6/7TEKA Category Theory, winter semester 2026/2027
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
Define a category to be arrow-small (respectively arrow-large) if the class of
its arrows forms a set (respectively does not form a set). How the notion of
small/large category we discussed in the class is related to the notion of
arrow-small/arrow-large category?
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
Is it possible that a functor between two categories is simultaneously covariant
and contravariant?
Homework 6
Let \(A\) 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
\(Hom_K(V,A)\) (all linear maps between \(V\) and \(A\)). Prove that this is a
functor. Is this functor covariant or contravariant? How this construction can
be generalized?
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 \(x^2 = 1\)).
Homeworks due October 20 12:00
Homework 8
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 9
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 10
Define the "projection functor" from the product of two categories
\(\mathbb C \times \mathbb D\) to the first factor \(\mathbb C\), by analogy
with projections for the direct product of groups, or direct sum of vector
spaces. Prove that this is a functor. Is it possible that for some categories
\(\mathbb C\), \(\mathbb D\) this projection functor is:
1) isomorphism;
2) equivalence?
Last modified: Wed Oct 14 2026 19:25:10 CEST