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In section 19.2.6 of Lurie's "Spectral Algebraic Geometry," he constructs the smooth projective space, which represents the derived version of [the dual of] the usual functor of points interpretation …

Let $\mathsf{A},\mathsf{B},$ and $\mathsf{C}$ be (ordinary) categories and $F : \mathsf{A}\to\mathsf{C}$ and $G : \mathsf{B}\to\mathsf{C}$ be functors such that $\mathsf{A}$ and $\mathsf{B}$ are comp …

Let $k$ be a commutative ring with $1$, and let $\mathsf{dgCat}_k$ be the category of $k$-linear dg-categories, as defined in [1, Section 2]. We may equip $\mathsf{dgCat}_k$ with the Morita model stru …

Conventions: In the below, unless otherwise stated, terms regarding derived algebraic geometry will follow the conventions of Yaylali. an algebraic stack will be a stack $\mathscr{S}$ over a base sch …

Suppose $k$ is a fixed commutative ring, and let $\mathsf{dgCat}_k$ denote the category of $k$-linear dg-categories. We will equip $\mathsf{dgCat}_k$ with the Morita model structure (see theorem 2.27 …

Throughout the literature, one can find many definitions of the Hochschild homology of various objects. However, the precise relationship between these definitions is not always so clear, at least to …

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{ …

This question is a follow-up to a previous question I asked. If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h …

I learned the following partial answer from Peter Haine (any errors are of course my own). In the following I will ignore any set-theoretic issues which may arise. Let $\mathsf{A}$ be an abelian cate …

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 …