Presentable Category

Definition Definition 1. Let $\kappa$ be a regular cardinal (for example $\kappa = \omega$ or $\kappa = \aleph_1$). A category $\mathcal{I}$ is called $\kappa$-filtered if for any $\kappa$-small category $\mathcal{K}$ and any functor $F\colon \mathcal{K} \to \mathcal{I}$, there exists an extension $F^{\rhd} \colon \mathcal{K}^{\rhd} \to \mathcal{I}$. When $\kappa = \omega$, we simply call a $\kappa$-filtered category a filtered category. Let $\mathcal{C}$ be a category admitting small colimits. An object $X \in \mathcal{C}$ is called $\kappa$-compact if for any $\kappa$-filtered diagram $(Y_i)_{i \in \mathcal{I}}$, there is an isomorphism of mapping spaces ...

What is \( n\mathsf{Pr} \) Recall that in the previous note we reviewed the basic theory of presentable categories. In this section, we introduce the notion of presentable \(n\)-categories. Let \(\mathcal{C} \in \mathsf{CAlg}(\mathsf{Pr}^L)\) be a commutative algebra object in \(\mathsf{Pr}^L\) with respect to the Lurie tensor product, and let \(A \in \mathsf{Alg}(\mathcal{C})\) be a commutative algebra object. We may then consider the category of \(A\)-modules \[ \mathsf{Mod}_A(\mathcal{C}). \]It is well known that \(\mathsf{Mod}_A(\mathcal{C})\) carries a natural symmetric monoidal structure, defined via the colimit of the bar construction. More precisely, for \(A\)-modules \(M\) and \(N\), consider the simplicial object ...

Stefanich Rings In the last note, we define \[ n\mathsf{Pr} \coloneqq \mathsf{Mod}_{(n-1)\mathsf{Pr}}(1\mathsf{Pr}), \qquad 1\mathsf{Pr} \coloneqq \mathsf{Pr}_{\aleph_1}^{L}. \] Now, we let $0\mathsf{Pr} \coloneqq \mathsf{An}$. Since we have \[ \mathsf{CAlg}(\mathcal{C}) \hookrightarrow \mathsf{CAlg}(\mathsf{Mod}_{\mathcal{C}}(\mathsf{Pr}_{\aleph_1}^L)), \quad A \mapsto \mathsf{Mod}_A(\mathcal{C}), \] and $1\mathsf{Pr} \in \mathsf{Pr}_{\aleph_1}^L$, we obtain a sequence \[ \mathsf{CAlg}(0\mathsf{Pr}) \hookrightarrow \mathsf{CAlg}(1\mathsf{Pr}) \hookrightarrow \cdots \hookrightarrow \mathsf{CAlg}(n\mathsf{Pr}) \hookrightarrow \cdots . \]In Presentable Categories, we know that $1\mathsf{Pr}$ admits all small colimits, which can be computed in $\widehat{\mathsf{Cat}}$ by passing to adjoint functors. Thus, we obtain the following definition. ...

Six-Functor Formalisms on \(n\mathsf{Pr}_{(-)}\) Let $\mathsf{CAlg}$ be the category of derived rings, and let $\mathsf{Aff} \coloneqq \mathsf{CAlg}^{\operatorname{op}}$ be the category of affine schemes. Quasicoherent Sheaves of Higher Categories