[Paper Review] Examples and non-examples of integral categories and the admissible intersection property

Souheila Hassoun, Amit Shah+1 more|arXiv (Cornell University)|May 22, 2020

Homotopy and Cohomology in Algebraic Topology8 citations

TL;DR

This paper investigates integral categories and the admissible intersection property in functional analytic categories, proving that a pre-abelian category is quasi-abelian if and only if it admits admissible intersections. It establishes that integral categories are not necessarily quasi-abelian, provides non-examples in topological and bornological vector spaces, and shows that most such categories lack enough projectives or injectives.

ABSTRACT

Integral categories form a sub-class of pre-abelian categories whose systematic study was initiated by Rump in 2001. In the first part of this article we determine whether several categories of topological and bornological vector spaces are integral. Moreover, we establish that the class of integral categories is not contained in the class of quasi-abelian categories, and that there exist semi-abelian categories that are neither integral nor quasi-abelian. In the last part of the article we show that a category is quasi-abelian if and only if it has admissible intersections, in the sense considered recently by Br{\\"u}stle, Hassoun and Tattar. This exhibits that a rich class of non-abelian categories having this property arises naturally in functional analysis.

Motivation & Objective

• To determine whether key categories in functional analysis—such as Banach, Fréchet, and bornological spaces—are integral.
• To resolve the open question of whether integral categories are necessarily quasi-abelian, showing they are not.
• To investigate the relationship between semi-abelian, integral, and quasi-abelian categories, showing they are not closed under union.
• To characterize quasi-abelian categories via the admissible intersection property, establishing a new intrinsic criterion.
• To analyze the existence of enough projectives and injectives in various functional analytic categories, showing most lack them.

Proposed method

• Analyzes the stability of monomorphisms under pushout and epimorphisms under pullback to test integrality in categories of topological and bornological vector spaces.
• Applies Rump’s framework of integral categories and compares it with quasi-abelian and semi-abelian structures via categorical diagrams and axiomatic definitions.
• Uses pullback diagrams of admissible monomorphisms to define and verify the admissible intersection property in pre-abelian categories.
• Applies results from Brüstle, Hassoun, and Tattar on exact structures with admissible intersections to prove a biconditional characterization.
• Employs duality and kernel-cokernel pair analysis to show that admissible intersections imply quasi-abelian structure.
• Leverages known results on projectivity and injectivity in categories like $\mathsf{BAN}$, $\mathsf{FRE}$, and $\mathsf{LCS}$ to derive non-existence theorems.

Experimental results

Research questions

• RQ1 Are standard categories of topological and bornological vector spaces, such as $\mathsf{BAN}$, $\mathsf{FRE}$, and $\mathsf{BOR}$, integral categories?
• RQ2 Is every integral category necessarily quasi-abelian, or do non-quasi-abelian integral categories exist?
• RQ3 Can a semi-abelian category fail to be either integral or quasi-abelian, and if so, what are the structural implications?
• RQ4 Is the admissible intersection property equivalent to quasi-abelian structure in pre-abelian categories?
• RQ5 Do most categories in functional analysis, such as $\mathsf{TVS}$, $\mathsf{LCS}$, and $\mathsf{BOR}$, have enough projectives or injectives?

Key findings

• The category $\mathsf{BAN}$ of Banach spaces is not integral, as monomorphisms are not stable under pushout.
• There exist integral categories that are not quasi-abelian, resolving an open question in the negative.
• Semi-abelian categories are not contained in the union of integral and quasi-abelian categories, as shown by a counterexample using Wengenroth’s construction.
• A pre-abelian category is quasi-abelian if and only if it has admissible intersections, establishing a new intrinsic characterization.
• Most categories studied—such as $\mathsf{TVS}$, $\mathsf{LCS}$, $\mathsf{BOR}$, and $\mathsf{HD-TVS}$—do not have enough projectives or injectives.
• The class of quasi-projective objects in $\mathsf{BOR}$ and $\mathsf{BAN} \times \mathsf{BOR}$ is insufficient to form a generating class, implying lack of enough quasi-projectives.