Embedding theorems in non-abelian categorical algebra

Jacqmin, Pierre-Alain [UCL]

The idea behind embedding theorems is to provide a representative element among a collection of categories, such that each category in that collection nicely embeds in (a power of) the representative category. The most famous examples are Yoneda’s, Barr’s and Lubkin’s embedding theorems for the collections of small, regular and abelian categories respectively. The aim of this thesis is to prove such embedding theorems for non-abelian categorical algebraic properties. We first treat the cases of Mal’tsev, n-permutable, unital, strongly unital, subtractive and protomodular regular categories. For each of those properties, we characterise essentially algebraic categories which satisfy it and construct among them our representative category. To prove the corresponding embedding theorems, we develop the theory of unconditional exactness properties, which we show to be preserved under the cofiltered limit completion. Using categories of partial algebras, we then prove embedding theorems for the classes of weakly Mal’tsev and weakly unital categories. As applications, we show on concrete examples how to use elements and operations to prove theorems for the above cited classes of categories. We also describe the bicategory of fractions with respect to weak equivalences between internal groupoids in a monadic category over a regular category where the axiom of choice holds. It suffices to replace internal functors by monoidal functors, which preserves the algebraic structures up to isomorphisms.