Draft:Neocategory

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• Comment: This may be a notable topic, but the article provides almost no background for readers who aren't versed in the topic. Please provide a general introduction as a lead section. WeirdNAnnoyed (talk) 22:08, 8 August 2026 (UTC)

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A neocategory (Ehresmann, who introduced this notion, called it a multiplicative graph [in French: graphe multiplicatif] ^[1]) is a generalization of an ordinary category in which associative law of composition is weakened to a partial law of composition. While an ordinary category is a concept combining a directed graph and a monoidal structure, a neocategory is a partial magma-like structure. Namely, it is a structure in one‑to‑one correspondence with the nodes of a directed graph, and is equipped with partial law of composition that satisfies only left and right identities.^[1] Cury is studying enriched neocategories under the term graphe multiplicatif enrichi.^[2] As a more general notion, there is the compositional graph, and neocategories can be seen as strongly identitive composition graphs.^[3]

This notion first appears in Ehresmann's book Catégories et structures. For the notion of a sketch which he himself introduced, Ehresmann needed to define a category-like structure that avoided redundant axioms as much as possible.^[4] This structure is the neocategory, and this is a type of relaxed notion of category, such as a semicategory.

A neocategory ${\displaystyle \Sigma }$ is couple formed by a set denoted by ${\displaystyle {\underline {\Sigma }}}$, and a partial law of composition ${\displaystyle \kappa }$ on ${\displaystyle {\underline {\Sigma }}}$ satisfying the following axioms:^[6][1]

1. ${\displaystyle \kappa }$ is a mapping from a subset of ${\displaystyle {\underline {\Sigma }}\times {\underline {\Sigma }}}$ (denoted by ${\displaystyle \Sigma *\Sigma }$ and called the set of composable couples) into ${\displaystyle {\underline {\Sigma }}}$; instead of ${\displaystyle \kappa (y,x)}$, we write ${\displaystyle y\circ x}$ and we call ${\displaystyle y\circ x}$ the composite of ${\displaystyle (y,x)}$.
2. There exists a graph ${\displaystyle ({\underline {\Sigma }},\beta ,\alpha )}$ (i.e. ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are retractions from ${\displaystyle {\underline {\Sigma }}}$ onto a subset of ${\displaystyle {\underline {\Sigma }}}$, denoted by ${\displaystyle \Sigma _{0}}$), such that:

(existence of units^[7][8]): For each element ${\displaystyle x}$ of ${\displaystyle {\underline {\Sigma }}}$, the composites ${\displaystyle x\circ \alpha (x)}$ and ${\displaystyle \beta (x)\circ x}$ are defined, and we have

${\displaystyle x\circ \alpha (x)=x=\beta (x)\circ x.}$

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

(coherence of dom/cod^[9][8]): If the composite ${\displaystyle y\circ x}$ is defined, then:

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

From the condition 2, the graph ${\displaystyle (\Sigma ,\beta ,\alpha )}$ is uniquely defined.

• An ordinary category is a neocategory in which all the couples ${\displaystyle (y,x)}$ where ${\displaystyle \alpha (y)=\beta (x)}$ are composable (so that ${\displaystyle \Sigma *\Sigma }$ is the pullback of ${\displaystyle (\alpha ,\beta )}$), the law of composition being furthermore associative.^[10][1]

• Partial groupoid
• Non-associative algebra

• 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 (in French). 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 (in French). Vol. 92. pp. 74–104. doi:10.1007/BFb0080766. ISBN 978-3-540-04611-0.
• Ehresmann, Charles (1965). Catégories et structures (in French).
• Coppey, L.; Lair, C. (1984). "Leçons de théorie des esquisses" (PDF). Diagrammes (in French). 12 (4). ISSN 0224-3911.
• Cury, F. (1979). "Systèmes de générateurs et relations pour les catégories enrichies" (PDF). Diagrammes (in French). 1.
• Cury, F. (1978). Graphes multiplicatifs enrichis (Thesis) (in French).
  • Cury, Florence (2004). "Graphes multiplicatis enrichis. Partie I" (PDF). Diagrammes (in French). 51: 1–46.
  • Cury, Florence (2005). "Graphes multiplicatis enrichis. Partie II" (PDF). Diagrammes (in French). 53: 47–95.
  • Cury, Florence (2006). "Graphes multiplicatis enrichis. Partie III" (PDF). Diagrammes (in French). 55: 96–120.
  • Cury, Florence (2007). "Graphes multiplicatis enrichis. Partie IV" (PDF). Diagrammes (in French). 57: 121–183.

• Tringali, Salvatore (2013). "Plots and Their Applications - Part I: Foundations". arXiv:1311.3524v1 [math.CT].