**Title**: Bifold algebras

**Abstract**:

Given V-enriched algebraic theories T and U for a system of arities J in the sense of [1], commuting pairs of T- and U-algebra structures on the same object may be described equivalently as bifunctors that preserve J-cotensors in each variable separately, and these we call bifold algebras. Bifold algebras may be described equivalently as algebras for a theory called the tensor product of T and U, provided that J and V satisfy certain conditions that we do not assume in this talk. Every bifold algebra has two underlying algebras, which we call its left and right faces. (By an algebra, here we mean a pair consisting of a theory T and a T-algebra A.)

In this talk, we construct a two-sided fibration [2] of bifold algebras over various theories, and we show that the notion of commutant for algebras [3, 4] arises via universal constructions in this two-sided fibration. Using this method, we develop a functorial treatment of commutants for algebras over various theories. On this basis, we study bifold algebras in which one face is the commutant of the other, and vice versa, and we discuss examples in algebra, order theory, and topology.

[1] R. B. B. Lucyshyn-Wright, Enriched algebraic theories and monads for a system of arities. Theory and Applications of Categories 31 (2016), 101-137.

[2] R. Street, Fibrations and Yoneda’s lemma in a 2-category. Lecture Notes in Mathematics 420 (1974), Springer.

[3] R. B. B. Lucyshyn-Wright, Commutants for enriched algebraic theories and monads. Applied Categorical Structures 26 (2018), 559-596.

[4] R. B. B. Lucyshyn-Wright, Functional distribution monads in functional-analytic contexts. Advances in Mathematics 322 (2017), 806-860.