6/7TEKA Category Theory, winter semester 2024/2025

HOMEWORKS

Rules for submitting homeworks
Scores

Homework 1
Does there exist a category consisting of: a) 4 objects and 3 arrows; b) 3 objects and 4 arrows?

Homework 2
For which natural values of \(n\) the following is true: there are "more" (in any conceivable sense) categories with \(n+1\) objects than categories with \(n\) objects?

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 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 6
Describe all covariant functors from the category \(\mathbf{C}_M\) associated to a fixed monoid \(M\), to \(\mathbf{Set}\).

Homework 7
Let \(U\) be a fixed vector space over a field \(K\). Define a functor \(\mathbf{Vect}_K \to \mathbf{Vect}_K\) sending a vector space \(V\) to \(Hom_K(V,U)\). Prove that this is a functor. Is this functor covariant or contravariant? How this construction can be generalized?

Homework 8
Prove that the forgetful functor from \(\mathbf{Group}\) to \(\mathbf{Set}\) is not an isomorphism.

Homework 9
Prove:
1. There is a functor \(\mathbf{Group} \to \mathbf{Set}\), sending each group \(G\) to its set of elements of order \(\le 2\) (i.e., \(x \in G\) such that \(x^2 = 1\)).
2. There is no functor \(\mathbf{Group} \to \mathbf{Set}\), sending each group \(G\) to its set of elements of order \(2\) (i.e., \(x \in G\) such that \(x \ne 1\) and \(x^2 = 1\)).

Homework 10
At the class we had an example of two equivalent categories (the category of finite-dimensional vector spaces and the category of matrices), one which is large, and another one is small with an infinite number of objects. Give further examples of two equivalent categories with a "different number" of objects. "Different number" should be interpreted in all possible remaining ways, namely:
1) Both categories are small, and the cardinalities of the sets of objects are different (the cardinalities could be finite of infinite).
2) One category is large, and another one is small with a finite number of objects.

Homework 11
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 12
At Homework 6, 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 the statement we used in the class to derive Cayley theorem from Yoneda embedding: any natural transformation between two functors \(A,B: \mathbf C_G \to \mathbf{Set}\) correspond 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 \(g \in G, x \in X\).

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 Dec 17 2024 17:15:01 CET