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Category Theory — 80-413/713 Homework #12 Due Wednesday 1:30pm, April 22

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Category Theory — 80-413/713 Homework #12 Due Wednesday 1:30pm, April 22
Category Theory — 80-413/713
Homework #12
Due Wednesday 1:30pm, April 22
Starred problems are for students enrolled in 80-713
1. (Kleisli category) Given a monad (T, η, µ) on a category C, in addition
to the Eilenberg-Moore category, we can construct another category CT
and an adjunction F a U , η : 1 → U F , ε : F U → 1 with U : CT → C
such that T = U F , the unit of the adjunction equals the unit η of the
monad, and µ = U (εF ). This category CT is called the Kleisli category
of the monad, and is defined as follows:
• the objects are the same as those of C,
• an arrow f : A →T B in CT is an arrow f : A → T B in C,
• the identity arrow 1TA : A →T A in CT is the unit arrow ηA : A →
T A in C,
• for composition, given f : A →T B and g : B →T C, the composite
g ◦T f : A →T C is defined to be the composite
µC ◦ T (g) ◦ f
in C as indicated in the following diagram:
g◦T f
A
TC
µC
f
TB
T (g)
T 2C
Verify that this indeed defines a category, and that there are adjoint
functors F : C → CT and U : CT → C giving rise to the monad
T = U F , as claimed.
2. Let P : Sets → Sets be a finitary polynomial functor,
P (X) = C0 + C1 × X + C2 × X 2 + · · · + Cn × X n
(thus the coefficient sets Ci , i = 0, . . . , n, are finite). Show that P
preserves ω-colimits.
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3. The notion of a coalgebra for an endofunctor P : S → S on an arbitrary
category S is exactly dual to that of a P -algebra. Determine the final
coalgebra for the endofunctor
P (X) = 1 + A × X
on Sets, parametrized by a set A. (Hint: recall that the initial algebra
consists of finite lists over A.)
4. (∗) Prove that a (strict) monadic functor U : D → C creates coequalizers
of U -split pairs, where a parallel pair f, g : D ⇒ D0 in D is called U -split
if
U (f )
U (D)
U (D0 )
U (g)
extends to a split coequalizer diagram in C (cf. problem 2 on the
midterm, the results of which you may use for this problem).
(To prove this, it suffices to look at U : CT → C the forgetful functor
from the Eilenberg-Moore category of a monad T on C. This accounts
for the “strict” qualification to match our strong notion of creation of
limits. Other sources, such as the nLab, defines creation of limits in a
slightly weaker way that is invariant under equivalence of categories, so
that the statement holds for any monadic functor with the same proof.)
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