[Paper Review] Categories vs. groupoids via generalised Mal'tsev properties

Nelson Martins-Ferreira, Tim Van der Linden|arXiv (Cornell University)|Jun 13, 2012

Advanced Algebra and Logic9 citations

TL;DR

This paper investigates the distinction between internal categories and internal groupoids in categories with generalized Mal'tsev properties, showing that in weakly Mal'tsev categories with kernel pairs and equalisers, internal categories and groupoids coincide precisely when every internal preorder is an equivalence relation—equivalently, when the variety is n-permutable. The key contribution is a new characterization of n-permutability via internal categorical structures, independent of binary products, and a construction of weakly Mal'tsev, Goursat but non-Mal’tsev varieties using implication algebras.

ABSTRACT

We study the difference between internal categories and internal groupoids in terms of generalised Mal'tsev properties---the weak Mal'tsev property on the one hand, and $n$-permutability on the other. In the first part of the article we give conditions on internal categorical structures which detect whether the surrounding category is naturally Mal'tsev, Mal'tsev or weakly Mal'tsev. We show that these do not depend on the existence of binary products. In the second part we focus on varieties of algebras.

Motivation & Objective

• To understand the difference between internal categories and internal groupoids in terms of generalized Mal'tsev properties.
• To characterize naturally Mal’tsev, Mal’tsev, and weakly Mal’tsev categories using internal categorical structures without relying on binary products.
• To investigate when internal categories become internal groupoids in weakly Mal’tsev categories, particularly in the context of varieties of universal algebras.
• To provide new characterizations of n-permutability in terms of internal structures, especially the condition that every internal preorder is an equivalence relation.
• To construct examples of categories that are weakly Mal’tsev and Goursat but not Mal’tsev, demonstrating independence of these properties.

Proposed method

• Uses kernel pairs and split pullbacks as foundational structures, avoiding reliance on binary products.
• Applies the notion of weakly Mal’tsev categories, where reflexive graphs admit at most one internal category structure.
• Employs the forgetful functor from internal categories to multiplicative graphs, showing it is an isomorphism in weakly Mal’tsev categories with equalisers and kernel pairs.
• Introduces a new quasi-identity condition to define weakly Mal’tsev quasivarieties, generalizing the Mal’tsev condition.
• Uses Hagemann’s theorem (Proposition 4.3) and its extension (Proposition 4.8) to construct weakly Mal’tsev quasivarieties from Goursat (3-permutable) ones.
• Constructs explicit examples using implication algebras with specific multiplication tables to demonstrate non-Mal’tsev behavior despite satisfying weak Mal’tsev and Goursat conditions.

Experimental results

Research questions

• RQ1 Under what conditions on a category do internal categories coincide with internal groupoids?
• RQ2 How can the weak Mal’tsev property be characterized using internal categorical structures such as multiplicative graphs and preorders?
• RQ3 What is the relationship between n-permutability and the property that every internal preorder is an equivalence relation in weakly Mal’tsev categories?
• RQ4 Can weakly Mal’tsev and Goursat categories exist that are not Mal’tsev, and how can such examples be systematically constructed?
• RQ5 Is the weak Mal’tsev property necessary for internal categories to be internal groupoids in n-permutable varieties?

Key findings

• In a weakly Mal’tsev category with kernel pairs and equalisers, the forgetful functor from internal categories to multiplicative graphs is an isomorphism.
• The forgetful functor from internal groupoids to internal categories is an isomorphism if and only if every internal preorder is an equivalence relation.
• In finitary quasivarieties, the condition that every internal preorder is an equivalence relation is equivalent to n-permutability for some n.
• The variety of implication algebras satisfying the quasi-identity (ba)x = (ab)c = (ba)x′ and (bc)x = (cb)a = (bc)x′ ⇒ x = x′ forms a weakly Mal’tsev quasivariety that is Goursat but not Mal’tsev.
• The construction in Proposition 4.8 allows for the systematic generation of weakly Mal’tsev, Goursat, but non-Mal’tsev quasivarieties, demonstrating that these properties are independent.
• The paper shows that in n-permutable varieties, the weak Mal’tsev property is redundant for ensuring that internal categories are internal groupoids, as n-permutability alone suffices.