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In ordinary categories $\mathcal{C}$, there are nice conditions under which a generator is also a colimit-generator for $\mathcal{C}$. In other words, under suitable conditions, if $\mathcal{C}$ conta …

The category $Mod^{E_n}_A(\mathcal{C})$ of $E_n$-modules for an $E_n$-algebra in a symmetric monoidal $\infty$-category $\mathcal{C}$ is defined in Lurie's Higher Algebra as a special case of a more g …

Let $k$ be a field and let $\mathcal{C}=\mathbf{StLin}_k$ be the $\infty$-category of stable infinity categories enriched over the $\infty$-category $\mathbf{Vect}_k$, regarded as a symmetric monoidal …

Let $\mathcal{C}$ be an additive category and let $N_{dg}(\mathcal{C})$ be the differential graded nerve of the differential graded category $Ch(\mathcal{C})$. This is a stable $\infty$-category. Not …

In ordinary category theory, the localization $C[S^{-1}]$ at a class of morphisms $S$ (with possibly some assumptions on $S$) is a category $C[S^{-1}]$ together with a map $L:C \to C[S^{-1}]$ such tha …

A groupoid object in an $(\infty,1)$ category $\mathcal{C}$ is a functor $G:N(\Delta)^{op} \to \mathcal{C}$ such that for any partition $[n]=S \cup S'$ intersecting in $s$, the object $G([n]$ is the p …