The Tangent Categories of Algebras over an Operad

Abstract

Tangent categories provide a categorical axiomatization of the tangent bundle. There are many interesting examples and applications of tangent categories in a variety of areas such as differential geometry, algebraic geometry, algebra, and even computer science. The purpose of this paper is to expand the theory of tangent categories in a new direction: the theory of operads. The main result of this paper is that both the category of algebras of an operad and its opposite category are tangent categories. The tangent bundle for the category of algebras is given by the semi-direct product, while the tangent bundle for the opposite category of algebras is constructed using the module of Kähler differentials, and these tangent bundles are in fact adjoints of one another. To prove these results, we first prove that the category of algebras of a coCartesian differential monad is a tangent category. We then show that the monad associated to any operad is a coCartesian differential monad. This also implies that we can construct Cartesian differential categories from operads. Therefore, operads provide a bountiful source of examples of tangent categories and Cartesian differential categories, which both recaptures previously known examples and also yield new interesting examples. We also discuss how certain basic tangent category notions recapture well-known concepts in the theory of operads.Acknowledgements. The authors would first like to thank Martin Frankland for providing a very useful result about adjoint tangent structure. The authors would also like to thank Martin Frankland (again), Geoff Cruttwell and Dorette Pronk for very useful discussions.

The differential bundles of the geometric tangent category of an operad

“…The author of this paper together with Sacha Ikonicoff and Jean-Simon Lemay extended the idea of studying the algebraic geometry of affine schemes with tangent categories to a new plethora of contexts. In [13], they showed that the category of algebras Alg 𝒫 of a (symmetric) operad 𝒫 over the category of 𝑅-modules (for a commutative and unital ring 𝑅) comes equipped with a tangent structure. In the following, we refer to this as the algebraic tangent structure of the operad 𝒫 which will be denoted by T (𝒫) , or simply by T when the operad 𝒫 is clear from the context.…”

“…In the aforementioned paper, it was proven that every operad comes with a coCartesian differential monad (cf. [13,Theorem 4.1.1]) and that this tangent category is precisely the tangent category of algebras of this monad.…”

“…We also assume the reader is knowledgeable about basic notions of tangent category theory (see [5] for reference). Even if we summarize in the őrst section the main results of the previous paper, we also recommend reading [13] to fully appreciate the whole story.…”

“…To properly appreciate the relevance of this result, notice that before the article [13], the most revelant available examples of tangent categories were differential geometry, synthetic differential geometry, algebraic geometry, commutative rings etc. In particular, there was no example of non-commutative geometry completely described by tangent category theory.…”

“…(𝒫) admits an adjoint tangent structure (cf. [13,Theorem 4.4.4]) which makes the opposite of the category of operadic algebras into a tangent category. In the following, we refer to this tangent structure as the geometric tangent structure of the operad 𝒫 which will be denoted by T (𝒫) , or simply by T when the operad 𝒫 is clear from the context.…”