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 7
Describe all covariant and all contravariant functors from the category \(\mathbf{C}_M\) associated to a fixed monoid \(M\), to \(\mathbf{Set}\).

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: Thu Nov 28 2024 21:59:56 CET