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UT creates ‘UT -special’ coequalisers. In fact this property characterises monadicity. Hence we arrive at our first attempt at a monadicity theorem:  G is monadic iff G creates G-special coequalisers. Look more closely at (). We want D to be like CT . e. objects of the form FX. e. that K is full and faithful. e. that K is essentially surjective. So does K hit all of the coequalisers? That is, can we find something in D which goes to each coequaliser? Well, if D has all the “special coequalisers” and G preserves them, then we can lift along U T , so seeing that K sends it to the right place.

To prove this, we shall first prove a series of propositions.  UT : C T C creates coequalisers for all UT -absolute-coequaliser pairs.  f A UT -absolute-coequaliser pair is a pair of morphisms A TA Tf Tg B such that TB ϕ θ A f “serially commutes”, and such that A g g f B g B has an absolute coequaliser A f g e B C in C. We aim to show that there is a unique lift to a fork TA Tf Tg TB Te TC ϕ θ f A g B ψ e C in CT , and that it is a coequaliser in CT .  Induce unique ψ by the universal property of coequaliser; the bottom fork is an absolute coequaliser, hence preserved by T; so the top fork is also a coequaliser.

E. ef = eg) with a splitting f A g e B C s t such that es = 1C , ft = 1B and gt = se.  A split coequaliser is a coequaliser.  Suppose we have a fork A exists a unique C k f g B h D, say, so that hf = hg. k h D commutes. Now consider hs : C D. We have hse = hgt = hft =h so hs certainly makes the diagram commute. And suppose k is any other such; then ke = h = hse ⇒ kes = hses  ⇒ k = hs so hs is the unique such.  An absolute coequaliser is a coequaliser that is preserved as a coequaliser by any functor.

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