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Intrinsic tensor products and a Ganea-type extension of the five-term exact sequence

Bo Shan Deval, Manfred Hartl, Tim Van der Linden

TL;DR

The paper develops a categorical framework for intrinsic tensor-like interactions in semi-abelian categories by systematizing cosmash products and two-nilpotent reflections. It proves a recognition theorem: any symmetric, bi-cocontinuous bifunctor on an abelian variety can be realized as a bilinear product in a surrounding two-nilpotent semi-abelian operad, tying classical tensor products to cosmash structures. It then establishes the right-exactness of cross-effects and derives a categorical version of Ganea’s six-term exact sequence in semi-abelian homology, with abelian extensions characterized via internal actions. A broad suite of examples (groups, loops, Lie/Leibniz algebras, modules, sheaves, crossed modules) demonstrates how familiar tensor products arise from this intrinsic construction and clarifies dependence on the surrounding category, opening avenues for operad-based generalizations and representation-theoretic applications.

Abstract

We establish a right-exactness theorem for the cross-effects of bifunctors, and consequently for cosmash products, in Janelidze-Márki-Tholen semi-abelian categories. This result motivates an intrinsic definition of a bilinear product, a tensor-like operation on objects of a category, constructed in terms of limits and colimits. Given two objects in the category, their bilinear product is the abelian object obtained as the cosmash product in the category of two-nilpotent objects of the reflections of these objects. In many concrete cases, this operation, applied to a pair of abelian objects, captures a classical tensor product. We explain this by proving a recognition theorem, which states that any symmetric, bi-cocontinuous bifunctor on an abelian variety of algebras can be recovered as the bilinear product within a suitable semi-abelian variety, namely of algebras over a certain two-nilpotent operad. In other words, the extra structure carried by such a bifunctor on the abelian variety (for instance, a tensor product, known in the literature) is encoded by means of a surrounding semi-abelian variety whose abelian core is the original variety. We illustrate the construction with several examples, develop its basic properties, and compare it to the semi-abelian analogue of the Brown-Loday non-abelian tensor product. As an application, we present a categorical version of Ganea's six-term exact homology sequence. Finally, we characterise abelian extensions via internal action cores, yielding explicit descriptions of cosmash products and bilinear products in certain categories of representations.

Intrinsic tensor products and a Ganea-type extension of the five-term exact sequence

TL;DR

The paper develops a categorical framework for intrinsic tensor-like interactions in semi-abelian categories by systematizing cosmash products and two-nilpotent reflections. It proves a recognition theorem: any symmetric, bi-cocontinuous bifunctor on an abelian variety can be realized as a bilinear product in a surrounding two-nilpotent semi-abelian operad, tying classical tensor products to cosmash structures. It then establishes the right-exactness of cross-effects and derives a categorical version of Ganea’s six-term exact sequence in semi-abelian homology, with abelian extensions characterized via internal actions. A broad suite of examples (groups, loops, Lie/Leibniz algebras, modules, sheaves, crossed modules) demonstrates how familiar tensor products arise from this intrinsic construction and clarifies dependence on the surrounding category, opening avenues for operad-based generalizations and representation-theoretic applications.

Abstract

We establish a right-exactness theorem for the cross-effects of bifunctors, and consequently for cosmash products, in Janelidze-Márki-Tholen semi-abelian categories. This result motivates an intrinsic definition of a bilinear product, a tensor-like operation on objects of a category, constructed in terms of limits and colimits. Given two objects in the category, their bilinear product is the abelian object obtained as the cosmash product in the category of two-nilpotent objects of the reflections of these objects. In many concrete cases, this operation, applied to a pair of abelian objects, captures a classical tensor product. We explain this by proving a recognition theorem, which states that any symmetric, bi-cocontinuous bifunctor on an abelian variety of algebras can be recovered as the bilinear product within a suitable semi-abelian variety, namely of algebras over a certain two-nilpotent operad. In other words, the extra structure carried by such a bifunctor on the abelian variety (for instance, a tensor product, known in the literature) is encoded by means of a surrounding semi-abelian variety whose abelian core is the original variety. We illustrate the construction with several examples, develop its basic properties, and compare it to the semi-abelian analogue of the Brown-Loday non-abelian tensor product. As an application, we present a categorical version of Ganea's six-term exact homology sequence. Finally, we characterise abelian extensions via internal action cores, yielding explicit descriptions of cosmash products and bilinear products in certain categories of representations.

Paper Structure

This paper contains 52 sections, 40 theorems, 126 equations, 5 figures, 1 table.

Table of Contents

  1. Cosmash products and Higgins commutators
  2. Binary and ternary cosmash products.
  3. The Higgins commutator.
  4. The ternary commutator.
  5. Two-nilpotency and the bilinear product
  6. Abelian objects.
  7. Two-nilpotent objects.
  8. Two-nilpotent categories.
  9. Products and coproducts
  10. Linear functors.
  11. Bilinear bifunctors.
  12. The bilinear product on a semi-abelian category.
  13. Symmetry, reducedness.
  14. Higher-order commutators and nilpotency
  15. Cosmash products and commutators
  16. ...and 37 more sections

Key Result

Lemma 1.2

There is a short exact sequence \xymatrix@=3em{ 0 \ar[r] & X\ensuremath{\diamond} Y\ensuremath{\diamond} Z \ar@{{ |>}->}[r] & (X+Y)\ensuremath{\diamond} Z \ar@{-{ >>}}[r] & (X\ensuremath{\diamond} Z)\times(Y\ensuremath{\diamond} Z) \ar[r] & 0}for all $X$, $Y$, $Z$ in $\mathbb{X}$.∎

Figures (5)

  • Figure 1: Alternate description of $X\diamond Y \diamond Z$
  • Figure 2: Comparing nilpotent reflections
  • Figure 3: Proving that $N_{X,Y}=[X+Y,X+Y,X+Y]\mathrel{\lhd} X\diamond Y$
  • Figure 4: Constructing the morphisms $\beta$ and $\delta$
  • Figure 5: The functor $F(X|-)$ applied to a reflexive graph

Theorems & Definitions (79)

  • Lemma 1.2
  • Lemma 1.3
  • Lemma 1.6
  • Proposition 2.6
  • proof
  • Proposition 2.8
  • Proposition 2.11
  • proof
  • Lemma 3.2
  • Lemma 3.5
  • ...and 69 more