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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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In mathematics, a multiplicative graph (in French:graphe multiplicatif or neocategory^[2] in some English-language papers) is an algebraic structure in category theory. It is a generalization of an ordinary category in the sense that neither the associativity of arrow composition nor the composibility of a pair of connected arrows are assumed. While an ordinary category is a notion combining a directed graph and a monoidal structure, a multiplicative graph 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.^[2]

Arrow composition in an ordinary category satisfies the following property: if arrows ${\displaystyle f}$ and ${\displaystyle g}$ connect in the sense that ${\displaystyle {\text{dom}}(g)={\text{cod}}(f),}$ their composition ${\displaystyle g\circ f}$ is defined, and, moreover, ${\displaystyle {\text{dom}}(g\circ f)={\text{dom}}(f)}$ and ${\displaystyle {\text{cod}}(g\circ f)={\text{cod}}(g).}$ In a multiplicative graph, however, condition ${\displaystyle {\text{dom}}(g)={\text{cod}}(f)}$ does not guarantee the existence of ${\displaystyle g\circ f}$ within that structure without further assumptions – but if this composition exists, it also satisfies ${\displaystyle {\text{dom}}(g\circ f)={\text{dom}}(f)}$ and ${\displaystyle {\text{cod}}(g\circ f)={\text{cod}}(g).}$^[3] When drawing a diagram for a multiplicative graph, it is almost always necessary to explicitly draw all existing arrows that play a role in the argument. For example, as shown in Coppey (1980), square diagrams in a multiplicative graph can take one of five types depending on which potential compositions in the diagram are actually defined.^[4]

This notion first appears in Ehresmann's book Catégories et structures.^[5] The French school bases its definition of sketch on the notion of a multiplicative graph,^[6] because this definition required a category-like structure that avoided redundant axioms as much as possible.^[7] This structure is the multiplicative graph, and this is a type of relaxed notion of category, such as a semicategory.^[8]

Cury is studying enriched multiplicative graph.^[9] As a more general notion, there is the compositional graph, and multiplicative graphs can be seen as strongly identitive compositional graphs.^[8]

A multiplicative graph ${\displaystyle \Sigma }$ 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:^[2][10]

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 reflexive graph^[6] ${\displaystyle ({\underline {\Sigma }},\beta ,\alpha )}$ (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^[11][12]): 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^[12][13]): 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 reflexive graph ${\displaystyle (\Sigma ,\beta ,\alpha )}$ is uniquely defined.

• An ordinary category is a multiplicative graph 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.^[2][12][14]
• For the two axioms above, a notion got by adding only the associativity axiom to a multiplicative graph, that is, a notion that an ordinary category without the composibility axiom, this is called a precategory.^[12] But, this is not standard terminology, a precategory is usually synonymous with a semigroupoid and does not require each object to have an identity morphism.

• 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.
• Coppey, L. (1990). "Actes des journées E.L.I.T. (Univ. Paris 7. 27 juin-2 juillet 1988)" (PDF). Diagrammes (in French). 24: 33–76.
• Coppey, L.; Lair, C. (1984). "Leçons de théorie des esquisses" (PDF). Diagrammes (in French). 12 (4). ISSN 0224-3911.
• 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.
• Cury, F. (1979). "Systèmes de générateurs et relations pour les catégories enrichies" (PDF). Diagrammes (in French). 1.
• Ehresmann, Charles (1965). Catégories et structures (in French).
• 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.
• 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.

• Tringali, Salvatore (2013). "Plots and Their Applications - Part I: Foundations". arXiv:1311.3524v1 [math.CT].
• Wells, Charles (2009). "Sketches: Outline with References" (PDF).