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Gejza Jenča - 2-categories in algebra and elsewhere

Gejza Jenča and Anna Jenčová - 2-categories in algebra and elsewhere

A category has objects and morphisms. A 2-category [2,3] generalizes this by including morphisms between morphisms, called 2-morphisms.

The principal example of a 2-category is Cat, the category of all small categories. The objects of Cat are categories, the morphisms are functors and morphisms between functors are natural transformations. There are composition operations between 2-morphisms in Cat, called horizontal and vertical composition, satisfying certain equational laws. These equational laws allow us to formulate (in an extrinsic way, i.e. without looking at the inside of the objects of Cat) many of the well-known notions from category theory, for example adjunctions, monads or Kan extensions.

The aim of these lectures is to show how the notions from the theory of 2-categories appear in other areas of mathematics than category theory. Of course, this would not be possible if Cat would be the only 2-category occurring in the wild. Fortunately, this is not the case; examples of 2-categories include the following.

  • Posets, isotone maps, comparisons of isotone maps.
  • Sets, relations, inclusions of relations.
  • Sets, spans, maps between spans.
  • Rings, bimodules, bimodule homomorphisms.
  • Relational monoids, lax homomorphisms, inclusions of relations.

We show how the notions from category theory in these 2-categories give rise to well-known notions.

We then focus on the case of relational monoids, that means, monoids in the category of sets and relations equipped with the direct product of sets. These structures include effect algebras and their generalizations. We show that certain definitions that appeared in the theory of effect algebras can be clearly motivated by considering (generalized) effect algebras as objects of the larger 2-category of relational monoids. Perhaps surprisingly, theory of 2-categories gives us some of the axioms of dimensional equivalence that appear in the classical paper [4] by Loomis. We also show how parts of the theory of test spaces [1] can be considered naturally within the framework of relational semigroups. Finally, we look at the (somewhat underexposed) fact that small categories themselves can be considered as relational monoids.

[1] D.J. Foulis and C.H. Randall. Operational quantum statistics. I. Basic concepts. J. Math. Phys., 13:1667-1675, 1972.

[2] G Max Kelly and Ross Street. Review of the elements of 2-categories. In Category seminar, pages 75-103. Springer, 1974.

[3] Stephen Lack. A 2-categories companion. In Towards higher categories, pages 105-191. Springer, 2010.

[4] Lynn H Loomis. The lattice theoretic background of the dimension theory of operator algebras. Number 18 in Memoirs of the AMS. American Mathematical Society, 1955.