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 18
In the class we proved that right adjoint to the functor
\(\mathbf{Set} \to \mathbf{Group}\) assigning to a set \(X\) the free group
freely generated by \(X\), is the forgetful functor
\(\mathbf{Group} \to \mathbf{Set}\). Formulate and prove the analogous statement
for the category of commutative rings with unit.
(Hint: what would the corresponding free object?)
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: Wed Oct 23 05:16:08 CEST 2024