Generalised Moore spectra in a triangulated category

TL;DR

This paper introduces a generalized construction in triangulated categories that extends Moore spectra concepts, enabling new embeddings of module categories and connecting to cluster categories.

Contribution

It develops a new functorial approach generalizing Moore spectra in triangulated categories, extending Jorgensen's work, and providing a novel embedding of module categories.

Findings

The functor is compatible with short exact sequences and triangles.

Recovers Keller's embedding of path algebra module categories into u-cluster categories.

Provides a new perspective on approximating objects in triangulated categories.

Abstract

In this paper we consider a construction in an arbitrary triangulated category T which resembles the notion of a Moore spectrum in algebraic topology. Namely, given a compact object C of T satisfying some finite tilting assumptions, we obtain a functor which "approximates" objects of the module category of the endomorphism algebra of C in T. This generalises and extends a construction of Jorgensen in connection with lifts of certain homological functors of derived categories. We show that this new functor is well-behaved with respect to short exact sequences and distinguished triangles, and as a consequence we obtain a new way of embedding the module category in a triangulated category. As an example of the theory, we recover Keller's canonical embedding of the module category of a path algebra of a quiver with no oriented cycles into its u-cluster category for u>1.