A brief introduction to 6-functor formalisms
Conventions. Category means $(\infty,1)$-category; $\mathsf{Cat}$ is the category of all categories. $2$-category means $(\infty,2)$-category; $\mathsf{Cat}_2$ is the $2$-category of all categories. $\mathsf{Pr}^L$ denotes presentable categories with left adjoints; $\mathsf{Pr}^L_{\mathrm{st}}$ is the stable version. $\mathsf{CAlg}(\mathcal M)$ denotes commutative algebra objects in a symmetric monoidal category $\mathcal M$. Given a geometric setup $(\mathcal C, E)$ — a category $\mathcal C$ with finite limits and a class of morphisms $E$ closed under base change, composition and diagonals — we write $\mathsf{Corr}(\mathcal C, E)$ for the span $(\infty, 2)$-category and $\mathsf{Span}_2(\mathcal C, E)_{I, P}$ for the CLL upgrade with biadjointability data. $\underline{\mathrm{Hom}}$ is internal hom; $\mathbf 1_{X}$ is the monoidal unit of $\mathsf D(X)$; $\mathbb 1$ is the terminal object of a slice $\mathcal C_{/Y}$. Part I · Six-functor formalisms 1. From cohomology to six functors A cohomology theory associates with every “space” $X$ a complex $\Gamma(X; \Lambda) \in \mathsf D(\Lambda)$ of $\Lambda$-modules — singular cohomology, étale cohomology, coherent cohomology, and many more. We then expect $\Gamma(X; \Lambda)$ to satisfy structural identities like the Künneth formula, Poincaré duality, proper base change, excision. Each is classically proved by hand, with most of the work intertwined with whatever specific sheaf-theoretic gadget produced $\Gamma(X; \Lambda)$. ...