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.
Bierman, G.M.: What is a categorical model of intuitionistic linear logic? In: International Conference on Typed Lambda Calculi and Applications, pp. 78–93. Springer (1995)
Blute, R.F., Cockett, J.R.B., Porter, T., Seely, R.A.G.: Kähler categories. Cahiers de Topologie et Géométrie Différentielle Catégoriques 52(4), 253–268 (2011)
Blute, R.F., Cockett, J.R.B., Seely, R.A.G.: Differential categories. Math. Struct. Comput. Sci. 16(06), 1049–1083 (2006)
Blute, R.F., Cockett, J.R.B., Seely, R.A.G.: Cartesian differential categories. Theory Appl. Categ. 22(23), 622–672 (2009)
Blute, R.F., Cockett, J.R.B., Seely, R.A.G.: Cartesian differential storage categories. Theory Appl. Categ. 30(18), 620–686 (2015)
Blute, R.F., Ehrhard, T., Tasson, C.: A convenient differential category. Cahiers de Top. et Géom Diff LIII, 211–232 (2012)
Blute, R.F., Lucyshyn-Wright, R.B.B., O’Neill, K.: Derivations in codifferential categories. Cahiers de Topologie et Géométrie Différentielle Catégoriques 57, 243–280 (2016)
Ehrhard, T., Regnier, L.: The differential lambda-calculus. Theor. Comput. Sci. 309(1), 1–41 (2003)
Ehrhard, T., Regnier, L.: Differential interaction nets. Theor. Comput. Sci. 364(2), 166–195 (2006)
Fiore, M.P.: Differential structure in models of multiplicative biadditive intuitionistic linear logic. In: International Conference on Typed Lambda Calculi and Applications, pp. 163–177. Springer (2007)
Guo, L.: An Introduction to Rota–Baxter Algebra, vol. 2. International Press Somerville, Somerville (2012)
Guo, L., Keigher, W.: On differential Rota–Baxter algebras. J. Pure Appl. Algebra 212(3), 522–540 (2008)
Joyal, A., Street, R.: The geometry of tensor calculus, i. Adv. Math. 88(1), 55–112 (1991)
Lang, S.: Algebra, revised 3rd ed. In: Graduate Texts in Mathematics, vol. 211 (2002)
Mac Lane, S.: Categories for the Working Mathematician. Springer, New York (1971, revised 2013)
Melliès, P.A.: Categorical models of linear logic revisited. https://hal.archives-ouvertes.fr/hal-00154229. Working paper or preprint (2003)
Schalk, A.: What is a categorical model of linear logic? Manuscript. http://www.cs.man.ac.uk/~schalk/notes/llmodel.pdf (2004)
Seely, R.A.G.: Linear Logic,*-Autonomous Categories and Cofree Coalgebras, vol. 92. American Mathematical Society, Providence (1989)
Selinger, P.: A survey of graphical languages for monoidal categories. In: New Structures for Physics, pp. 289–355. Springer (2010)
Zhang, S., Guo, L., Keigher, W.: Monads and distributive laws for rota-baxter and differential algebras. Adv. Appl. Math. 72, 139–165 (2016)
The authors would like to thank the anonymous referee for very helpful and constructive comments in their review, especially regarding the overall structure of this paper.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first and second author are supported by NSERC. The third author is supported by Kellogg College, the Clarendon Fund, and the Oxford-Google DeepMind Graduate Scholarship. The fourth author is supported by FRQNT.
About this article
Cite this article
Blute, R.F., Cockett, J.R.B., Lemay, JS.P. et al. Differential Categories Revisited. Appl Categor Struct 28, 171–235 (2020). https://doi.org/10.1007/s10485-019-09572-y