Download problem set 6

PHI324/MAT313 PROBLEM SET 6
Instructions: Complete three of the following exercises, including either #4 or #5. Due Monday, April 14
by 3:30pm.
Exercise 1. Given functors G : D → C and K : C → D and natural transformations : KG ⇒ 1D and
ρ : 1C ⇒ GK such that G · ρG = 1G : G ⇒ GKG ⇒ G. prove that K · Kρ : K ⇒ K is an idempotent in
DC .
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Exercise 2. Let hT, η, µi be a monad on C and let T -Alg be the category of T -algebras. Show that if T A → A
is an initial object in T -Alg then h is an isomorphism in C.
Exercise 3. Let U be the forgetful functor Monoids → Set.
(1) Briefly describe the left adjoint F to U . Hence describe the monad on Set induced by W0 = U F . In
particular, describe the unit and multiplication for this monad.
(2) Show that a W0 -algebra consists of the following data: a set M together with a string m0 , m1 , ... of
n-ary operations mn , where m0 : {∗} → M is to be thought of as the unit of the monoid M and mn
is to be thought of as the n-fold multiplication in the monoid M .
Exercise 4. Let A be a commutative ring, A-Alg the category of A-algebras and A-Mod the category of Amodules. Use Beck’s monadicity theorem to show that the forgetful functor U : A-Alg → A-Mod is monadic.
(NB: This exercise requires some non-trivial commutative algebra. In particular, the fact that the coequalizer
of two module homomorphisms f, g is the cokernel of the map f − g might come in handy.)
Exercise 5 (Ultrafilters as a monad). Recall that an ultrafilter F on a set X is a family of subsets of X
such that
(i) ∅ ∈
/F
(ii) If B ∈ F and B ⊆ A then A ∈ F
(iii) If A, B ∈ F then A ∩ B ∈ F
(iv) For every subset A of X, either A ∈ F or X \ A ∈ F
We call an ultrafilter F principal if there is an element x ∈ A such that x ∈ A for all A ∈ F . For any subset
of A ⊆ X let [A] denote the set of all ultrafilters on X that contain A. Show the following:
(1) For any set X let UX denote the set of ultrafilters on X. Show that U defines a functor Set → Set
(2) For any set X, take any F ∈ UUX (an ultrafilter of ultrafilters.) Show that the set µ(F) = {A|[A] ∈
F} is an ultrafilter.
(3) Show that U defines a monad on Set as follows: its unit ηX : X → UX sends an element of X to
the principal ultrafilter generated by that element and its multiplication µX : UUX → X is defined
as in part (2).
(4) Recall that a lattice homomorphism between (Boolean) P, Q is a map h : P → Q that preserves meets
(∧) and joins (∨). Show that ultrafilters on X correspond to lattice homomorphisms PX → 2 (with
2 regarded as the two element ordered Boolean algebra 0 → 1.) Briefly explain how this fact can be
used to characterize the adjunction that gives rise to the ultrafilter monad.
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