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.