Comparing $\mathrm{Add}(M)$ with $\mathrm{Prod}(M)$

TL;DR

The paper studies when Add$(M)$ is contained in Prod$(M)$ and vice versa in locally finitely presented categories and, analogously, in compactly generated triangulated categories, using a generalized Chase-type lemma. It proves that Add$(M)\subseteq\text{Prod}(M)$ is equivalent to $M$ being $\Sigma$-pure-injective under the assumption that $\mathrm{Hom}(X,M)$ is not $\omega$-measurable for finitely presented $X$, and it characterizes Prod$(M)\subseteq$Add$(M)$ as the property that $M$ is product-complete (equivalently, Add$(M)$ is (pre)enveloping). The results extend to the triangulated setting, yielding triangulated analogues of Enochs’ conjecture for Prod$(M)$ and triangulated versions of Krause/Angeleri–Hügel type descriptions, with a Chase-type lemma as a key technical tool. Set-theoretic assumptions about $\omega$-measurable cardinals influence the strength of the equivalences and the existence of (pre)covering properties. Overall, the work unifies purity, definability, and approximation properties to provide sharp criteria for when direct-sum and direct-product constructions of a single object align in two broad categorical contexts.

Abstract

We present characterizations for the inclusions $\mathrm{Add}(M)\subseteq \mathrm{Prod}(M)$ and $\mathrm{Prod}(M)\subseteq \mathrm{Add}(M)$ in locally finitely presented categories and in compactly generated triangulated categories. As applications, we describe the situations when the classes of the form $\mathrm{Prod}(M)$ and $\mathrm{Add}(M)$ are (pre)covering, respectively (pre)enveloping.

Theorems & Definitions (48)

• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Remark 2.5
• Proposition 2.6
• proof
• Lemma 3.1