DUALITY AND DEFINABILITY IN FIRST ORDER LOGIC SECTION 1 5
where the upper horizontal part of the diagram is the same as (1), the lower part the same as (3),
u.
the vertical arrows are isomorphisms, and i and j are defined so that the corresponding
quadrangles commute, making i the transpose of i . By the naturality of e , two
quadrangles on the right commute, and also, f = j°i . Since F as a left adjoint preserves
colimits, e is the equalizer of the two arrows out of its codomain. Hence, j is an equalizer of
the two arrows out of its codomain as well, and the upper half is the coregular factorization of
f.
j is automatically injective (since the forgetful functor A—Set preserves
[]1.
equalizers), i is injective by Z3. Hence, by Z2., i is surjective.
As we said in the above proof, A
o p
is essentially a full subcategory of S. Since we
are proving something about the category A , we should ask to what extent the larger category
S is used in the proof. The answer is that the object 5 that appears in (2) is the one (and only
one) object possibly outside A that is used in the proof. Below, we apply the setup to Boolean
algebras as A ; this application will however be redundant in the sense that in that case, the
object S will, in fact, be in (the image of) A . On the other hand, in our main application of a
2-categorical version of the setup, (the analog of) the object 5 will definitely not (aside from
extreme cases for the initial data f: A B ) come from A .
2. Proposition. The category Boole of Boolean algebras is part of a Zawadowski
setup (Boole,...).
Proof (sketch). An ultrafilter {J, U) is a Boolean homomorphism 2 —2 , from the
ordinary J-th power 2 of the 2-element Boolean algebra 2 , to 2 itself. We let L be the
(large) similarity type consisting of all ultrafilters (J, U) , each (J, U) meant to be an J-ary
operation symbol. An algebra of type L is an entity S = { \S\ ,
{UK
' ) , .
f
) ,
with 151 a small set, and with cr : 151 151 , an J-ary operation on 151 for each
ultrafilter (J, 17) . S is defined to be the category whose objects are the algebras of type L ,
and whose arrows are the L-homomorphisms.
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