Recent zbMATH articles in MSC 18C15https://www.zbmath.org/atom/cc/18C152021-06-15T18:09:00+00:00WerkzeugThe order-sobrification monad.https://www.zbmath.org/1460.180012021-06-15T18:09:00+00:00"Jia, Xiaodong"https://www.zbmath.org/authors/?q=ai:jia.xiaodongSummary: We investigate the so-called \textit{order-sobrification} monad proposed by \textit{W. K. Ho} et al. [Log. Methods Comput. Sci. 14, No. 1, Paper No. 7, 19 p. (2018; Zbl 1459.06003)] for solving the Ho-Zhao problem, and show that this monad is commutative. We also show that the Eilenberg-Moore algebras of the order-sobrification monad over dcpo's are precisely the \textit{strongly complete dcpo's} and the algebra homomorphisms are those Scott-continuous functions preserving suprema of irreducible subsets. As a corollary, we show that this monad gives rise to the free strongly complete dcpo construction over the category of posets and Scott-continuous functions. A question related to this monad is left open alongside our discussion, an affirmative answer to which might lead to a uniform way of constructing non-sober complete lattices.Non-abelian Galois cohomology via descent cohomology.https://www.zbmath.org/1460.180022021-06-15T18:09:00+00:00"Mesablishvili, Bachuki"https://www.zbmath.org/authors/?q=ai:mesablishvili.bachukiIntegral and differential structure on the free \(C^\infty\)-ring modality.https://www.zbmath.org/1460.180102021-06-15T18:09:00+00:00"Cruttwell, Geoffrey"https://www.zbmath.org/authors/?q=ai:cruttwell.geoffrey"Lemay, Jean-Simon Pacaud"https://www.zbmath.org/authors/?q=ai:pacaud-lemay.jean-simon"Lucyshyn-Wright, Rory B. B."https://www.zbmath.org/authors/?q=ai:lucyshyn-wright.rory-b-bThe first notion of integration in a differential category was introduced in [\textit{T. Ehrhard}, Math. Struct. Comput. Sci. 28, No. 7, 995--1060 (2018; Zbl 1456.03097)] with the introduction of differential categories with antiderivatives. \textit{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.
\begin{itemize}
\item \textit{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 [\textit{E. J. Dubuc} and \textit{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.
\item Although this key example of integral categories does not possess a \textit{codereliction} [\textit{R. F. Blute} et al., Appl. Categ. Struct. 28, No. 2, 171--235 (2020; Zbl 07181514); Math. Struct. Comput. Sci. 16, No. 6, 1049--1083 (2006; Zbl 1115.03092)], it does possess many significant features of codereliction.
\item An integral category obeys certain Rota-Baxter axiom [\textit{G. Baxter}, Pac. J. Math. 10, 731--742 (1960; Zbl 0095.12705); \textit{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.
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)