Multiplicative graph

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 vertices of a directed graph, and each object has left and right identity morphisms, but composability is partial, and, moreover, associativity is not required.

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. The French school bases its definition of sketch on the notion of a multiplicative graph, because this definition required 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 studied enriched multiplicative graph. As a more general notion, there is the compositional graph, and multiplicative graphs can be seen as strongly identitive compositional graphs.