Subject: categories: RE: co-

From: Fred E J Linton <FEJLINTON/[email-masked]>
Date: Wed, 08 Jul 1998 14:39:02 -0500 (EST)

Hi, all,

Jurgen asks:

> Why are certain categorical notions preferred over their dual
> counterparts?  E.g., hardly anyone talks about the Yoneda embedding
> of A into [A,Set]^op.

One point of my old (alas still unpublished) remarks "Sur les choix de variance
predestinees" was exactly why one "should" only see those Yoneda maps in the
forms  A ---> [A,Set]^op  -- and  A ---> [A^op, Set]  -- but no others (!). 
[First Ehresmann conf., Paris/Fontainebleau, 197?.] 

-- Fred

---

From: Vaughan Pratt <[email-masked]>
Date: Wed, 08 Jul 1998 21:04:34 -0700

From: Fred E J Linton <FEJLINTON/[email-masked]>
>[First Ehresmann conf., Paris/Fontainebleau, 197?.] 

Mildly apropos of this, the two maps can be rolled into one, in a sense,
to give the "bi-Yoneda embedding" F:C->Chu(Set,|C|) that I presented at
the Barrfest, where |C| denotes the set of arrows of C.

This embedding represents each object b of C as the Chu space F(b) =
(A,r,X), r:AxX->K, where A is the set of arrows f:a->b over all a,
X is the set of arrows h:b->c over all c, and r(f,h) = hf.

Each morphism g:b->b' is represented as the pair F(g) = (j,k) of functions
j:F_A(b)->F_A(b'), k:F_X(b')->F_X(b) defined by j(f) = gf, k(h) = hg
for each point f:a->b in F_A(b) and state h:b'->c in F_X(b').  (j,k) is
a Chu transform (= continuous, = satisfies the adjointness condition).

F is full, faithful, concrete with respect to U:C->Set defined by U(b)
{f:a->b} ("left" Yoneda), and co-concrete with respect to V:C->Set^op
defined by V(b) = {h:b->c} ("right" Yoneda) (or the other way round
depending on which way you're facing).

Regrettably this didn't go in the proceedings, being already committed
to TCS.

Vaughan Pratt