[Paper Review] Algebraically coherent categories

Alan S. Cigoli, James Richard Andrew Gray+1 more|arXiv (Cornell University)|Sep 15, 2014

Homotopy and Cohomology in Algebraic TopologyMathematics19 citations

TL;DR

This paper introduces the concept of algebraically coherent categories—finitely complete categories where change-of-base functors along the fibration of points preserve finite limits and jointly strongly epimorphic pairs. The key contribution is establishing that such categories satisfy strong structural properties in semi-abelian contexts, including the Smith is Huq condition, normality of Higgins commutators, and fibre-wise algebraic cartesian closedness, with broad examples including categories of interest and compact Hausdorff algebras.

ABSTRACT

We call a finitely complete category algebraically coherent when the change-of-base functors of its fibration of points are coherent, which means that they preserve finite limits and jointly strongly epimorphic pairs of arrows. We give examples of categories satisfying this condition; for instance, coherent categories, categories of interest in the sense of Orzech, and (compact) Hausdorff algebras over a semi-abelian algebraically coherent theory. We study equivalent conditions in the context of semi-abelian categories, as well as some of its consequences: including amongst others, strong protomodularity, and normality of Higgins commutators for normal subobjects, and in the varietal case, fibre-wise algebraic cartesian closedness.

Motivation & Objective

• To define and investigate a new categorical property—algebraic coherence—extending the notion of coherent categories to the fibration of points.
• To establish that algebraic coherence implies key structural properties in semi-abelian categories, such as the Smith is Huq condition and normality of Higgins commutators.
• To provide a broad class of examples, including categories of interest and compact Hausdorff algebras, showing that algebraic coherence is a natural and widespread condition.
• To clarify the relationship between algebraic coherence and fibre-wise algebraic cartesian closedness in varieties.
• To demonstrate that algebraic coherence implies strong protomodularity and the coincidence of two-nilpotent object definitions in semi-abelian categories.

Proposed method

• Define algebraic coherence as the property that change-of-base functors along the fibration of points preserve finite limits and jointly strongly epimorphic pairs.
• Characterize algebraic coherence via the kernel functor, showing equivalence with preservation of certain pullbacks and jointly strongly epimorphic pairs.
• Establish closure properties: algebraic coherence is preserved under slicing, taking points, and regular epi-reflections.
• Prove that all categories of interest in the sense of Orzech are algebraically coherent, using their internal Mal’cev term structure.
• Show that (compact) Hausdorff algebras over a semi-abelian algebraically coherent theory remain algebraically coherent.
• Demonstrate that in algebraically coherent semi-abelian categories, the two natural definitions of two-nilpotent objects coincide: $[X,X,X] = [[X,X],X]$.

Experimental results

Research questions

• RQ1 What conditions on a finitely complete category ensure that its change-of-base functors along the fibration of points preserve finite limits and jointly strongly epimorphic pairs?
• RQ2 How does algebraic coherence relate to known categorical properties such as the Smith is Huq condition, normality of Higgins commutators, and strong protomodularity?
• RQ3 Which classical algebraic categories—such as groups, rings, Lie algebras, and Poisson algebras—satisfy algebraic coherence?
• RQ4 In what contexts does algebraic coherence imply fibre-wise algebraic cartesian closedness?
• RQ5 Do the two natural definitions of two-nilpotent objects in a semi-abelian category coincide when the category is algebraically coherent?

Key findings

• Algebraic coherence implies the Smith is Huq condition (SH), the strong version (SSH), and the normality of Higgins commutators (NH) in semi-abelian categories.
• In any algebraically coherent semi-abelian category, the two-nilpotent object conditions $[X,X,X] = [[X,X],X]$ hold, showing the coincidence of two natural definitions.
• All categories of interest in the sense of Orzech—such as groups, Lie algebras, rings, associative algebras, and Poisson algebras—are algebraically coherent.
• Categories of (compact) Hausdorff algebras over a semi-abelian algebraically coherent theory are algebraically coherent, extending the class of examples.
• Algebraic coherence implies strong protomodularity in pointed Mal’tsev categories, and fibre-wise algebraic cartesian closedness in varietal contexts.
• The change-of-base functors along the fibration of points preserve Higgins and Huq commutators, normal closures, and cokernels in algebraically coherent semi-abelian categories.