Download EXAMPLE SHEET 1 1. If k is a commutative ring, prove that b k

EXAMPLE SHEET 1
TOPICS IN CATEGORY THEORY
PART III – LENT TERM 2016
1. If k is a commutative ring, prove that bk endows the category Alg of k-algebras with
a monoidal structure. If f : A Ñ C Ð B : g are two morphisms of algebras, we say
that f commutes with g if f paqgpbq “ gpbqf paq for all a P A and b P B. We claim that
the morphisms i1 : A Ñ A bk B Ð B : i2 given by i1 paq “ a b 1 and i2 pbq “ 1 b b,
exhibit A bk B as the universal algebra in which A and B commute. Write a precise
statement and a proof of this claim.
/
2. (a) Prove that the free abelian group adjunction Ab o J Set is a monoidal adjunction (where Set is monoidal with the cartesian product and Ab with the usual
tensor product). Let R be a commutative ring.
(b) Prove that, if R is a commutative monoid, the free R-module adjunctions
R-Mod o
3.
4.
5.
6.
J
/
Ab
and
R-Mod o
J
/
Set
are monoidal adjunctions.
Show that the category Mon` p1, Cq of monoidal functors 1 Ñ C into a monoidal
category C and monoidal natural transformations is isomorphic to the category of
monoids in C.
Show that the category of small strict categories and strict monoidal functors is isomorphic to the category CatpMonoidq of internal categories in the cartesian category
of monoids.
Show that the category Mons of (small) monoidal categories and strict monoidal
functors is monadic over Cat.
Express the adjunction pX b ´q % rX, ´sr in terms of the unit and counit (known as
the evaluation):
η
ev
Y
Y ÝÝÑ
rX, X b Y s
Y
X b rX, Y s ÝÝÑ
Y.
7. Prove that if both Y and Z are right duals of X, then there exists a unique isomorphism Y – Z that is compatible with the evaluations and coevaluations in the
appropriate sense.
8. Let V be a monoidal category. Describe a monoidal structure on the functor
Vp´, ´q : V op ˆ V ÝÑ Set.
Deduce that, if C is a comonoid and A is a monoid in V the functors
Vp´, Aq : V op ÝÑ Set
and
Send corrections and comments to [email protected].
1
VpC, ´q : V ÝÑ Set
2
TOPICS IN CT – PART III 2016
have monoidal structures, and VpC, Aq has a monoid structure. (The multiplication
of VpC, Aq is called the convolution product.)
9. If V is a monoidal category, let EpVq the category whose objects are endofunctors S
of V equipped with an natural isomorphism σX,Y : X b SpY q – SpX b Y q, and whose
morphisms pS, σq Ñ pT, τ q are natural transformations ω : S ñ T such that
τX,Y ¨ pX b ωY q “ ωXbY ¨ σX,Y .
Equip EpVq with the strict monoidal structure given on objects by pS, σq ˝ pT, τ q “
pT S, σ ˝ τ q, where
τX,SpY q
T σX,Y
pσ ˝ τ qX,Y : X b T SpY q ÝÝÝÝÝÑ T pX b SpY qq ÝÝÝÝÑ T SpX b Y q.
Finally, consider the functor N : V Ñ EpVq given by N pXq “ p´ b X, nX q where
´1
pnX qY,Z “ αY,Z,X
: Y b pZ b Xq Ñ pY b Zq b X.
Prove that the following conditions are equivalent for X P V:
(a) X has a left dual.
(b) p´ b Xq has a right adjoint rX, ´s` , with unit η and counit ε, and rX, ´s` is an
object of EpVq and η, ε are morphisms of EpVq.
(c) There exists a morphism e : Y b X Ñ I such that the functions
VpZ, X b W q ÝÑ VpY b Z, W q
f ÞÑ pe b W q ¨ pY b f q
are bijections.
(d) p´ b Xq has a right adjoint rX, ´s` and the comparison morphism rX, Is` b Y Ñ
rX, Y s` (to be defined) is an isomorphism for all Y .
Write down a further equivalent condition, dual to 9.c.
10. If R is a commutative ring, prove that an R-module has a left (and a right) dual if
and only if it is projective and finitely presentable.
11. If X is a set, define a monoidal structure on SetXˆX that has as product the “matrix
multiplication”
ÿ
pA b Bqpx, yq “
Apx, zq ˆ Bpz, yq,
x, y P X.
12.
13.
14.
15.
Describe monoids in this monoidal category. Describe the monoidal structure on
Set{X ˆ X that makes the canonical equivalence with SetXˆX a strong monoidal
functor.
Prove that the forgetful functor from the category of (co)monoids in a monoidal
category V to V creates (co)limits. Deduce that the category of (co)monoids is
(co)complete if V is so.
Prove that a subspace V Ă C of a coalgebra C is a subcoalgebra if and only if it is a
left and a right coideal.
Let V and W be a k-vector spaces and
ř U Ă W a subspace. Every time one writes an
element x P V b U Ă V b W as x “ ni“1 vi b wi with the tvi u linearly independent,
one necessarily has wi P U for all i.
Prove the fundamental theorem of comodules: if M is a C-comodule over a field k,
then M is the union of its finite dimensional subcomodules. (Hint: if ρ : M Ñ M b C
EXAMPLE SHEET 1
3
ř
is a C-comodule, write ρpxq “ ni“1 xi b ci with the ci linearly independent, and
proceed in a similar way to the proof of the fundamental theorem of coalgebras).
16. Suppose that k is a field, and consider the functor Setop
f Ñ Vect given on objects
X
by X ÞÑ k (Setf is the category of finite sets). Prove that this functor is strong
monoidal, and describe:
(a) the monoid structure on k X induced by the comonoid structure on X;
(b) if G is a monoid, the coalgebra structure on k G induced by it.
17. Let A be an algebra over a field k. A right A-module M is rational if for each m P M
the orbit m ¨ A “ tm ¨ x : x P Au is finite dimensional. Given a coalgebra C, prove
that the functor ComodpCq Ñ Mod-C ˚ is an isomorphism into the full subcategory of
rational right C ˚ -modules. If A is finite dimensional, there is an isomorphism between
the categories of finite dimensional C-comodules and finite dimensional C ˚ -modules.
18. Use the adjoint functor theorem to prove that the forgetful functor from Coalg “
ComonpVectq to Vect has a right adjoint. Deduce that this functor is comonadic.