1. BETHS THEOREM FOR PROPOSITIONAL LOGIC
In any category that is sufficiently complete and cocomplete (precisely, that has
pullbacks and coequalizers of equivalence relations), we may consider the regular factorization
of an arrow f:A—B as follows: take first
K
P
0
t A ^ ^ B ,
7 f
-B
the kernel-pair of f (that is, p- T i f is a pullback-square); then take
1
I
Po
\
K
^A
K
1 A
- ^ C ,
the quotient (coequalizer) of the pair (pQ, p- ) ; in the commutative diagram
i is given by the universal property of s ; the factorization f = i o s so obtained is the
regular factorization of f. It is determined up to isomorphism (in a straightforward precise
sense), since the ingredients used, all defined by universal properties, are so determined.
In Set , the category of (small) sets and functions, the regular factorization is the same
as the surjective/injective factorization. Many concrete categories (categories equipped with a
conservative functor to Set ) inherit this property from Set ; e.g., the ones that are monadic
over Set , regular-epi-reflective subcategories of the latter, etc. (the reason is the fact that epis
split in Set ).
We will talk about the coregular factorization of an arrow in a category when we mean
the regular factorization of the corresponding arrow in the opposite category. We will show that
Received by the editor February 11, 1991 and, in revised form, April 7, 1992.
Supported by NSERC Canada and FCAR Quebec.
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