By Joachim Lambek

Lecture notes in arithmetic No.24

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T (HoF)/y]. [H(A), T], by Proposition F/y]. H preserves (2). Let U 9 oA -+ Consider ~ [H(A), H(F(y))] t(y) goes to It is easily verified that F/y] is if the latter exists. That it does exist Under this isomorphism, of [ ~ , = T. 1. Therefore, We must still show that any as in Section 2. [T, H -] -~ T in [A, Ens] ~ Assuming this for the moment, Ens T . 1 below). Moreover, inf all very proper A to Ens. and diagram Now the functor infs, We shall temporarily for the opposite c a t e g o r y of inf-preserving inf (3).

From this we deduce LEMMA complete objects. 1. ) object mg = he, following: has m exists Assume that A a representative an e x t r e m a l a unique g d is inf- set of sub- monomorphism such that and m d = h. > J e d 4 ~ J f / f j h Proof L decompositions. canonical g = gmg e Let Then x and h = hmh e (mg m) ge = h m (hee) decompositions an i s o m o r p h i s m '> such of that fo be the c a n o n i c a l = f, say, By u n i q u e n e s s , Xge = hee and are there two exists h m X = mg m. - Take d d = -mU x -1 he, then 46 - Since md = h.

J e d 4 ~ J f / f j h Proof L decompositions. canonical g = gmg e Let Then x and h = hmh e (mg m) ge = h m (hee) decompositions an i s o m o r p h i s m '> such of that fo be the c a n o n i c a l = f, say, By u n i q u e n e s s , Xge = hee and are there two exists h m X = mg m. - Take d d = -mU x -1 he, then 46 - Since md = h. m is mono, is unique with this property. > L ~[ gm I -> x I I m ! > m h PROPOSITION is also sup-complete CASE I. consisting of A O A Let CASE 2. A_ be inf-complete. _A A contains two cases: subcategory Ao, and arbitrary sums a generator A.

### Completions of Categories by Joachim Lambek

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4.2