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Let $\mathcal{A}$ be an additive category, and a morphism $e: X \rightarrow X$ in it an idempotent, i.e. $e \circ e = e$. We say that $e$ splits, if we can decompose $X = \operatorname{ker}(e) \oplus …

Regard the following two bicategories: $\operatorname{dg-\mathcal{B}imod}$, with objects dg categories, and morphisms categories from $C$ to $D$ being the categories of $C$-$D$-bimodules. Composition …

I have a question concerning the proof of Corollary 7.3.6.5 in Luries "Higher Topos Theory" (the same issue also occurs in the proof of 7.3.6.10, but it is clearer here). Given is a continuous map $p: …

In Higher Algebra Lemma 4.8.4.2, Lurie shows that for $\mathcal{K}$ a small set of simplicial sets, the $\infty$-category $\mathcal{C}at(\mathcal{K})$ of small $\infty$-categories with $K$-shaped coli …

Note that by HTT 5.4.1.2 since $\tau$ is an uncountable regular cardinal, an $\infty$-category is $\tau$-compact iff it is $\tau$-small. Our first step is to show that the inclusion $\mathcal{C}at(\ma …

Let $\operatorname{CAlg}$ be the category of commutative rings (with unit) and $\operatorname{S-CAlg}$ the category of supercommutative $\mathbb{Z}/2$-graded rings. Then we have an adjoint triple (as …