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In [1], Wehrheim and Woodward construct a category from the "partially defined" Weinstein category of Lagrangian correspondences. This seems to be an instance of a more general construction. I would summarize it as follows:

Let's say that a partially defined category is exactly like a category, except that composition is only defined on some subsets of the morphisms $$\circ : S_{X,Y,Z} \subset \operatorname{Hom}(X, Y) \times \operatorname{Hom}(Y, Z) \to \operatorname{Hom}(X, Z).$$ Then we make a category out of that by taking the same objects, and the morphisms are now finite sequences of morphisms $(f_1, f_2, f_3, \ldots, f_n)$ up to the equivalence relation that two adjacent morphisms are combined if they are composable (e.g. $(f,g,h) \sim (g \circ f, h)$ if $g \circ f$ is defined). All compositions are now well-defined by concatenation.

Questions: Did this construction appear elsewhere? Does it have a name? Did anyone study a 2-category version of this?

[1] Wehrheim, K. and Woodward, C. T. Functoriality for Lagrangian correspondences in Floer theory. Quantum topology 1, no. 2 (2010): 129-170.

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  • $\begingroup$ In semigroup theory we call this the universal semigroup of a partial semigroup. However, you should state clearly what is your definition of a partial category. There are a number of ways to do it. Stallings gave an axiomatization of a pregroup and his notion of associativity is sufficiently strong to guarantee that the elements looks like sequences with the equivalence relation you described. But under weaker notions of associativity, you cannot do something so simple and it is in general undecidable if a partial semigroup embeds in its universal semigroup (under the most general notion). $\endgroup$
    – Benjamin Steinberg
    Commented May 19, 2023 at 15:11

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A reference is Hermida–Mateus's Paracategories I: Internal Paracategories and Saturated Partial Algebras. There the notion of partially-defined categories you describe appear as "precategories" and are subsumed by the more general notion of paracategory (which comes with partial $n$-ary composition operators for each natural number $n$). The forgetful functor from the category of small categories with distinguished morphisms to the category of small paracategories admits a fully faithful left adjoint (Proposition 2.1 ibid.), called the enveloping category: this is exactly the construction you describe.

More generally, the authors work in the context of paracategories and categories internal to a finitely complete category with a class of monomorphisms. It seems plausible one could recover a 2-categorical (or double categorical) version by working internal to $\mathrm{Cat}$.

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