Differential categories were introduced to provide a minimal categorical doctrine for differential linear logic. Here we revisit the formalism and, in particular, examine the two different approaches to defining differentiation which were introduced. The basic approach used a deriving transformation, while a more refined approach, in the presence of a bialgebra modality, used a codereliction. The latter approach is particularly relevant to linear logic settings, where the coalgebra modality is monoidal and the Seely isomorphisms give rise to a bialgebra modality. Here, we prove that these apparently distinct notions of differentiation, in the presence of a monoidal coalgebra modality, are completely equivalent. Thus, for linear logic settings, there is only one notion of differentiation. This paper also presents a number of separating examples for coalgebra modalities including examples which are and are not monoidal, as well as examples which do and do not support differential structure. Of particular interest is the observation that-somewhat counter-intuitively-differential algebras never induce a differential category although they provide a monoidal coalgebra modality. On the other hand, Rota-Baxter algebras-which are usually associated with integration-provide an example of a differential category which has a non-monoidal coalgebra modality.

Differential categories are now an established abstract setting for differentiation. However not much attention has been given to the process which is inverse to differentiation: integration. This paper presents the parallel development for integration by axiomatizing an integral transformation, s A : !A − → !A ⊗ A, in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies these two theorem is called a calculus category.Modifying an approach to antiderivatives by T. Ehrhard, we define having antiderivatives as the demand that a certain natural transformation, K : !A − → !A, is invertible. We observe that a differential category having antiderivatives, in this sense, is always a calculus category.When the coalgebra modality is monoidal, it is natural to demand an extra coherence between integration and the coalgebra modality. In the presence of this extra coherence we show that a calculus category with a monoidal coalgebra modality has its integral transformation given by antiderivatives and, thus, that the integral structure is uniquely determined by the differential structure.The paper finishes by providing a suite of separating examples. Examples of differential categories, integral categories, and calculus categories based on both monoidal and (mere) coalgebra modalities are presented. In addition, differential categories which are not integral categories are discussed and vice versa.The proof follows the form of (i): we show that J −1 n+1 := δJ n d • (ε ⊗ e) by first showing, with this definition, that J −1 n+1 J n+1 = 1 and then showing that J n+1 J −1 n+1 = J −1 n+1 J n+1 . First, by naturality of J n , [J.7], and [cd.7] we have:=== e δ ε = J −1 n ε === e δ J n = ε === e δ = Proof: These are mostly straightforward calculations by using the properties of J: [J −1 .1]: Here we use the property [J.1]: (n + 1) · 0 J −1 n = 0 J n J −1 n = 0 = J −1 n J n 0 = (n + 1) · J −1 n 0 [J −1 .2]: Here we use [J.2]: (n + 1) · e J −1 n = e J n J −1 n = e [J −1 .3]: Here we use [J.3]: (n + 2) · ε J −1 n = ε J n J −1 n = ε 42 [J −1 .4] Here we use [J.6]: J −1 n === = J −1 n === Jn+1 J −1 n+1 = J n === J −1 n+1 J −1 n = J −1 n+1 === [J −1 .5] Here we use [J.7]: J −1 n+1 === = J −1 n+1 === Jn J −1 n = J n+1 === J −1 n J −1 n+1 = J −1 n === [J −1 .6]: Here we use [J.12]: === === J −1 n = Jn === === J −1 n J −1 n = Jn === === J −1 n J −1 n = J −1 n === === [J −1 .7]: Here we use [J −1 .6]: === === J −1 n = === === J −1 n + J −1 n = J −1 n === === + J −1 n = === === J −1 n [J −1 .8]: Here we use [L.6] and that, as K = L + !(0), the differential of K and L are equal: === [J −1 .10]: Here we use [J −1 .4] and [d.5]: J −1 n === === = === === J −1 n+1 = === === J −1 n+1 = === === J −1 n [J −1 .11]: Here we use [J −1 .5] and [cd.6]:n ✷ Proposition 6.6 In a differential category with antiderivatives and a monoidal coalgebra modality, the followi...

Differential categories were introduced to provide a minimal categorical doctrine for differential linear logic. Here we revisit the formalism and, in particular, examine the two different approaches to defining differentiation which were introduced. The basic approach used a deriving transformation, while a more refined approach, in the presence of a bialgebra modality, used a codereliction. The latter approach is particularly relevant to linear logic settings, where the coalgebra modality is monoidal and the Seely isomorphisms give rise to a bialgebra modality. Here, we prove that these apparently distinct notions of differentiation, in the presence of a monoidal coalgebra modality, are completely equivalent. Thus, for linear logic settings, there is only one notion of differentiation.This paper also presents a number of separating examples for coalgebra modalities including examples which are and are not monoidal, as well as examples which do and do not support differential structure. Of particular interest is the observation that -somewhat counter-intuitively -differential algebras never induce a differential category although they provide a monoidal coalgebra modality. On the other hand, Rota-Baxter algebras -which are usually associated with integration -provide an example of a differential category which has a non-monoidal coalgebra modality.

We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

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.