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Integral and differential structure on the free \(C^\infty\)-ring modality. (English. French summary) Zbl 1460.18010

The first notion of integration in a differential category was introduced in [T. Ehrhard, Math. Struct. Comput. Sci. 28, No. 7, 995–1060 (2018; Zbl 1456.03097)] with the introduction of differential categories with antiderivatives. J. R. B. Cockett and the second author [Math. Struct. Comput. Sci. 29, No. 2, 243–308 (2019; Zbl 1408.18012)] have provided the full story of integral categories, calculus categories and differential categories with antiderivatives, presenting an integral category of polynomial functions where it was not at all clear from its definition that the formula for its deriving transformation could be generalized to yield an integral category of arbitrary smooth functions. The principal objective in this paper is to present an integral category structure on the free \(C^{\infty}\)-ring monad that is compatible with the known differential structure.
The following results in the paper are also to be noticed.
R. Blute et al. [Cah. Topol. Géom. Différ. Catég. 57, No. 4, 243–279 (2016; Zbl 1364.13026)] defined derivations for codiferential categories. It is shown in this paper that derivations in this general sense, when applied to the \(C^{\infty}\)-ring considered here, correspond precisely to derivations of the Fermat theory of smooth functions in [E. J. Dubuc and A. Kock, Commun. Algebra 12, 1471–1531 (1984; Zbl 1254.51005)], which provides additional evidence that the Blute/Lucyshyn-Wright/O’Nei definition is the appropriate generalization of derivations in the context of codifferential categories.
Although this key example of integral categories does not possess a codereliction [R. F. Blute et al., Appl. Categ. Struct. 28, No. 2, 171–235 (2020; Zbl 1465.18011); Math. Struct. Comput. Sci. 16, No. 6, 1049–1083 (2006; Zbl 1115.03092)], it does possess many significant features of codereliction.
An integral category obeys certain Rota-Baxter axiom [G. Baxter, Pac. J. Math. 10, 731–742 (1960; Zbl 0095.12705); G. C. Rota, Bull. Am. Math. Soc. 75, 325–329 (1969; Zbl 0192.33801); Bull. Am. Math. Soc. 75, 330–334 (1969; Zbl 0319.05008)], so that free \(C^{\infty}\)-rings as examples of integral categories are Rota-Baxter algebras.

MSC:

18F40 Synthetic differential geometry, tangent categories, differential categories
18M05 Monoidal categories, symmetric monoidal categories
18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
26B12 Calculus of vector functions
13N15 Derivations and commutative rings
13N05 Modules of differentials
26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
03F52 Proof-theoretic aspects of linear logic and other substructural logics
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