THE MONADICITY THEOREM

BEN ELIAS AND ALEX ELLIS

Consider two R E  B. I ∼ I and R are adjoint if HomE (IM,N) = HomB(M,RN) naturally in M and N. Examples: forget (1) Grp  Set. free Res (2) A -mod  B -mod for B ⊂ A. Ind f∗ f (3) Shv(Y )  Shv(X) for X ← Y . f ∗ ∼ Equivalent description: fro all N ∈ Ob(E) we have HomB(RN,RN) = Hom(IRN,N). Looking at the of the identity, we get a counit

 : IR → 1E . ∼ Similarly, by looking at the image of the identity under HomE (IM,IM) = HomB(M,RIM) we get the unit

η : 1B → RI. They satisfy certain snake diagrams. Examples: forget (1) Grp  Set. In this case IR(G) is the free on elements free of G. Then the counit IR(G) → G is given by multiplication. However, there is no functorial map G → IR(G). Similarly, RI(X) are words in x, x−1. Then the map X → RI(X) is the inclusion of singletons. Res (2) A -mod  B -mod. The counit IR → 1 is the multiplication Ind A ⊗B A → A (this is a map of A-bimodules). The unit B → A is the inclusion (the morphism as B-bimodules). Definition. A on C is an endofunctor T : C → C equipped with the multiplication T ◦ T → T and the unit 1 2 BEN ELIAS AND ALEX ELLIS

η : 1 → T satisfying the associativity and right and left unit axioms. Dually, one can define a comonad. Definition. An over the monad T is an object M ∈ C equipped with the action map T ◦ M → M satisfying the associativity and the unit axioms. The monad of an adjunction: let (I,R) be an adjunction. Here R : E → B and I : B → E. Adjunction gives the counit  and unit η. Define a monad in B by T = RI. The unit for the adjunction η gives the unit η for the monad. The multiplication map µ : T ◦ T → T comes from RIRI → RI, where we used the counit IR → 1. One can check that the unit and associativity axioms follow from the snake diagrams for the adjunction. Given a monad (T, η, µ) in C define CT be the of T - in C and functors IT : C → CT and RT : CT → C. Data of a T -algebra is (X, h), where X ∈ C and h is a morphism T x → x. Then IT (x) = (T X, µx) and RT (x, h) = x. Theorem 1. The functors (IT ,RT ) form an adjunction and the result- ing monad is (T, η, µ). Consider T E  B  B , where T is the monad coming from E  B. T Theorem 2. Let B  B be the adjunction of the previous theorem, where T is the monad of B  E. Then there is a unique E →K BT , such that RT K = R and KI = IT . K is called the comparison functor. BT is final among all categories which are adjoint to B in a way giving rise to T . Note, there is also the initial category called the BT . Definition. We say R : E → B is monadic if there is an adjunction (I,R) with the corresponding comparison functor an equivalence. Given an object x ∈ E we have a map T Rx = RIRx → RX. This gives the T -algebra structure on Kx. Key trick: TTX ⇒ TX → X. THE MONADICITY THEOREM 3

The two maps are given either by multiplication in the monad or the action map. We also have the maps X → TX and TX → TTX coming from the unit map in the monad.

a e Definition. A fork is M ⇒ N → P with ea = eb. b Forks form a category. is the initial object of that cate- gory (i.e. a colimit). Definition. Coequalizer is absolute if FM ⇒ FN → FP is coequal- izer for any F . Definition. Coequalizer is split if we have maps P → N → M which split the maps. Note, that split are absolute. φ Definition. F creates coequalizers if FM FN → X is a coequalizer, e ⇒ then there is a unique map M ⇒ N → P , such that F sends it to the previous diagram and P is a coequalizer. Example. Consider M ∈ E. Then we have IRIRM ⇒ IRM → M. Although we have a splitting IRM → IRIRM, we don’t have a map M → IRM. However, if we apply R we get TTRM ⇒ TRM → RM, which is split. So, if R creates coequalizers, then M is a coequalizer. Theorem 3 (Barr-Beck). The following are equivalent: (1) E →K BT is an equivalence. (2) E →R B creates coequalizers for (a) absolute (b) split. Proof. 1 ⇒ 2. Just check. 2b ⇒ 1 Construct K−1. If we apply R to IRIRX ⇒ IX, then we get TTX ⇒ TX → X. −1 Since R is split, we get IRIRX ⇒ IX → K (X).  Exercise: consider the adjunction between semigroups and sets. Un- wind what SetT is and convince yourself that Barr-Beck holds. discrete Non-example: B = Set  . Then R is not monadic. forget Suppose f : Y → X is an open covering, i.e. Y = tiUi. f ∗ ∗ Consider the adjunction Shv(X)  Shv(Y )..Then f preservers col- f∗ imits and f∗ preserves limits. 4 BEN ELIAS AND ALEX ELLIS

Then the category of data is equivalent to T Shv(Y ) (coalge- bras for T ). By the Barr-Beck, we get an equivalence T Shv(Y ) ∼= Shv(X).