5
$\begingroup$

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?

$\endgroup$
3
  • $\begingroup$ Could you give a citation as to where this is called the join? I'd like to know because this is not what I've been calling the join of two categories--I hope I'm not in trouble! $\endgroup$ Sep 22, 2012 at 19:51
  • $\begingroup$ @tanaka:thats the name they give it in [nlab](ncatlab.org/nlab/show/join+of+categories), but as Todd points out below its a special case of a collage or cograph. Out of interest what have you been calling it? $\endgroup$ Sep 23, 2012 at 3:38
  • 1
    $\begingroup$ This is the name Joyal calls it in his notes on quasi-categories, see for instance p. 26 of math.uchicago.edu/~may/IMA/JOYAL/JoyalDec08.pdf. There are some other notes by Joyal which deal with that construction in slightly more details, if I am not mistaken, but I cannot recall whether he states a universal property or not. Anyway, he uses it to define slices of quasi-categories. $\endgroup$ Sep 23, 2012 at 6:47

1 Answer 1

11
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.