On the 2-category of symmetric 2-rigs

The notion of a 2-rig provides one possible way to categorify the notion of a rig, i.e. a ring without negatives. Here, colimits play the role of sums, tensor products play the role of products, and preservation of colimits by the tensor product corresponds to the distributive law of rings. Explicitly, a 2-rig is a monoidally cocomplete category, i.e. a cocomplete category equipped with a monoidal structure such that the tensor product preserves colimits in each variable. Symmetric 2-rigs, where the monoidal structure is symmetric, categorify commutative rigs.

The aim of the talk is to discuss some steps in the program of developing the theory of symmetric 2-rigs in analogy with commutative ring theory, as suggested and investigated by André Joyal. The motivation for this comes from several areas, including algebraic topology (via operads), geometry (via categories of quasi-coherent sheaves), mathematical logic and theoretical computer science (via analytic functors). Indeed, examples of symmetric 2-rigs abound. In particular, we have free 2-rigs (obtained by taking presheaves on free symmetric monoidal categories), semi-free 2-rigs (obtained by taking presheaves over symmetric monoidal categories), operadic 2-rigs (obtained by taking presheaves over symmetric monoidal categories associated to operads), and ‘symmetric algebra’ 2-rigs (obtained by freely adding a symmetric tensor product to a locally presentable category).

First, I will describe some properties of the 2-category of symmetric 2-rigs, which are quite analogous to those of the category or commutative rings, although unavoidably more subtle, as they involve 2-dimensional monad theory rather than standard monad theory. I will also illustrate how the operation of ‘taking points’ of a symmetric 2-rig gives rise to a duality.

Secondly, I will consider two full sub-2-categories of 2-category of symmetric 2-rigs and identify them with the bicategory of categorical symmetric sequences [2] and the bicategory of operads and bimodules [4]. The latter is of particular interest since, as will be explained, the Boardman-Vogt tensor product of operads can be shown to act also on bimodules [5, 6], extending work of Dwyer and Hess [1].

Finally, I will outline how the analogy with commutative ring theory suggests the idea of exploring counterparts of the notions of a derivation and of the module of Kähler differentials, showing how the work in [3] fits in this framework and can be extended further.