Monads T T T T T. Dually (by inverting the arrows in the given definition) one can define the notion of a comonad. T T

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1 Monads Definition A monad T is a category C is a monoid in the category of endofunctors C C. More explicitly, a monad in C is a triple T, η, µ, where T : C C is a functor and η : I C T, µ : T T T are natural transformations which make the following diagrams commute: T T T µt T T T µ µ T T µ T T ηt T T id T T η id T Dually (by inverting the arrows in the given definition) one can define the notion of a comonad.

2 Monads Definition A monad T is a category C is a monoid in the category of endofunctors C C. More explicitly, a monad in C is a triple T, η, µ, where T : C C is a functor and η : I C T, µ : T T T are natural transformations which make the following diagrams commute: T T T µt T T T µ µ T T µ T T ηt T T id T T η id T Dually (by inverting the arrows in the given definition) one can define the notion of a comonad.

3 Monads Definition A monad T is a category C is a monoid in the category of endofunctors C C. More explicitly, a monad in C is a triple T, η, µ, where T : C C is a functor and η : I C T, µ : T T T are natural transformations which make the following diagrams commute: T T T µt T T T µ µ T T µ T T ηt T T id T T η id T Dually (by inverting the arrows in the given definition) one can define the notion of a comonad.

4 Monads 1) Trivial monad In any category C the triple I C, id, id forms a monad. 2) Topological closure The functor T : Top Top defined by T (X) = X together with the natural transformations η : I T, η X : X X µ : T T T, µ X = id X forms a monad.

5 Monads 1) Trivial monad In any category C the triple I C, id, id forms a monad. 2) Topological closure The functor T : Top Top defined by T (X) = X together with the natural transformations η : I T, η X : X X µ : T T T, µ X = id X forms a monad.

6 Monads 3) The free category monad Let Graphs be a category of oriented graphs and morphisms between them. There is a functor Graphs SmallCats which sends a graph Γ to its free category F Γ

7 Monads 3) The free category monad Let Graphs be a category of oriented graphs and morphisms between them. There is a functor Graphs SmallCats which sends a graph Γ to its free category F Γ morphisms in F Γ = edges of Γ objects in F Γ = vertices of Γ; + identity morphisms + arrows induced by finite paths in Γ.

8 Monads 3) The free category monad Let Graphs be a category of oriented graphs and morphisms between them. There is a functor Graphs SmallCats which sends a graph Γ to its free category F Γ and the forgetful functor SmallCats Graphs. The composition F of these two functors yields a monad in Graphs.

9 Monads 3) The free category monad Let Graphs be a category of oriented graphs and morphisms between them. There is a functor Graphs SmallCats which sends a graph Γ to its free category F Γ and the forgetful functor SmallCats Graphs. The composition F of these two functors yields a monad in Graphs. On X Graphs the natural transformation η : I F is the injection X F (X) and µ X : (F F )(X) F (X) evaluates the paths compositions.

10 Algebras over monads Definition An algebra over a monad T = T, η, µ (or simply a T -algebra) in a category C consists of an object X C and a morphism h : T X X such that the following diagrams commute: (T T )X µ X T X T h h T X h X X η X T X id h X

11 Algebras over monads 1) Any object X C is an algebra over the trivial monad in C. 2) If T is a monad in C and X C, then T X is automatically a T -algebra. 3) Any closed topological space is an algebra over the topological closure monad.

12 Algebras over monads 1) Any object X C is an algebra over the trivial monad in C. 2) If T is a monad in C and X C, then T X is automatically a T -algebra. 3) Any closed topological space is an algebra over the topological closure monad.

13 Algebras over monads 1) Any object X C is an algebra over the trivial monad in C. 2) If T is a monad in C and X C, then T X is automatically a T -algebra. 3) Any closed topological space is an algebra over the topological closure monad.

14 Algebras over monads 4) Let G be a group. We define the left G-action monad T = T, η, µ in Sets as follows: T (X) = G X η X :X G X x (e, x) µ X :G (G X) G X (g 1, (g 2, x)) (g 1 g 2, x). Then a T -algebra is a set Y with the map h : G Y Y satisfying certain compatibility conditions. It turns out that a T -algebra Y is simply a set with a left G-action.

15 Algebras over monads 4) Let G be a group. We define the left G-action monad T = T, η, µ in Sets as follows: T (X) = G X η X :X G X x (e, x) µ X :G (G X) G X (g 1, (g 2, x)) (g 1 g 2, x). Then a T -algebra is a set Y with the map h : G Y Y satisfying certain compatibility conditions. It turns out that a T -algebra Y is simply a set with a left G-action.

16 Algebras over monads 4) Let G be a group. We define the left G-action monad T = T, η, µ in Sets as follows: T (X) = G X η X :X G X x (e, x) µ X :G (G X) G X (g 1, (g 2, x)) (g 1 g 2, x). Then a T -algebra is a set Y with the map h : G Y Y satisfying certain compatibility conditions. It turns out that a T -algebra Y is simply a set with a left G-action.

17 Adjoint functors produce monads Let F, G, η, ɛ be an adjuction C G F D. By definition, for the unit η : 1 D GF and counit ɛ : F G 1 C transformations, the compositions F F η F GF ɛf F G η GF G Gɛ G are the identity transformations 1 F and 1 G respectively.

18 Adjoint functors produce monads The composition T = GF is an endofunctor in D and there is a natural transformation from T T to T : µ = GɛF : GF GF GF Moreover, the following diagrams commute: GF GF GF GF GɛF GF GF GɛF GF GɛF GF GF GɛF GF ηgf I C GF GF GF id GF GF η id GF I D Thus, T, µ, η is a monad in D.

19 Adjoint functors produce monads The composition T = GF is an endofunctor in D and there is a natural transformation from T T to T : µ = GɛF : GF GF GF Moreover, the following diagrams commute: GF GF GF GF GɛF GF GF GɛF GF GɛF GF GF GɛF GF ηgf I C GF GF GF id GF GF η id GF I D Thus, T, µ, η is a monad in D.

20 Adjoint functors produce monads The composition T = GF is an endofunctor in D and there is a natural transformation from T T to T : µ = GɛF : GF GF GF Moreover, the following diagrams commute: GF GF GF GF GɛF GF GF GɛF GF GɛF GF GF GɛF GF ηgf I C GF GF GF id GF GF η id GF I D Thus, T, µ, η is a monad in D.

21 Adjoint functors produce monads Remarks 1) In a similar way, from an adjunction C construct a comonad S, ɛ, δ in the category C. G F D one can 2) In fact, any (co-)monad is induced by a certain adjunction. This is a theorem due to Eilenberg and Moore [1965].

22 The universal property of We equip the simplicial category with a monoidal structure as follows. Let the monoidal product in be the bifunctor + :, which sends a pair ([m], [n]) to [n + m] and a pair of morphisms f : [n] [n ], g : [m] [m ] to the morphism { f(i), 0 i n 1 (f + g)(i) = n + g(i n), n i n + m 1. The unit object in is [0]. The object [1] together with the (unique) morphisms µ : [2] [1] and η : [0] [1] is a monoid [1], µ, η in. Theorem Let B,, I be a (strictly) monoidal category and C, µ, η be a monoid in B. Then there exists a unique morphism of monoidal categories F :, +, 0 B,, I which sends the monoid [1], µ, η to C, µ, η.

23 The universal property of We equip the simplicial category with a monoidal structure as follows. Let the monoidal product in be the bifunctor + :, which sends a pair ([m], [n]) to [n + m] and a pair of morphisms f : [n] [n ], g : [m] [m ] to the morphism { f(i), 0 i n 1 (f + g)(i) = n + g(i n), n i n + m 1. The unit object in is [0]. The object [1] together with the (unique) morphisms µ : [2] [1] and η : [0] [1] is a monoid [1], µ, η in. Theorem Let B,, I be a (strictly) monoidal category and C, µ, η be a monoid in B. Then there exists a unique morphism of monoidal categories F :, +, 0 B,, I which sends the monoid [1], µ, η to C, µ, η.

24 The universal property of We equip the simplicial category with a monoidal structure as follows. Let the monoidal product in be the bifunctor + :, which sends a pair ([m], [n]) to [n + m] and a pair of morphisms f : [n] [n ], g : [m] [m ] to the morphism { f(i), 0 i n 1 (f + g)(i) = n + g(i n), n i n + m 1. The unit object in is [0]. The object [1] together with the (unique) morphisms µ : [2] [1] and η : [0] [1] is a monoid [1], µ, η in. Theorem Let B,, I be a (strictly) monoidal category and C, µ, η be a monoid in B. Then there exists a unique morphism of monoidal categories F :, +, 0 B,, I which sends the monoid [1], µ, η to C, µ, η.

25 The universal property of We equip the simplicial category with a monoidal structure as follows. Let the monoidal product in be the bifunctor + :, which sends a pair ([m], [n]) to [n + m] and a pair of morphisms f : [n] [n ], g : [m] [m ] to the morphism { f(i), 0 i n 1 (f + g)(i) = n + g(i n), n i n + m 1. The unit object in is [0]. The object [1] together with the (unique) morphisms µ : [2] [1] and η : [0] [1] is a monoid [1], µ, η in. Theorem Let B,, I be a (strictly) monoidal category and C, µ, η be a monoid in B. Then there exists a unique morphism of monoidal categories F :, +, 0 B,, I which sends the monoid [1], µ, η to C, µ, η.

26 Simplicial objects from comonads. Let T be a monad in D, that is, T is a monoid in the category D D. By the previous theorem, there is a unique morphism D D which sends [1] to T. Thus, a monad T gives a cosimplicial object in D D. By duality, a comonad L, δ, ɛ in D yields an augmented simplicial object op D D, which we denote by Smp(L): where I ɛ s 0 s 0 s 1 L d L 2 0 d L 3 0 L 4... d 1 d 1 d 2 d n i = L i ɛl n i : L n+1 L n s n i = L i δl n i 1 : L n L n+1

27 Simplicial objects from comonads. Let T be a monad in D, that is, T is a monoid in the category D D. By the previous theorem, there is a unique morphism D D which sends [1] to T. Thus, a monad T gives a cosimplicial object in D D. By duality, a comonad L, δ, ɛ in D yields an augmented simplicial object op D D, which we denote by Smp(L): where I ɛ s 0 s 0 s 1 L d L 2 0 d L 3 0 L 4... d 1 d 1 d 2 d n i = L i ɛl n i : L n+1 L n s n i = L i δl n i 1 : L n L n+1

28 Simplicial objects from comonads. Let T be a monad in D, that is, T is a monoid in the category D D. By the previous theorem, there is a unique morphism D D which sends [1] to T. Thus, a monad T gives a cosimplicial object in D D. By duality, a comonad L, δ, ɛ in D yields an augmented simplicial object op D D, which we denote by Smp(L): where I ɛ s 0 s 0 s 1 L d L 2 0 d L 3 0 L 4... d 1 d 1 d 2 d n i = L i ɛl n i : L n+1 L n s n i = L i δl n i 1 : L n L n+1

29 Simplicial objects from comonads. Now, if D is an additive category (or there is a functor F from D D to an additive category), then for any X D, the simplicial identities would imply that the alternating sums n = d n 0 d n ( 1) n d n n : L n+1 (X) L n+1 (X) satisfy n n+1 = 0. That gives a chain complex L(X) L 2 (X)... L n (X)... with the augmentation ɛ X : L(X) X. This complex is called the standard or bar resolution of X.

30 Simplicial objects from comonads. Now, if D is an additive category (or there is a functor F from D D to an additive category), then for any X D, the simplicial identities would imply that the alternating sums n = d n 0 d n ( 1) n d n n : L n+1 (X) L n+1 (X) satisfy n n+1 = 0. That gives a chain complex L(X) L 2 (X)... L n (X)... with the augmentation ɛ X : L(X) X. This complex is called the standard or bar resolution of X.

31 Simplicial objects from comonads. Now, if D is an additive category (or there is a functor F from D D to an additive category), then for any X D, the simplicial identities would imply that the alternating sums n = d n 0 d n ( 1) n d n n : L n+1 (X) L n+1 (X) satisfy n n+1 = 0. That gives a chain complex L(X) L 2 (X)... L n (X)... with the augmentation ɛ X : L(X) X. This complex is called the standard or bar resolution of X.

32 Simplicial objects from comonads. Adjoint pair (Co)monad Bar-resolution Let G be a group and G Mod be the category of left G-modules. Step 1. The forgetful functor U : G Mod Ab has a left adjoint: U G Mod ZG Ab

33 Simplicial objects from comonads. Adjoint pair (Co)monad Bar-resolution Let G be a group and G Mod be the category of left G-modules. Step 1. The forgetful functor U : G Mod Ab has a left adjoint: U G Mod ZG Ab

34 Group cohomology Step 2. The composition G Mod Ab G Mod induces a comonad L, δ, ɛ in G Mod, where L = ZG U( ) and ɛ A : ZG U(A) A x a xa δ A : ZG U(A) ZG U(ZG U(A)) x a x 1 a

35 Group cohomology Step 3. The category G Mod is additive. Consider Z as the trivial G-module. Then ZG U(Z) ZG and the simplicial object Smp(L)(Z) looks like: s 0 s 0 s 1 ZG ZG 2 d ZG 3 0 ZG d 1 d 1 d 2 Using this resolution, for a G-module A, we can compute the right derived functor of Hom G Mod (, A). That what is called the cohomology of the group G.

36 Group cohomology Step 3. The category G Mod is additive. Consider Z as the trivial G-module. Then ZG U(Z) ZG and the simplicial object Smp(L)(Z) looks like: s 0 s 0 s 1 ZG ZG 2 d ZG 3 0 ZG d 1 d 1 d 2 Each tensor power ZG n+1 is generated by the elements of the form x x 1 x n = x[x 1... x n ], where x, x 1,..., x n G. Using this resolution, for a G-module A, we can compute the right derived functor of Hom G Mod (, A). That what is called the cohomology of the group G.

37 Group cohomology Step 3. The category G Mod is additive. Consider Z as the trivial G-module. Then ZG U(Z) ZG and the simplicial object Smp(L)(Z) looks like: s 0 s 0 s 1 ZG ZG 2 d ZG 3 0 ZG d 1 d 1 d 2 On the generators the face maps act as xx 1 [x 2... x n ], i = 0 d i (x[x 1... x n ]) = x 1 [x 2... x i x i+1... x n ], 0 < i < n x[x 1... x n 1 ], i = n Using this resolution, for a G-module A, we can compute the right derived functor of Hom G Mod (, A). That what is called

38 Group cohomology Step 3. The category G Mod is additive. Consider Z as the trivial G-module. Then ZG U(Z) ZG and the simplicial object Smp(L)(Z) looks like: s 0 s 0 s 1 ZG ZG 2 d ZG 3 0 ZG d 1 d 1 d 2 The alternating sums of d i are the boundary operators of the augmented chain complex Z ZG ZG 2 (X)... ZG n (X)... This is the standard or bar resolution of the trivial G-module Z. Using this resolution, for a G-module A, we can compute the right derived functor of Hom G Mod (, A). That what is called

39 Group cohomology Step 3. The category G Mod is additive. Consider Z as the trivial G-module. Then ZG U(Z) ZG and the simplicial object Smp(L)(Z) looks like: s 0 s 0 s 1 ZG ZG 2 d ZG 3 0 ZG d 1 d 1 d 2 Using this resolution, for a G-module A, we can compute the right derived functor of Hom G Mod (, A). That what is called the cohomology of the group G.