The definition is “unbiased” in that it comes with basic (partial) -ary composition operations for all , 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.
A paracategory is a quiver together with the following structure. We write for the iterated pullback , the set of length- strings of composable arrows.
Partial functions , over
is total, so all identity arrows exist.
is the identity.
If is defined, then (the equality being Kleene equality).
A functor between paracategories is a quiver morphism such that if is defined, then so is and it equals . A Kleene functor is a functor such that if is defined, then so is ; this is equivalently a quiver morphism such that is a Kleene equality.
If is any category and is a subclass of arrows in containing all the identities, then becomes a paracategory whose objects are the objects of , whose arrows are the arrows in , and where the composite of a string of arrows is defined iff the composite of that string in happens to lie in .
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 consists of “unnatural transformations.”
In fact, if denotes the category of categories equipped with a subclass of arrows containing the identities, then the functor 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, and more general “partial algebras,” can be considered as a special case of generalized multicategories; see the papers of Hermida.
The definition is due to Peter Freyd in apparently unpublished work.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”