6TEKA Category Theory, winter semester 2022/2023
HOMEWORKS
Homework 1
Prove that for any two
vector spaces \(V\) and \(W\), it holds \(V \otimes W \simeq W \otimes V\).
Homework 2
Which of the following
sets form a) group b) semigroup?
(i) \(\mathbb Z\) with respect to the subtraction \(a - b\);
(ii) \(\mathbb Z\) with respect to the operation \(a * b = a + b + ab\);
(iii) All subsets of a given set with respect to the symmetric difference
\(A \triangle B = (A \backslash B) \cup (B \backslash A)\) .
Homework 3
Give examples of categories from the "real world". (Think of people,
geographical places, books, ...). Remember you have to define both objects and
morphisms!
Homework 4
Does there exist a category consisting of:
a) 4 objects and 3 arrows;
b) 3 objects and 4 arrows?
Homework 5
For which natural values of n the following is true: there are "more"
(in any conceivable sense) categories with n+1 objects than categories
with n objects?
Homework 6
Denote by \(\mathbf S_X\) the category associated to a set \(X\) (by considering
\(X\) as the set of objects, and identity maps on elements of \(X\) as the set
of arrows), and by \(\mathbf C_M\) the category associated to a monoid \(M\),
as discussed in the class. Describe all the cases when one of these categories
is a subcategory of the other.
Homework 7
Describe all covariant and all contravariant functors from the category
\(\mathbf{C}_M\) associated to a fixed monoid \(M\), to \(\mathbf{Set}\).
Homework 8
Is it possible that a functor between two categories is simultaneously covariant
and contravariant?
Homework 9
Let \(U\) be a fixed vector space over a field \(K\). Define a functor
\(\mathbf{Vect}_K \to \mathbf{Vect}_K\) sending a vector space \(V\) to
\(Hom_K(V,U)\). Prove that this is a functor. Is this functor covariant or
contravariant? How this construction can be generalized?
Homework 10
Prove:
1. There is a functor \(\mathbf{Group} \to \mathbf{Set}\), sending
each group \(G\) to its set of elements of order \(\le 2\)
(i.e., \(x \in G\) such that \(x^2 = 1\)).
2. There is no functor \(\mathbf{Group} \to \mathbf{Set}\), sending each group
\(G\) to its set of elements of order \(2\)
(i.e., \(x \in G\) such that \(x \ne 1\) and \(x^2 = 1\)).
Homework 11
Enumerate, up to isomorphism, all categories containing exactly 3 arrows, and
for each of them find all subcategories, up to isomorphism.
Homework 12
Recall that a constant functor between two categories \(\mathbb C\) and
\(\mathbb D\) is a functor sending each object of \(\mathbb C\) to a fixed
object \(A\) in \(\mathbb D\), and each arrow in \(\mathbb C\), to \(1_A\).
Describe all situations when a constant functor is isomorphism of categories.
Homework 13
Prove that the forgetful functor from \(\mathbf{Group}\) to \(\mathbf{Set}\) is
not an isomorphism.
Homework 14
Which of the three categories: \(\mathbb C_M\) for a fixed monoid \(M\),
\(\mathbb S_X\) for a fixed set \(X\), and \(\mathbf{Set}\) are equivalent?
Homework 15
Give examples of two equivalent categories, such that one of them has finite
number of objects, and another one:
1) is small with infinite number of objects;
2) is large.
Homework 16
Prove that there exists a natural transformation between the identity functor
\(id_{\mathbf{Set}}: \mathbf{Set} \to \mathbf{Set}\), and the power set functor
\(P: \mathbf{Set} \to \mathbf{Set}\). Is this natural transformation a natural
equivalence?
Homework 17
Prove that for any cardinality \(\kappa\) there is a functor
\(F: \mathbf{Set} \to \mathbf{Set}\) such that there are exactly \(\kappa\)
different natural transformations between \(F\) and the identity functor
\(id_{\mathbf{Set}}\). (Hint: use Yoneda lemma).
Homework 18
In the class we proved that right adjoint to the functor
\(\mathbf{Set} \to \mathbf{Group}\) assigning to a set \(X\) the free group
freely generated by \(X\), is the forgetful functor
\(\mathbf{Group} \to \mathbf{Set}\). Formulate and prove the analogous statement
for the category of commutative rings with unit.
(Hint: what would the corresponding free object?)
Homework 19
Give an example of a functor for which left or right adjoint does not exist.
Homework 20
In the class and in homeworks we have described all possible functors between
the following categories:
1) \(\mathbb C_M\) and \(\mathbb C_N\) for two monoids \(M\) and \(N\).
2) \(\mathbb S_X\) and \(\mathbb S_Y\) for two sets \(X\) and \(Y\).
3) \(\mathbb C_M\) and \(\mathbb S_X\) for a monoid \(M\) and a set \(X\).
4) \(\mathbb C_M\) and \(\mathbf{Set}\).
5) \(\mathbb S_X\) and \(\mathbf{Set}\).
In each of these cases, for each functor describe the left/right adjoint
functor.
Homework 21
Does there exist a category \(\mathbb D\) such that for any category \(\mathbb C\),
and any functor \(F: \mathbb D \to \mathbb C\), the direct limit of \(F\) exists?
Last modified: Mon Oct 30 12:13:59 CET 2023