is there a universal property that characterises the join of two categories?

Question

Let A & B be two categories, the join A*B is created by stipulating its class of object is the disjoint union of the objects of A & B, the morphisms remain the 'same', but we throw in an extra morphism for every object a in A, and b in B.

that is:

A*B[a,a']=A[a,a'] if a,a' are in A

A*B[b,b']=B[b,b'] if b,b' are in B

A*B[a,b]=1 if a is in A, and b in B

A*B[b,a]=0 if b is in B, and a in A

it seems like a pretty ad-hoc construction (its obviously based on a construction coming from algebraic topology), is there a more categorical way of defining this?

Answer 1

It's a special case of what's called a collage or cograph construction. Recall that a profunctor or bimodule between categories $B$, $A$ is a functor $R: A^{op} \times B \to Set$. The cograph of $R$ is the category $\bar{R}$ where $Ob(\bar{R}) = Ob(A) \sqcup Ob(B)$, and where $\bar{R}(a, a') = A(a, a')$ if $a, a' \in Ob(A)$, $\bar{R}(b, b') = B(b, b')$ if $b, b' \in Ob(B)$, $\bar{R}(a, b) = R(a, b)$ if $a \in Ob(A), b \in Ob(B)$, and $\bar{R}(b, a) = \emptyset$ if $a \in Ob(A), b \in Ob(B)$. Compositions and identities are the obvious ones.

A cograph of the terminal object in the category of bimodules from $B$ to $A$ is the join of $A$ and $B$. In turn, the cograph of $R$ is a lax colimit, in the bicategory of small categories, bimodules, and bimodule homomorphisms of the diagram consisting just of $R$ itself. The nLab is a good source of information on this (as it is for many categorical questions).