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.
Homeworks due October 14, 2025
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?
Homeworks due October 21, 2025
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 the Yoneda embedding).
Last modified: Tue Oct 14 2025 19:38:43 CEST