Unbiased Algebra

Abstract: The main trust of the unbiased approach to algebra is that operations on a set X form a category whose objects are (finite) families I → X and are indexed by functions I → J. Furthermore, operations can be reindexed along pullbacks in finite sets.

This inevitably leads to the notion of unbiased symmetric multicategory [1]: a double category M with parallel sums, along with a sum-preserving double functor M → Pb (to the double category of pullback squares in finite sets) which is a discrete fibration in the tight direction. The loose arrows of M should be thought of as the abstract operations of the algebraic theory, while tight arrows and cells give reindexing.

Operads are the single-sorted theories, namely those M with just an object over a terminal set. If S is a species of structure in the sense of Joyal, an S-function is a function endowed with an S-structure on each of its fibers. For many basic species, S-functions can be naturally composed, so that they form the operations of an operad; in fact all operads arise in this way.

An algebra for M is itself a symmetric multicategory A with a double functor A → M which is a discrete opfibration in the loose direction. The loose arrows of A are the (concrete) operations of the algebra, indexed by the abstract operations in M. For instance, an algebra for the terminal operad Pb → Pb is (the unbiased form of) a commutative monoid X: a concrete operation evaluates a family I → X along a function I → J (the underlaying abstract operation) to give a family J →X. The algebras for the operad Tot, associated to the species of total orders, are plain monoids: operations evaluate families along functions with totally ordered fibers.

It is well-known that operads (or more generally symmetric multicategories) do not capture all kinds of ordinary algebraic theories since, roughly, one cannot duplicate or delete variables. To this end one introduces cartesian multicategories, which have a very neat definition in the present context. Namely, they are the algebras for a natural monad cart on symmetric multicategories: the loose arrows of cart M are spans (actually, "enhanced" spans [1]) formed by a tight arrow and a loose arrow in M. These are composed in the usual way for spans, except that we use M-cells in place of pullbacks. Thus, a cartesian structure cart M → M provides a way to "sum" an operation along a function to give a new operation.

Multicategories which are both cartesian and cocartesian (a.k.a. sequential [2]) amount to categories enriched in commutative monoids. In particular, cartesian and cocartesian operads amount to rigs. In the operad associated to a rig R, an operation α over a function I → J, is a family I → R, and composition is given by multiplication in the underlying monoid (as in the discrete distribution monad). The sum of α along f : I → K is obtained by adding the elements in the fibers of f.

Algebras for a cartesian operad M are required to satisfy a further compatibility condition, corresponding to the fact that they can be seen as morphisms of cart-algebras from M to the endomorphism operad End(X). The algebras for the cartesian operad associated to a rig R are the R-modules. For an operad M, the free M-algebra monad is given again by spans in M but, in a sense, dually to cart. If M is cartesian, free M-algebras are obtained by quotienting with respect to a natural equivalence on spans. Similarly, given N → M, the relatively free M-algebras are obtained by quotienting spans in M. We also illustrate the connections between cartesian operads and Lawvere theories.

The overall purpose of the talk is to show how the unbiased approach, by avoiding the awkwardness which inevitably arises from an unnatural skeletal indexing, turns out to be rather effective and opens up new perspectives.

[1] C. Pisani, Unbiased multicategory theory, Theory and Applications of Categories 44 (2025), 826–868. [2] C. Pisani, Sequential multicategories, Theory and Applications of Categories 29 (2014), 496–541.

category theory

Audience: researchers in the topic

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Third Virtual Workshop on Double Categories

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