Recollements in stable $\infty$-categories

TL;DR

This paper extends the theory of recollements to stable -categories, generalizing classical results and exploring their properties in a higher categorical setting, with applications to stratified spaces and gluing of t-structures.

Contribution

It develops a comprehensive framework for recollements in stable -categories, including new properties of associated t-structures and their applications to stratified geometric contexts.

Findings

Recollements induce a -categorical t-structure up on .

The properties of factorization systems are analyzed in the  setting.

A generalized associative property for n-fold gluing of t-structures is established.

Abstract

We develop the theory of recollements in a stable $\infty$-categorical setting. In the axiomatization of Beilinson, Bernstein and Deligne, recollement situations provide a generalization of Grothendieck's "six functors" between derived categories. The adjointness relations between functors in a recollement $\mathbf{D}^0\leftrightarrow \mathbf{D} \leftrightarrow \mathbf{D}^1$ induce a "recoll\'ee" $t$-structure $\mathfrak{t}_0\uplus\mathfrak{t}1$ on $\mathbf{D}$ , given $t$-structures $\mathfrak{t}_0,\mathfrak{t}_1$ on $\mathbf{D}^0, \mathbf{D}^1$. Such a classical result, well-known in the setting of triangulated categories, is recasted in the setting of stable $\infty$-categories and the properties of the associated ($\infty$-categorical) factorization systems are investigated. In the geometric case of a stratified space, various recollements arise, which "interact well" with the combinatorics of the intersections of strata to give a well-defined, associative $\uplus$ operation. From this we deduce a generalized associative property for $n$-fold gluing $\mathfrak{t}_0\uplus\cdots\uplus \mathfrak{t}_n$, valid in any stable $\infty$-category.