6/7TEKA Category Theory, winter semester 2024/2025
HOMEWORKS
Homework 4
Describe all functors between the following pairs of categories:
\(\mathbf C_M\) to \(\mathbf C_N\), \(\mathbf C_M\) to \(\mathbf S_X\),
\(\mathbf S_X\) to \(\mathbf C_M\) for arbitrary monoids \(M\) and \(N\), and
arbitrary set \(X\).
Homework 5
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 coincides with some
constant functor (sending each object to a fixed object \(A\), and each arrow to
\(1_A\), as discussed in the class)?
Homework 8
Prove that the forgetful functor from \(\mathbf{Group}\) to \(\mathbf{Set}\) is
not an isomorphism.
Homework 13
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 14
In the class we proved that the 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 algebras with unit.
Homework 15
In the class we proved that the right adjoint to the functor
\(\mathbf{Set} \to \mathbf{Top}\) assigning to each set the topological space
with discrete topology on this set, is the forgetful functor
\(\mathbf{Top} \to \mathbf{Set}\). Formulate and prove the analogous statement
for the functor \(\mathbf{Set} \to \mathbf{Top}\), assigning to each set the
topological space with trivial topology on this set. (Hint: what should be left
and what should be right?)
Last modified: Tue Oct 14 2025 19:20:48 CEST