Ellerman, David (2025) Where do Adjunctions Come From? Chimera Morphisms and Adjoint Functors in Category Theory. [Preprint]

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Abstract

Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunction seeming to be the primary lens. Our topic is a theory showing “where adjoints come from”. The theory is based on object-to-object “chimera morphisms”, “heteromorphisms”, or “hets” between the objects of different categories (e.g., the insertion of generators as a set-to-group map). After showing that heteromorphisms can be treated rigorously using the machinery of category theory (bifunctors), we show that all adjunctions between two categories arise (up to an isomorphism) as the representations (i.e., universal models) within each category of the heteromorphisms between the two categories. The conventional treatment of adjunctions eschews the whole concept of a heteromorphism, so our purpose is to shine a new light on this notion by showing its origin as a het between categories being universally represented within each of the two categories. This heteromorphic treatment of
adjunctions shows how they can be split into two separable universal constructions. Such universals can also occur without being part of an adjunction. We conclude with the idea that it is the universal constructions (adjunctions being an important special case) that are really the foundational concepts to pick out what is important in mathematics and perhaps  in other sciences, not to mention in philosophy.

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Item Type:	Preprint
Creators:	Creators Email ORCID Ellerman, David david@ellerman.org 0000-0002-5718-618X
Keywords:	adjunctions; adjoint functors; heteromorphism; universal mapping properties; representable functors
Subjects:	Specific Sciences > Mathematics > Foundations Specific Sciences > Cognitive Science > Action Specific Sciences > Mathematics Specific Sciences > Cognitive Science > Perception
Depositing User:	Dr. David Ellerman
Date Deposited:	02 Apr 2025 17:41
Last Modified:	02 Apr 2025 17:41
Item ID:	24961
DOI or Unique Handle:	10.3390/foundations5010010
Subjects:	Specific Sciences > Mathematics > Foundations Specific Sciences > Cognitive Science > Action Specific Sciences > Mathematics Specific Sciences > Cognitive Science > Perception
Date:	March 2025
URI:	https://philsci-archive.pitt.edu/id/eprint/24961

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