A protocategory is a way to present a category that makes formal the idea that a single datum (a “protomorphism”) can represent many different morphisms with different sources and targets. For instance, in material set theory, a given set of ordered pairs can represent functions with many different codomains.
As compared to the definition of category with one set of objects and one set of morphisms, a protocategory removes the requirement that each morphism have a unique source and target. As compared to the definition of category with separate hom-sets $\hom(A,B)$ for each pair of objects, a protocategory requires all these homsets to be (not necessarily disjoint) subsets of one set of all “protomorphisms”.
It is important to recognize, however, that unlike those two definitions of category, a protocategory is not a definition of a category. It retains information about “which morphisms in distinct hom-sets are ‘equal’”, which is not relevant category-theoretic information: in a true category it is never ‘meaningful’ to compare two morphisms for equality unless one already knows for other reasons that their domains and codomains coincide. A protocategory is rather a way of presenting a category, analogous to a group presentation, formalizing the fact that in some cases morphisms in different hom-sets can be “equal” until we “forget about that fact” in a structured way.
A protocategory consists of:
satisfying the axioms
Let the objects be the sets in material set theory (such as ZFC), and let the protomorphisms be sets $f$ of ordered pairs such that for any $x$ there is at most one $y$ such that $(x,y)\in f$. The domain of such an $f$ is the set of all $x$ such that there is such a $y$, and its range is the set of all $y$ such that there exists an $x$ with $(x,y)\in f$. We say $f:A\to B$ if $A$ is the domain of $f$ and $B$ is a superset of the range of $f$. With a suitable definition of composition, this yields a protocategory that generates the category Set.
More generally, if we take the protomorphisms to be all sets of ordered pairs, and $f:A\to B$ to mean that $A$ is a superset of the domain of $f$ in addition to $B$ being a superset of the codomain of $f$, then we get a protocategory that generates the category Rel.
Let the objects be groups, let the protomorphisms be functions, and say that $f:G\to H$ when $f$ is a function between the underlying sets of $G$ and $H$ that is a group homomorphism. This protocategory generates the category Grp.
Any category can be presented by a protocategory in a trivial way, by taking the protomorphisms to be the disjoint union of all the sets of morphisms (i.e. the set $C_1$ in the one-set-of-morphisms definition of category), with the source-target relation being simply a function from protomorphisms to pairs of objects.
It may seem that, at least in a material set theory, we should be able to modify the previous example by taking a non-disjoint union of the sets of morphisms (assuming given a category in the many-sets-of-morphisms definition). However, this is not true, because the composition predicate in a protocategory makes no reference to objects. For example, consider the following category:
Then if we take unions of homsets we have $g\circ f = h$ and also $g\circ f = h'$, but then the protocategory composition axiom fails: we have $f:0\to 1$ and $g:1\to 2$, but there does not exist a unique $k$ such that $k:0\to 2$ and $g \circ f = k$.
The notion is due to Freyd and Scedrov, Categories, Allegories.
It is recalled in A1.1 of Sketches of an Elephant.
Last revised on April 16, 2018 at 05:34:17. See the history of this page for a list of all contributions to it.