Gluing simple-minded collections in triangulated categories

TL;DR

This paper introduces a new technique for combining simple-minded collections in triangulated categories, ensuring compatibility with existing structures and preserving key properties like order and mutation.

Contribution

It develops a novel gluing method for simple-minded collections that aligns with known techniques for $t$-structures, silting objects, and co-$t$-structures, enhancing the understanding of their interplay.

Findings

The gluing technique is compatible with existing structures.

It preserves partial order among collections.

It commutes with mutation operations.

Abstract

We provide a technique to glue simple-minded collections along a recollement of Hom-finite Krull-Schmidt triangulated categories over a field. This gluing technique for simple-minded collections is shown to be compatible with those for gluing bounded $t$-structures, silting objects, and co-$t$-structures in the literature. Furthermore, it also enjoys the properties of preserving partial order and commuting with the operation of mutation.