The first obstructions to enhancing a triangulated category

TL;DR

This paper investigates the fundamental obstructions to enhancing triangulated categories with algebraic or topological structures, linking them to cohomology and providing criteria for various categorical enhancements.

Contribution

It introduces initial obstructions to enhancements, demonstrates their non-vanishing in certain cases, and characterizes enhancements via cohomology for different categorical frameworks.

Findings

Obstructions prevent enhancements in some triangulated categories.

Cohomological criteria characterize pre-triangulated DG, A-infinity, and spectral categories.

Explicit examples show obstructions do not always vanish.

Abstract

In this paper we relate triangulated category structures to the cohomology of small categories and define initial obstructions to the existence of an algebraic or topological enhancement. We show that these obstructions do not vanish in an example of triangulated category without models. We also obtain cohomological characterizations of pre-triangulated DG, A-infinity, and spectral categories.