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The Lawvere condition

Nelson Martins-Ferreira

TL;DR

This paper generalizes the Lawvere condition by formulating a relative version with respect to a chosen class of spans and proves a comprehensive equivalence theorem that unifies multiple diagrammatic and categorical formulations. It shows that the Lawvere condition, sections of forgetful functors, compatibility conditions, and (di)kite admissibility are all equivalent under mild ambient assumptions, and specializes these results to weakly Mal’tsev, Mal’tsev, and naturally Mal’tsev categories via canonical span classes. By extending beyond finite limits, it also connects to Janelidze–Pedicchio pseudogroupoids and broader internal structure theories. The results yield both conceptual simplifications of local product notions and new avenues for applying Mal’tsev-like ideas across diverse categorical contexts.

Abstract

The original Lawvere condition asserts that every reflexive graph admits a unique natural structure of internal groupoid. This property was identified by P. T. Johnstone, following a question by A. Carboni and a suggestion by F. W. Lawvere, and it plays a central role in the characterization of naturally Mal'tsev categories. A broad and conceptually rich generalization emerges when the condition is formulated relative to a chosen class of spans. In this setting, the familiar Mal'tsev situation is recovered when the class consists of internal relations (that is, jointly monic spans) in which case the condition states that every internal reflexive relation is an equivalence relation. The purpose of this paper is to establish a comprehensive equivalence theorem that unifies the various categorical and diagrammatic formulations of the relative Lawvere condition. Furthermore, this formulation retains its significance even beyond the context of categories with finite limits, extending, for example, to categories admitting pullbacks of split epimorphisms along split epimorphisms. In addition to providing a fresh perspective on previously established results, we present a new characterization involving Janelidze-Pedicchio pseudogroupoids.

The Lawvere condition

TL;DR

This paper generalizes the Lawvere condition by formulating a relative version with respect to a chosen class of spans and proves a comprehensive equivalence theorem that unifies multiple diagrammatic and categorical formulations. It shows that the Lawvere condition, sections of forgetful functors, compatibility conditions, and (di)kite admissibility are all equivalent under mild ambient assumptions, and specializes these results to weakly Mal’tsev, Mal’tsev, and naturally Mal’tsev categories via canonical span classes. By extending beyond finite limits, it also connects to Janelidze–Pedicchio pseudogroupoids and broader internal structure theories. The results yield both conceptual simplifications of local product notions and new avenues for applying Mal’tsev-like ideas across diverse categorical contexts.

Abstract

The original Lawvere condition asserts that every reflexive graph admits a unique natural structure of internal groupoid. This property was identified by P. T. Johnstone, following a question by A. Carboni and a suggestion by F. W. Lawvere, and it plays a central role in the characterization of naturally Mal'tsev categories. A broad and conceptually rich generalization emerges when the condition is formulated relative to a chosen class of spans. In this setting, the familiar Mal'tsev situation is recovered when the class consists of internal relations (that is, jointly monic spans) in which case the condition states that every internal reflexive relation is an equivalence relation. The purpose of this paper is to establish a comprehensive equivalence theorem that unifies the various categorical and diagrammatic formulations of the relative Lawvere condition. Furthermore, this formulation retains its significance even beyond the context of categories with finite limits, extending, for example, to categories admitting pullbacks of split epimorphisms along split epimorphisms. In addition to providing a fresh perspective on previously established results, we present a new characterization involving Janelidze-Pedicchio pseudogroupoids.

Paper Structure

This paper contains 10 sections, 11 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Basic definitions of internal categorical structures
  3. Proof of the Main Theorem
  4. Proof of the Corollaries
  5. Weakly Mal'tsev
  6. Mal'tsev
  7. Naturally Mal'tsev
  8. Extending the scope of the main theorem
  9. A few further applications
  10. Conclusion

Key Result

Theorem 1

Let $\mathbb{C}$ be a category with pullbacks of split epimorphisms along split epimorphisms, and let $\mathcal{M}$ be a class of spans in $\mathbb{C}$ that is closed under the kernel pair construction and contains the span part of every local product. Then the following conditions are equivalent:

Theorems & Definitions (19)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Definition 3
  • Proposition 1
  • Theorem 2
  • proof
  • ...and 9 more