This page used to be bigger, but I've chopped it up. This list is an index to the resulting fragments.

Early summer, 1998: I've organised that lot enough that I can state and prove the Yoneda Lemma. This pleases me.

The following stray and garbled section is waiting for me to work out where to put it, which will probably involve a rewrite !.

Cartesian Products and Disjoint Unions

Note: this uses wrong namings.

As an illustration of duality, consider now the definition of a Cartesian product, which I shall not state; instead, I shall define its dual, the disjoint union and leave the reader to infer the definition of the Cartesian product.

A pair of arrows, (f, h), in a category are said to form a disjoint union precisely if their values under Post are equal and, whenever (d, j) is a pair of arrows having Post(d) = Post(j) with Prior(d) contained in Prior(f) and Prior(j) contained in Prior(h) [this will be clearer to readers who draw the picture], there is a unique arrow d+j in Post(f) with Post(d+j) = Post(d), which is equal to Post(j), for which (d+j) o f = d and (d+j) o h = j. In particular, f+h is an identity composable before each of f, h.

When we have such a pair with identities e and c in Prior(f) and Prior(h), respectively, and a unique identity in Post(f) (which is Post(h)), it is conventional (by a slight but unambiguous abuse of notation) to call this identity e+c. Likewise, it is usual to refer to f and h as embeddings f = ie and h = ic. Strictly, e+c should be denoted ie+ic.

For Cartesian products, the operator + is replaced by × and the pair of arrows forming the product are known as projections πe and πc, analogously to embeddings; ie the dual of + is × and that of i is π.

Maintained by Eddy.