Neocategory

A neocategory (also called graphes multiplicatifs) is a generalization of a ordinary category where composition is replaced by a partial magma, that is, a one-to-one correspondence with the nodes of a directed graph that satisfies only left and right identities. In a neocategory, composition is still defined, but it does not satisfy the associative law. As a more general notion, there is the compositional graph, and neocategories can be seen as strongly identitive composition graph.

Definition

A neocategory $\mathrm{\Sigma }$ is couple formed by a set denoted by $\underset{\_}{\mathrm{\Sigma }}$, and a partial law of composition $\kappa$ on $\underset{\_}{\mathrm{\Sigma }}$ satisfying the following axioms:

1. $\kappa$ is a mapping from a subset of $\underset{\_}{\mathrm{\Sigma }}\times \underset{\_}{\mathrm{\Sigma }}$ (denoted by $\mathrm{\Sigma }*\mathrm{\Sigma }$ and called the set of composable couples) into $\underset{\_}{\mathrm{\Sigma }}$; instead of $\kappa (y,x)$, we write $y\circ x$ and we call $y\circ x$ the composite of $(y,x)$.
2. There exists a graph $(\underset{\_}{\mathrm{\Sigma }},\beta ,\alpha )$ (i.e. $\alpha$ and $\beta$ are retractions from $\underset{\_}{\mathrm{\Sigma }}$ onto a subset of $\underset{\_}{\mathrm{\Sigma }}$, denoted by ${\mathrm{\Sigma }}_0$), such that:

(unit axiom): For each element $x$ of $\underset{\_}{\mathrm{\Sigma }}$, the composites $x\circ \alpha (x)$ and $\beta (x)\circ x$ are defined, and we have

$x\circ \alpha (x)=x=\beta (x)\circ x$;

(coherence axiom): If the composite $y\circ x$ is defined, then:

$\alpha (y)=\beta (x),\alpha (y\circ x)=\alpha (x),\beta (y\circ x)=\beta (y)$

Reference

• Bastiani, Andrée; Ehresmann, Charles (1972). "Categories of sketched structures" (PDF). Cahiers de Topologie et Géométrie Différentielle Catégoriques. 13 (2). ISSN 1245-530X.
• Coppey, L. (1980). "Quelques problèmes typiques concernant les graphes multiplicatifs" (PDF). Diagrammes. 3. ISSN 0224-3911.
• Mateus, Paulo; Sernadas, Amílcar; Sernadas, Cristina (1999). "Precategories for Combining Probabilistic Automata". Electronic Notes in Theoretical Computer Science. 29: 169–186. doi:10.1016/S1571-0661(05)80315-9.

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