Recollements in stable $\mathrm{\infty }$-categories

We develop the theory of recollements in a stable $\mathrm{\infty }$-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 $\mathrm{\infty }$-categories and the properties of the associated ($\mathrm{\infty }$-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 $\mathrm{\infty }$-category.