Multiplicative graph

• 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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For comparison, this diagram shows a typical arrow composition in an ordinary category. (Without arrow composition, it is simply a directed graph.) The arrows $f$ and $g$ in the diagram are consecutive; they connect in $B=\text{cod}(f)=\text{dom}(g).$ In an ordinary category, the composition of any pair of consecutive arrows exists, whereas in a multiplicative graph, a pair of consecutive arrows is not necessarily composable.

In mathematics, a multiplicative graph (in French:graphe multiplicatif or neocategory 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.

Arrow composition in an ordinary category satisfies the following property: if arrows $f$ and $g$ connect in the sense that $\text{dom}(g)=\text{cod}(f),$ their composition $g\circ f$ is defined, and, moreover, $\text{dom}(g\circ f)=\text{dom}(f)$ and $\text{cod}(g\circ f)=\text{cod}(g).$ In a multiplicative graph, however, condition $\text{dom}(g)=\text{cod}(f)$ does not guarantee the existence of $g\circ f$ within that structure without further assumptions – but if this composition exists, it also satisfies $\text{dom}(g\circ f)=\text{dom}(f)$ and $\text{cod}(g\circ f)=\text{cod}(g).$ 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.

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

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

Definition

Template:Group-like structures A multiplicative graph $\mathrm{\Sigma }$ 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 reflexive 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:

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

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

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:

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

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

Example

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

See also

• Partial groupoid
• Non-associative algebra

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

External link

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

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