6/7TEKA Category Theory, winter semester 2025/2026
HOMEWORKS
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 6
Enumerate, up to isomorphism, all categories containing exactly 3 arrows, and
for each of them find all subcategories, up to isomorphism.
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 Yoneda lemma).
Homework 12
Prove that for any cardinality \(\kappa\) there is a functor
\(F: \mathbf{Set} \to \mathbf{Set}\) such that there are exactly \(\kappa\)
different natural transformations between \(F\) and the identity functor
\(id_{\mathbf{Set}}\). (Hint: use Yoneda lemma).
Homework 13
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 \(\mathbf{Set}\).
In each of these cases, for each functor describe the left/right adjoint
functor.
Homework 14
Describe the left/right adjoint functors to the functor
\(\mathbf{Set} \to \mathbf{Top}\) assigning to each set the topological space
with:
1) the discrete topology
2) the trivial topology
on this set.
Homework 15
Try to find other other embeddings in algebra, similar to the Cayley theorem,
and give a categorical-theoretic proof of them.
Homework 16
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 colimit
of \(F\) exists?
Homework 17 (Mac Lane, p.72, Ex. 3)
Prove that if category \(\mathbb D\) has an initial object, than every functor
\(F: \mathbb D \to \mathbb C\) to any category \(\mathbb C\) has a limit (which
one?). Dualize.
Homework 18 (Mac Lane, p.112, Ex. 5)
Prove that every functor from a small category to \(\mathbf{Cat}\) (the category
of small categories) has a limit.
Homework 19 (Bergman, Ex. 8.6:2)
Describe limit and colimit of a functor from \(\mathbb C_M\) (\(M\) is a monoid)
to \(\mathbf{Set}\).
Homework 20
Determine whether each of the two functors:
1) the free group functor \(\mathbf{Set} \to \mathbf{Group}\) sending a set
\(X\) to the free group freely generated by \(X\);
2)
the forgetful functor \(\mathbf{Group} \to \mathbf{Set}\)
is a) continuous b) cocontinuous.
Last modified: Wed Oct 14 2026 19:17:37 CEST