6/7TEKA Category Theory, winter semester 2025/2026
HOMEWORKS
Rules for submitting homeworks
Scores
Homework 1
Does there exist a category consisting of:
a) 4 objects and 3 arrows;
b) 3 objects and 4 arrows?
Homework 2
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 3
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 4
Describe all covariant and all contravariant functors from the category
\(\mathbf{C}_M\) associated to a fixed monoid \(M\), to \(\mathbf{Set}\).
Homework 5
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 6
Enumerate, up to isomorphism, all categories containing exactly 3 arrows, and
for each of them find all subcategories, up to isomorphism.
Homework 7
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 8
1) 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?
2) Is it true that for any two monoids \(M\) and \(N\), the categories
\(\mathbf{C}_M\) and \(\mathbf{C}_N\) are equivalent if and only if they are
isomorphic?
Homework 9
In the class we had an example of two equivalent categories (the category of
finite-dimensional vector spaces and the category of matrices), one of which is
large, and another one is small with an infinite number of objects. Give
further examples of two equivalent categories with a "different number" of
objects. "Different number" should be interpreted in all possible remaining ways, namely:
1) Both categories are small, and the cardinalities of the sets of objects are
different (the cardinalities could be finite of infinite).
2) One category is large, and another one is small with a finite number of
objects.
Homework 10
Prove that there exists a natural transformation between the identity functor
\(id_{\mathbf{Set}}: \mathbf{Set} \to \mathbf{Set}\), and the power set functor
\(P: \mathbf{Set} \to \mathbf{Set}\) (i.e., the functor assigning to any set
\(X\) its power set \(P(X)\)).
Homework 11
In Homework 4, we proved that functors \(\mathbf C_G \to \mathbf{Set}\), where
\(G\) is a group, are, essentially, homomorphisms \(A: G \to S(X)\) for some set
\(X\), or, in other words, actions of \(G\) on a set \(X\). Prove that any
natural transformation between two functors
\(A,B: \mathbf C_G \to \mathbf{Set}\) corresponds to \(G\)-equivariant map
between the corresponding actions, i.e., to a map \(\sigma: X \to Y\) such that
\(\sigma(A(g)(x)) = B(g)(\sigma(x))\) for any \(g \in G, x \in X\). (This will
be used later to derive the Cayley theorem in group theory from Yoneda lemma).
Homework 12
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 13
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 \(\mathbf{Set}\).
In each of these cases, for each functor describe the left/right adjoint
functor.
Homework 14
Describe the left/right adjoint functors to the functor
\(\mathbf{Set} \to \mathbf{Top}\) assigning to each set the topological space
with:
1) the discrete topology
2) the trivial topology
on this set.
Homeworks due November 18, 2025
Homework 15
Try to find other other embeddings in algebra, similar to the Cayley theorem,
and give a categorical-theoretic proof of them.
Homework 16
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?
Homeworks due November 25 December 2, 2025
Homework 17 (Mac Lane, p.72, Ex. 3)
Prove that if category \(\mathbb D\) has an initial object, than every functor
\(F: \mathbb D \to \mathbb C\) to any category \(\mathbb C\) has a limit (which
one?). Dualize.
Homework 18 (Mac Lane, p.112, Ex. 5)
Prove that every functor from a small category to \(\mathbf{Cat}\) (the category
of small categories) has a limit.
Homework 19 NEW (Bergman, Ex. 8.6:2)
Describe limit and colimit of a functor from \(\mathbb C_M\) (\(M\) is a monoid)
to \(\mathbf{Set}\).
Last modified: Mon Nov 24 2025 19:38:45 CET