Free gs-monoidal categories and free Markov categories (Q6102159)

Closely related to Markov categories are gs-monoidal categories, which differ from Markov ones only by dropping an axiom in correspondence with the normalization of probability [\textit{K. Cho} and \textit{B. Jacobs}, Math. Struct. Comput. Sci. 29, No. 7, 938--971 (2019; Zbl 1452.18008)]. In the strict case, these go back to [\textit{A. Corradini} and \textit{F. Gadducci}, Appl. Categ. Struct. 7, No. 4, 299--331 (1999; Zbl 0949.68121)], where they were considered in the context of term graphs and term graph rewriting. Their intended application had suggested to study gs-monoidal categories freely generated by a given collection of morphisms, which had already been constructed and characterized in the single-sorted case with generators of coarity one [\textit{A. Corradini} and \textit{F. Gadducci}, Appl. Categ. Struct. 7, No. 4, 299--331 (1999; Zbl 0949.68121)] and later defined in general [\textit{M. Coccia} et al., Electron. Notes Theor. Comput. Sci. 69, 83--100 (2003; Zbl 1270.68123); \textit{R. Bruni} et al., Fundam. Inform. 134, No. 3--4, 287--317 (2014; Zbl 1320.68125)]. Apparently unaware of the previous work, free gs-monoidal categories appeared in the context of probability in [\textit{B. Fong}, ``Causal theories: a categorical perspective on Bayesian networks'', Preprint, \url{arXiv:1301.6201}] with the aim of providing a categorical framework for Bayesian networks. Another independent appearance of free gs-monoidal categories is in [\url{https://dario-stein.de/thesis.pdf}], where a construction as syntactical categories was given. This paper constructs free gs-monoidal categories generated by a collection of morphisms of arbitrary arity and coarity in the form of an explicit combinatorial description of their morphisms as structured cospans of labeled hypergraphs as well as free Markov categories. The synopsis of the paper goes as follows. \begin{itemize} \item[\S 2] states the main definitions concerning gs-monoidal categories and Markov categories. \item[\S 3] explicitly constructs free gs-monoidal categories generated by morphisms of arbitrary arity and coarity and with arbitrarily many sorts. \item[\S 4] establishes (Theorem 4.1) that the free gs-monoidal categories abide by the right 2-categorical universal property rather than merely a 1-categorical one and without any strictness requirement on the target categories. \item[\S 5] shows that every free gs-monoidal category comes equipped with a canonical factorization system (the \textit{bloom-circuitry factorization}) claiming that every gs-monoidal string diagram factors uniquely into a \textit{pure bloom}, where all its boxes appear and every wire gets copied so as to connect a unique output interface, followed by \textit{pure circuitry}, where wires get copied and discarded. \item[\S 6] explains how the characterization is to be adapted from free gs-monoidal categories to free Markov categories. \item[\S 7] provides a sketch of how the morphisms in a free gs-monoidal category are to be viewed and utilized in statistics as causal models generalizing the DAGs underlying Bayesian networks. \end{itemize}