nLab
paracategory

Idea

A paracategory is a category where composition is only partially defined.

The definition is “unbiased” in that it comes with basic (partial) $n$ -ary composition operations for all $n$ , and unlike in the total case these cannot always be reduced to binary and nullary operations. A paracategory in which all the composites are generated from binary and nullary ones is sometimes called a precategory.

Definition

A paracategory is a quiver $C_{1} ⇉ C_{0}$ together with the following structure. We write $C_{n}$ for the iterated pullback $C_{1} \times_{C_{0}} \dots \times_{C_{0}} C_{1}$ , the set of length- $n$ strings of composable arrows.

Partial functions $\circ_{n} : C_{n} ⇀ C_{1}$ , over $C_{0} \times C_{0}$
$\circ_{0} : C_{0} \to C_{1}$ is total, so all identity arrows exist.
$\circ_{1} : C_{1} \to C_{1}$ is the identity.
If $\circ_{n} \overset{⇀}{y}$ is defined, then $\circ_{m + 1 + k} (\overset{⇀}{x}, \circ_{n} \overset{⇀}{y}, \overset{⇀}{z}) = \circ_{m + n + k} (\overset{⇀}{x}, \overset{⇀}{y}, \overset{⇀}{z})$ (the equality being Kleene equality).

A functor between paracategories is a quiver morphism $f : C \to D$ such that if $\circ_{n} \overset{⇀}{x}$ is defined, then so is $\circ_{n} \overset{⇀}{f (x)}$ and it equals $f (\circ_{n} \overset{⇀}{x})$ . A Kleene functor is a functor such that if $\circ_{n} \overset{⇀}{f (x)}$ is defined, then so is $\circ_{n} \overset{⇀}{x}$ ; this is equivalently a quiver morphism such that $\circ_{n} \overset{⇀}{f (x)} = f (\circ_{n} \overset{⇀}{x})$ is a Kleene equality.

Examples

If $D$ is any category and $C$ is a subclass of arrows in $D$ containing all the identities, then $C$ becomes a paracategory whose objects are the objects of $D$ , whose arrows are the arrows in $C$ , and where the composite of a string of arrows is defined iff the composite of that string in $D$ happens to lie in $C$ .
extranatural transformations and dinatural transformations form paracategories, since they are not always composable. This is exploited in the definition of extraordinary 2-multicategory?. This example can also be regarded as a case of the previous example, where the ambient category $D$ consists of “unnatural transformations.”
In fact, if ${Cat}_{P}$ denotes the category of categories equipped with a subclass of arrows containing the identities, then the functor ${Cat}_{P} \to ParCat$ defined above is actually a coreflection, i.e. it has a fully faithful left adjoint. In particular, any paracategory is isomorphic to one obtained from a class of arrows in some category, and moreover in a universal way.

Paracategories as generalized multicategories

Paracategories, and more general “partial algebras,” can be considered as a special case of generalized multicategories; see the papers of Hermida.

References

The definition is due to Peter Freyd in apparently unpublished work. It has been studied further in the papers:

Hermida and Mateus, “Paracategories I: Internal Paracategories and Saturated Partial Algebras”
Hermida and Mateus, “Paracategories II: Adjunctions, fibrations and examples from probabilistic automata theory”