By Julia Goedecke

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Example: Consider Set lr has Set lr ❑ P, Top . The monad this induces on Set is the identity monad, which D U 1 ❑ P, Set as its Eilenberg-Moore adjunction. 1 But the Eilenberg-Moore adjunction is a terminal object in the category of adjunctions inducing T. We will make this more precise. Definition: Given a monad T ✏ ♣T, η, µq on C , let Adj♣Tq be the category whose objects are ✶ D D D L L L adjunctions F ✪ G inducing the monad T, and whose morphisms F ✪ G ÝÑ F ✶ ✪ G✶ are functors C C C H : D ÝÑ D ✶ such that HF ✏ F ✶ and G✶ H ✏ G.

Let f : A ÝÑ B be a morphism in A . Let k : K 1 ,P 1 ,P I be the cokernel of k. Then as f k ✏ 0, we have p: A 1 K P, k f ,P A v 5a,P B vvv rrr r v r p A! r i I ,P A be the kernel of f and 56 5. ABELIAN CATEGORIES We will show that i is monic by showing that ix ix ✏ 0. ☎ l yÔ 1 K ,P X ✏ 0 implies x ✏ 0. So consider x : X ÝÑ I with h k x ,P A ,P B ~9h L ~ i ~ ~ p r ~~~~ c 1 ,P P, I dl f ☎ s We get a unique r such that r coker x ✏ i. Now as both p and c ✏ coker x are epis, cp is an epi and so the cokernel of some h.

So if f is a mono and an epi, it is a regular mono and an epi and so an iso (Proposition 8 in Section 2C). 41 Lemma: (“Preadditive equalisers via kernels”) Let A be preadditive. Then the pair A f g ,P,P B has an equaliser iff the kernel of f ✁ g exists, and then they coincide. Proof. The equaliser of f and g and the kernel of f given h : C ÝÑ A, we have f h ✏ gh ô ♣f ✁ g qh ✏ 0. ✁ g have the same universal property: Notice that in general normal ñ regular ñ strong ñ mono. This lemma shows that in a preadditive category, normal ô regular; and in an abelian category we have normal ô mono, so all steps coincide.

### Category Theory [Lecture notes] by Julia Goedecke

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