Draft:Neocategory

A neocategory (also called graphe multiplicatif ) is a generalization of an ordinary category where associative law of composition is weakened to partial magma. Namely, it is the following structure:a one‑to‑one correspondence with the nodes of a directed graph, equipped with partial law of composition that satisfies only left and right identities. As a more general notion, there is the compositional graph, and neocategories can be seen as strongly identitive composition graphs.

Definition

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

1. $\kappa$ is a mapping from a subset of ${\underline {\Sigma }}\times {\underline {\Sigma }}$ (denoted by $\Sigma *\Sigma$ and called the set of composable couples) into ${\underline {\Sigma }}$; 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 $({\underline {\Sigma }},\beta ,\alpha )$ (i.e. $\alpha$ and $\beta$ are retractions from ${\underline {\Sigma }}$ onto a subset of ${\underline {\Sigma }}$, denoted by $\Sigma _{0}$), such that:

(existence of units): For each element $x$ of ${\underline {\Sigma }}$, the composites $x\circ \alpha (x)$ and $\beta (x)\circ x$ are defined, and we have

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

Here, $\alpha (x)$ is the right identity of $x$ and is called the source of $x$, while $\beta (x)$ is the left identity of $x$ and is called the target of $x$;

(coherence of dom/cod): If the composite $y\circ x$ is defined, then:

$\alpha (y\circ x)=\alpha (x),\ \beta (y\circ x)=\beta (y),\ {\text{and}}\ \alpha (y)=\beta (x).$

Example

• An ordinary category is a neocategory in which all the couples $(y,x)$ where $\alpha (y)=\beta (x)$ are composable (so that $\Sigma *\Sigma$ is the pullback of $(\alpha ,\beta )$), the law of composition being furthermore associative.

See also

• Sketch (mathematics)
• Partial groupoid

References

• 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 (2). 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.
• Ehresmann, Charles (1969). "Construction de structures libres". Category Theory, Homology Theory and their Applications II. Lecture Notes in Mathematics. Vol. 92. pp. 74–104. doi:10.1007/BFb0080766. ISBN 978-3-540-04611-0.
• Ehresmann, Charles (1965). Catégories et structures.
• Coppey, L.; Lair, C. (1984). "Leçons de théorie des esquisses" (PDF). Diagrammes. 12 (4). ISSN 0224-3911.

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Categories:

• Algebraic structures
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