Categories and Functors

Construct a contravariant functor from the category of all subsets of a given set (again the morphisms are the imbeddings) into itself.

Does there exist a universal repelling object in: the category of groups, the category of linear spaces over a given field, the categories dual to the categories of groups and linear spaces?

Let $G_1$ be the category of abelian groups with a distinguished generator (the morphisms are the group homomorphisms preserving the distinguished generator.) Determine universal objects in $G_1$.

Let $G_2$ be the category of groups with two distinguished generators (the morphisms are the group homomorphisms preserving the distinguished generators). Prove that $G_2$ has a universal object. This object is called the free group with two generators.

Let $AG_2$ be the subcategory of $G_2$ (see Problem 46) whose objects are the abelian groups with two distinguished generators. Prove that there is a universal object in $AG_2$. This object is called the free abelian group with two generators.

Let $A_n(K)$ be the category of associative algebras with $n$ distinguished generators over a field $K$. Prove that there is a universal object in $A_n(K)$. This universal object is the so-called tensor algebra over an $n$-dimensional linear space over the field $K$.

Prove that there exists a universal object for the subcategory $CA_n(K)$ of $A_n(K)$ consisting of the commutative algebras.

Prove that there exists a universal object for the category $LA_n(K)$ of Lie algebras with $n$ distinguished generators over the field $K$. The corresponding object is called the free Lie algebra with $n$ generators.

Let $\mathfrak G$ be a Lie algebra over a field $K$ of characteristic zero. We consider the category $K(\mathfrak G)$ whose objects are those linear mappings $\varphi$ from $\mathfrak G$ into associative algebras (depending on the object) having the property that

A morphism of an object $\varphi \colon \mathfrak G\to A$ to an object $\psi \colon \mathfrak G\to B$ is defined to be a homomorphism $\chi \colon A\to B$ which makes the following diagram commute:

Prove that $K(\mathfrak G)$ has a universal object $\varphi_0\colon \mathfrak G\to U(\mathfrak G)$. The algebra $U(\mathfrak G)$ is called the associative hull or enveloping algebra of $\mathfrak G$.

Prove that the associative hull (see Problem 51) of the free Lie algebra with $n$ generators is isomorphic to the tensor algebra over an $n$-dimensional space.

Let $\left\{ X_\alpha \right\}_{\alpha \in A}$ be a family of objects in the category $K$. We consider the category $\widetilde K$ whose objects are the sets of morphisms $\varphi_\alpha \in \operatorname {Mor}(X_\alpha ,Y)$, $\alpha \in A$ ($Y$ is some object in $K$ depending on the object in $\widetilde K$). The morphisms in $\widetilde K$ are the sets of commutative diagrams of the form

If $\widetilde K$ has a universal repelling object, then the corresponding object in $K$ is called the sum of the objects and is denoted by $\coprod_{\alpha \in A}X_\alpha$. The morphisms $i_\alpha \colon X_\alpha \to \coprod_{\alpha \in A}X_\alpha$ are called the canonical imbeddings of the terms in the sum. Prove that in the category of sets and in the category of linear spaces over a given field, the sum of any family of objects is defined.

The definition of the product of a family $\left\{ X_\alpha \right\}_{\alpha \in A}$ of objects in a category $K$ is obtained from the definition of the sum (see Problem 53) by reversing the arrows. Namely, the product $\prod_{\alpha \in A}X_\alpha$ is defined to be the sum of the objects $X_\alpha$ in the dual category $\widetilde K^0$. The morphisms $p_\alpha \colon \prod_{\alpha \in A}X_\alpha \to X_\alpha$ are called the canonical projections of the product onto the factors. Prove that in the categories of sets and of linear spaces over a given field, the product of any family of objects is defined.

Prove that the sum $\coprod_{k=1}^nL_k$ and the product $\prod_{k=1}^nL_k$ of finitely many objects of the category of linear spaces over a given field are isomorphic.

Let $L_1$ and $L_2$ be two linear spaces over a field $K$. We consider the category whose objects are bilinear mappings $\varphi \colon L_1\times L_2\to L$, where $L$ is a linear space (depending on the object $\varphi$). A morphism from an object $\varphi \colon L_1\times L_2\to L$ to an object $\psi \colon L_1\times L_2\to M$ is defined to be a linear mapping $\chi \colon L_1\to M$ for which the diagram

commutes. Prove that the resulting category has a universal repelling object $\pi \colon L_1\times L_2\to L_1\otimes_K L_2$. The space $L_1\otimes_K L_2$ is called the tensor product of the spaces $L_1$ and $L_2$ over the field $K$.

Let $G_1$ and $G_2$ be finite abelian groups. We consider the category of all mappings $\varphi \colon G_1\times G_2\to G$, where $G$ is a finite abelian group (depending on the object), which are homomorphisms in each variable. The morphisms are the commutative diagrams of the form

where $\chi$ is a homomorphism. Prove that this category has a universal object $G_1\times G_2\to \operatorname {Tor}(G_1,G_2)$ (the so-called torsion product of the two groups). Compute $\operatorname {Tor}(C_m,C_n)$, where $C_m$ is the cyclic group of order $m$.

Let $A$ be a directed set, and let $K$ be a category. Suppose that for each $\alpha \in A$ an object $X_\alpha \in \operatorname {Ob}K$ has been singled out, and for each pair $\alpha <\beta$ a morphism $\varphi_{\alpha \beta }\in \operatorname {Mor}(X_\alpha ,X_\beta )$ has been chosen in such a way that for any $\alpha <\beta <\gamma$ the diagram

commutes. Consider the category $K_A$ whose objects are families $\left\{ \varphi_\alpha \colon X_\alpha \to X\right\}_{\alpha \in A}$ of morphisms compatible with the $\varphi_{\alpha \beta }$, where $X$ is an object of $K$ depending on the family. A morphism from the object $\left\{ \varphi_\alpha \colon X_\alpha \to X\right\}_{\alpha \in A}$ to the object $\left\{ \psi_\alpha \colon X_\alpha \to {}\right\}_{\alpha \in A}$ is defined to be a morphism $\chi \in \operatorname {Mor}(X,Y)$ such that the diagram

commutes for any $\alpha \in A$. A universal object in $K_A$ (if it exists) is called the inductive limit of the family $\left\{ X_\alpha \right\}$. The dual concept of a projective limit is defined to be a universal object in $(K_A)^0$. Prove that:

Each complex linear space can be regarded as a real space, and each complex linear mapping as a real linear mapping. Prove that the correspondence described is a covariant functor from the category $L(\mathbb {C})$ of linear spaces over $\mathbb {C}$ to the category $L(\mathbb {R})$ of linear spaces over $\mathbb {R}$.

Prove that the mapping $L\mapsto L\otimes_{\mathbb {R}}\mathbb {C}$ (tensor product over $\mathbb {R}$ in the sense of Problem 56) generates a covariant functor from $L(\mathbb {R})$ to $L(\mathbb {C})$.

Prove that the categories $L(\mathbb {R})$ and $L(\mathbb {C})$ (see Problem 59) are not equivalent.

Let $G$ be a group and $K$ a field. Consider the collection $K[G]$ of formal linear combinations of elements of $G$ with coefficients in $K$. It is an algebra over $K$ with respect to the natural operations of addition, multiplication, and multiplication by elements of $K$. Prove that: