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Measuring coalgebras
Measuring coalgebras
Idea
Measuring coalgebras are an enrichment of the category of commutative rings (or commutative -algebras) in the cartesian closed category Cocomm Coalg of cocommutative coalgebras (which we will write simply as ), given a ground field .
The starting point is the observation that the category acts on the category Alg of commutative algebras: there is a functor
where, given a coalgebra and an algebra , is the abelian-group hom of additive homomorphisms , made into an algebra whose multiplication is given by
where is the coalgebra comultiplication and is the algebra multiplication. That this is an “action” here means that there is a natural isomorphism
of algebras; here is sometimes described as an actegory over .
Definition
Definition
Given two algebras , the measuring coalgebra is by definition the representing object of the functor
so that there is an isomorphism, natural for coalgebras , of the form
Assume the existence of equalizers in , and of a right adjoint
to the forgetful functor (the cofree cocommutative coalgebra construction). We let
denote the counit of the adjunction .
We construct explicitly as the equalizer in of a pair of maps of the form
where we denote the internal hom in Vect by exponentiation (and we recall here that the cartesian product in is given by tensor product at the level of ). The first of these maps is
where is the multiplication on and is the unit. The second is given by a pair of maps
which we now describe separately.
The map is the unique coalgebra map such that lifts the map
through . Here denotes the comultiplication (same as the diagonal map as seen in ), and indicates the structure of enriched functoriality for .
The map is the unique coalgebra map such that lifts the map
through . Here denotes the counit (same as the unique map to the terminal object as seen in ).
Enrichment of algebras in coalgebras
Proposition
The measure coalgebra indeed gives an enrichment
Here the composition law in
(recalling that the product in is the tensor product of the underlying additive groups) is derived by universality from a composition of maps:
Literature
- M. E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, 1969
- Mathieu Anel, André Joyal, Sweedler Theory for (co)algebras and the bar-cobar constructions, arXiv:1309.6952
A dual notion to measuring coalgebra is Tambara's universal coacting bialgebra from
- Daisuke Tambara, The coendomorphism bialgebra of an algebra, J. Fac. Sci. Univ. Tokyo Sect. IA Math, 37, 425-456, 1990 pdf
- Martin Hyland, Ignacio López Franco?, Christina Vasilakopoulou, Hopf measuring comonoids and enrichment, Proc. London Math. Soc. 115:5 (2017) 1118-1148, doi
- Christina Vasilakopoulou, Enrichment of categories of algebras and modules, arXiv:1205.6450
- Marjorie Batchelor, Measuring coalgebras, quantum group-like objects, and non-commutative geometry, In: Bartocci, C., Bruzzo, U., Cianci, R. (eds) Differential Geometric Methods in Theoretical Physics. Lecture Notes in Physics 375 doi; Measuring comodules - their applications, J. Geom. Physics 36:3-4 (2009) 251-269 (doi); Difference operators, measuring coalgebras and quantum group like objects, Adv. Math. 105, 190-218 (1994) pdf
- Maximilien Péroux, The coalgebraic enrichment of algebras in higher categories, J. Pure Appl. Alg. 226:3 (2022) 106849 doi
- Paige Randall North, Maximilien Péroux, Coinductive control of inductive data types, arXiv:2303.16793