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When working with homotopy coherent diagrams in an $(\infty,1)$-category $\mathcal{C}$ (viewing $(\infty,1)$-categories as quasi-categories), we can make sense of them as objects in $\operatorname{Map …

I think the answer I want is given by Johnson-Freyd and Scheimbauer's paper "(Op)lax natural transformations, twisted quantum field theories, and 'even higher' Morita categories". Here is a summary fo …

Let $B\leftarrow A\to C$ be a diagram of commutative rings, and let $\mathcal{D}(A)$ be the derived $\infty$-category of $A$-modules (as in Lurie's "Higher Algebra"). Then is there an equivalence $$\m …

Suppose we are given categories $\mathsf{C},\mathsf{D},\mathsf{E},$ equipped with collections of weak equivalences $\mathcal{W}_{\mathsf{C}},\mathcal{W}_{\mathsf{D}},$ and $\mathcal{W}_{\mathsf{E}},$ …

As Harrison notes in the comments, we may define $\mathcal{D}_{\mathsf{B}}(\mathsf{A})$ as the full subcategory of $\mathcal{D}(\mathsf{A})$ consisting of objects $X$ such that $\pi_0(X[n])\in\mathsf{ …

In this question, it is asked why we like to consider $\mathsf{D}_\textrm{qc}(X)$ rather than $\mathsf{D}(\mathsf{QCoh}(X)).$ Professor Cisinski answers rather convincingly that the $\infty$-categoric …