Introduction to extensive and distributive categories
Aurelio Carboni, Stephen Lack, and R.F.C. Walters
This appeared in Journal of Pure and
Applied Algebra 84:145-158, 1993.
In recent years, there has been considerable discussion as to the
appropriate definition of distributive categories. Three definitions
which have had some support are:
- A category with finite sums and products such that the canonical
map from AxB+AxC to Ax(B+C) is an isomorphism (Walters).
- A category with finite sums and products such that the canonical
functor + from
A/A x A/B to A/(A+B)
is an equivalence. (Monro)
- A category with finite sums and finite limits such that the
canonical functor of (2) is an equivalence. (Lawvere-Schanuel)
There has been some confusion as to which of these was the natural
notion to consider. This resulted from the fact that there are
actually two elementary notions being combined in the above three definitions.
The first, to which
we give the name distributivity, is exactly that of (1).
The second notion, which we shall call extensivity, is that
of a category with finite sums for which the canonical functor + of
definitions (2) and (3) is an equivalence. Extensivity, although it
implies the existence of certain pullbacks, is essentially a property
of having well-behaved sums. It is the existence of these pullbacks
which has caused the confusion.
The connections between definition (1) and definitions
(2) and (3) above are that any extensive category with products is
distributive in the first sense, and that any category satisfying (3)
satisfies (1) locally.
The purpose of this paper is to
present some basic facts about extensive and distributive categories,
and to discuss the relationships between the two notions.
Steve Lack
Last modified: Fri Aug 30 09:13:53 EST 2002