Does anyone know if there exists a name in the literature for the data of

1) a class of objects,

2) for each pair of objects $(x, y)$ a set $hom(x, y)$

3) for each triple of objects $(x, y, z)$ a morphism of sets $hom(x, y) \times hom(y, z) \to hom(x, z)$.

I don't impose any conditions on this data (if I were to impose the usual associativity and identity axioms this would be the definition of a category).

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    $\begingroup$ If there's only one object this is a magma. I'd call it magma with several objects. $\endgroup$ Dec 3 '11 at 1:17
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    $\begingroup$ Magmoid? $\endgroup$ Dec 3 '11 at 1:32
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    $\begingroup$ @name: Could you provide a little bit more context? Why do you consider this structure? Where does it appear? What are typical examples? Thanks. @Darij: 1+ :D $\endgroup$ Dec 3 '11 at 11:05
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    $\begingroup$ @name: If you drop composition then you have what Mac Lane calls a precategory (and everyone else, a multidigraph). If you have composition and identities then you have what Lambek and Scott calls a deductive system. $\endgroup$
    – Zhen Lin
    Dec 3 '11 at 12:20
  • $\begingroup$ @Martin: Hi Martin, I'm in the process of defining a category, and to prove that the composition is associative I want to use a lemma that is most clearly stated and proved in the context of these kinds of structures. $\endgroup$
    – name
    Dec 3 '11 at 17:16

As Zhen Lin already noted in his comment, this is called a deductive system in section I.1 of

J. Lambek, P. J. Scott, Introduction to higher order categorical logic

I think that the idea is the following: We treat a morphism $A \to B$ as a deduction from $A$ to $B$. The identity morphism is the trivial deduction $A \to A$ and the composition $A \to B$, $B \to C$ $\leadsto$ $A \to C$ is a rule of inference, namely the hypothetical syllogism.


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