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 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 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: Tue Sep 30 2025 17:18:19 CEST