segal's category $\Gamma^\mathsf{op}\overset{\mathrm{def}}{=}\mathsf{Sk}(\mathrm{FinSets}_*)$ to the $\infty$-category of fi …

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 …

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 …

Lately there has been a lot of progress on the foundations of $(\infty,2)$-categories (for example, all currently-known models for them were shown to be equivalent and finally we have a construction o …

$\newcommand{\op}{\mathrm{op}}\newcommand{\Un}{\mathrm{Un}}$Lurie's relative nerve functor $\mathrm{N}^{F}_{\bullet}$ is a simpler version of the unstraightening functor $$\Un_\phi:\mathrm{sPSh}(\math …

A functor $N\colon\mathrm{Cat}_{A_\infty}\longrightarrow\mathrm{Cat}_\infty$ is constructed in a paper [1] by Faonte. This gives a way to get an $\infty$-category by starting with an $A_\infty$-categ …

For ordinary categories, the assignment $\mathcal{C}\mapsto\mathsf{Mon}(\mathcal{C})$ defines a functor $\mathsf{Mon}\colon\mathsf{Alg}_{\mathbb{E}_{k}}(\mathsf{Cats})\to\mathsf{Alg}_{\mathbb{E}_{k-1} …

It is well-known that when passing to $\infty$-categories the notion of commutativity gets replaced by an infinite array of notions of commutativity: $\mathbb{E}_{1}$, $\mathbb{E}_{2}$, ..., $\mathbb …

In Coalgebras in symmetric monoidal categories of spectra, Péroux and Shipley prove the following (Lemma 2.4): Let $\mathcal{C}=\mathsf{Sets},\mathsf{Top}$, or $\mathsf{sSets}$. The free basepoint fu …

The group $\mathbb{Z}/2$ corepresents the functor $\mathrm{Inv}\colon\mathsf{Mon}\to\mathsf{Sets}$ sending a monoid $A$ to its set of involutory elements (those satisfying $a^2=1_A$). A similar story …

$\newcommand{\unsim}{{\sim}}$The set of conjugacy classes of a group $G$ is the quotient of $G$ by the equivalence relation $\sim_1$ obtained by declaring $a\sim_1b$ if there exists some $g\in G$ such …

This is the one categorical level higher version of the question Delooping monoidal $\infty$-groupoids into $\infty$-categories. The classical, bicategorical, setting. Given a monoidal category $(\mat …

There are several notions of nerves, including nerves of categories, $2$-categories, and simplicial categories. These define functors \begin{align*} \mathrm{N} &\colon \mathrm{Cats}_{(2,1)} …

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� …

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 …