The Heteromorphic Approach to Adjunctions: Theory and History

Round 1

Reviewer 1 Report

Ellerman's paper broadens the concept of morphisms, previously restricted to homomorphisms, to include what he terms ‘heteromorphisms’ which, unlike homomorphisms, extend across categories. In other words, it's analogous to the usual bifunctoral relation of morphisms, only he's relating objects across different categories rather than the usual relation of morphisms across different categories

Ellerman successfully shows that one can decompose adjunctions between a pair of categories into the more granular 'heteromorphisms' across those categories. The advantage here is that one can begin with  morphisms relating objects within each category and from those, inductively generate the 'heteromorphic' mappings relating objects across the categories. So from the adjunctive relations of relations across the categories, one can induce a map of the relations of objects across the categories.

More specifically, Ellerman formalizes his concept of heteromorphisms via a theory of adjunctions via a pair of adjoint functors SPANNING two categories. Thus homomorphisms (aka conventional morphisms) and heteromorphisms together make two different categorical species of structure, thus generalizing the well-known notion of a homomorphism / morphism.

Crucially, Ellerman develops both the theoretical and historical aspects of this idea allowing in this way to figure out the roots of his approach. The heteromorphic theory of adjunctions is developed using het-bifunctors between two different categorical species, which are valued in the category of sets. In this way, the representability of a Het-bifunctor within each of these species (left and right representation) leads naturally and universally to a pair of adjoint functors between the corresponding categories.

The idea of a het-bifunctor has been first employed by Pareigis in his book, called a connection. The author demonstrates the strength of his approach in a series of examples including the important case of the Tensor-Hom adjunction.

Ellerman’s approach is very elegant and conceptually clear. The heteromorphic treatment of adjunctions leads to significant simplifications in comparison to the standard treatments. It also looks very promising in relation to the functorial approach to physical geometry introduced by Mallios and Zafiris. This is especially interesting when applied to topological interpretations of quantum mechanics when the latter is applied to distilling object information from topological phase information in non-local QM measurement.

The most important conclusion is that all adjunctions arise as bi-representations of het-bifunctors according to the fundamental categorical notion of a universal mapping property.

Ellerman’s paper is a valuable addition to the literature on the subject of adjunctions and his  proposed approach seems well suited to problems in mathematical physics, viz. the QM measurement issue I mentioned above. I strongly recommend publication of this article in its present form.

Reviewer 2 Report

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