On the arrow category of partial semigroups

In the theory of semigroups of atomic charts, I described how categories are like partial semigroup homomorphisms from the partial semigroup of a category to the complete brandt semigroup of atomic charts. This is part of the partial algebraic theory of categories, which we can now extend here to deal with functors.

Definition. the category $PS$ of partial semigroups has two components:

• Objects: all maps $*: R \to X$ with $R \subseteq X^2$ that form partial semigroups so that if $(ab)c$ exists then so does $a(bc)$ and whenever they both exist they coincide $a(bc) = (ab)c$
• Morphisms: homomorphisms $f: (*_X: R \to X) \to (*_Y: S \to Y)$ defined by mappings of the form $f: X \to Y$ with the property that whenever $ab$ exists then $f(a)f(b)$ exists and $f(ab) = f(a)f(b)$

Then the arrow category of $PS$ is the standard arrow category defined in category theory: $PS^{\to}$. We will show that every category is associated to a partial semigroup homomorphism and every functor is associated to a morphism of partial semigroup homomorphisms in the arrow category. This leads to a functor: \[ A : Cat \to PS^{\to} \] This functor is the main way that we will form the partial algebraic theory of categories. In doing so, we will see that categories are just special types of actions in partial algebra.

Definition. the functor $A: Cat \to PS^{\to}$ has two components:

• the object part takes any category $C$ to a partial semigroup homomorphism $A(C): (Arrows(C),\circ) \to S_{Ob(C)}$ where $(Arrows(C),\circ)$ is the composition partial semigroup of the category and $S_{Ob(C)}$ is the complete brandt semigroup of partial transformations on the object set $Ob(C)$. Let $m: X \to Y$ be a morphism in $C$ then $A(C)(m)$ maps to the atomic action $(x,y) \in S_{Ob(C)}$.
• the morphism part takes any functor $F: C \to D$ to a morphism of partial semigroup homomorphisms $A(F) : A(C) \to A(D)$ which as a member of an arrow category has two components: (1) the arrow part of the functor $F$ which is a partial semigroup homomorphism from $(C,\circ)$ to $(D,\circ)$ and (2) a partial semigroup homomorphism of atomic partial transformation semigroups from $S_{Ob(C)}$ to $S_{Ob(D)}$ that maps $(x,y)$ to $(f(x),f(y))$.

This defines the object and morphism parts of the functor $A: Cat \to PS^{\to}$ and it describes how categories can be mapped to semigroup homomorphisms, while functors can be mapped to morphisms of partial semigroup morphisms but it doesn't definitively prove that this is a valid functor.

Theorem. the mapping $A: Cat \to PS^{\to}$ is a functor.

Proof. $A$ is a mapping from $Cat$ to $PS^{\to}$ which a functor to a morphism of partial semigroup homomorphisms. This forms a commutative diagram of the following form: Let $m : A \to B$ be an arrow in $C$ then by this commutative diagram we want to show that $A(F_M(m)) = F^*_O(A(m))$. In the first place $F(m) : F(A) \to F(B)$ so that $A(F(m)) = (F(A),F(B))$. On the other hand, $A(m) = (a,b)$ and $F(A(m)) = (f(a),f(b))$. It follows that this is a valid morphism of partial semigroups, so that $A : Cat \to PS^{\to}$ is a functor. $\square$

This demostrates the usefulness of the partial algebra construction, as every category can now be associated to partial semigroup homomorphism. We see that in general, all algebra should be done with a partial algebraic perspective in mind because that is how categories work. Categories are partially defined on composable morphisms, and so they are related to a number of interesting constructions in partial algebra.

References:
Semigroups of trivial charts