Differential algebras in codifferential categories.(English)Zbl 1420.18019
Derivations valued in modules over a commutative algebra are generalized to the context of a codifferential category in [R. Blute et al., Cah. Topol. Géom. Différ. Catég. 57, No. 4, 243–279 (2016;Zbl 1364.13026)] with the notion of \(\top\)-algebra. A \(\top\)-differential algebra in a codifferential category is a \(\top\)-algebra endowed with a \(\top\)-derivation of type \(D:A\rightarrow A\). \(\top\)-differential algebras, which are the main concept of study in this paper, are the appropriate generalization of differential algebras in codifferential categories. The paper consists of 8 sections. It is shown in §5 that every \(\top\)-differential algebra is a differential algebra in the classical sense, satisfying not only the higher-order Leibniz rule but also the Faà di Bruno formula for a higher-order chain rule. §6 is devoted to three basic examples of codifferential categories and their \(\top \)-differential algebras. §7 demonstrates that, given a codifferential category with countable coproducts, its category of \(\top\)-differential algebras is monadic over the codifferential category, while §8 shows that, given a codifferential category with countable coproducts, the category of \(\top\)-differential algebras is comonadic over the category of \(\top\)-algebras.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 12H05 | Differential algebra |
Citations:
Zbl 1364.13026Cite
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