izak

complete categories

To prove that the forgetful functor \( U: \text{Vect}_k \to \text{Set} \) is monadic, we need to show that it satisfies the conditions of monadicity, as given by Beck's Monadicity Theorem. The forgetful functor \( U \) maps a vector space over a field \( k \) to its underlying set and maps linear transformations to their underlying functions.

The forgetful functor \( U: \text{Vect}_k \to \text{Set} \) has a left adjoint \( F: \text{Set} \to \text{Vect}_k \), which is the free vector space functor. For a set \( X \), \( F(X) \) is the vector space whose basis is \( X \). The adjunction is given by the natural isomorphism: \[ \text{Hom}_{\text{Vect}_k}(F(X), V) \simeq \text{Hom}_{\text{Set}}(X, U(V)), \] where \( V \) is a vector space and \( X \) is a set. This shows that \( U \) is a right adjoint.

The adjunction \( F \dashv U \) induces a monad \( T = UF \) on \( \text{Set} \). First, for a set \( X \), \( T(X) = U(F(X)) \) is the underlying set of the free vector space generated by \( X \). This is the set of formal linear combinations of elements of \( X \) with coefficients in \( k \). Second, the unit \( \eta_X: X \to T(X) \) maps an element \( x \in X \) to the corresponding basis vector in \( F(X) \). Third, the multiplication \( \mu_X: T(T(X)) \to T(X) \) is the linear map that "flattens" formal linear combinations of formal linear combinations into formal linear combinations.

Beck's Monadicity Theorem states that a functor \( U \) is monadic if: \((1.)\) \( U \) has a left adjoint, \((2.)\) \( U \) reflects isomorphisms, and \((3.)\) \( U \) preserves and reflects coequalizers of \( U \)-split pairs.

This third condition might be least familiar. Wikipedia's descriptions for this are not as subpar as they often are. First, (\( U \)-split pairs are those parallel pairs of morphisms in $\text{Vect}_k$, which $U$ sends to pairs having a split coequalizer in $\text{Set} $). Second, in the category of sets, the coequalizer of two functions \( f, g: X \to Y \) is the quotient of \( Y \) by the smallest equivalence relation \( \sim \) satisfying \( f(x) \sim g(x) \) for every \( x \in X \). For example, if \( R \) is an equivalence relation on a set \( Y \), and \( r_1, r_2: R \to Y \) are the natural projections (where \( R \subseteq Y \times Y \)), then the coequalizer of \( r_1 \) and \( r_2 \) is the quotient set \( Y / R \).

Before looking at \( U \)-split pair in \( \text{Vect}_k \), let's verify the first two conditions.

For $(1)$, we already showed \( U \) has a left adjoint.

For $(2)$, a linear transformation \( f: V \to W \) is an isomorphism in \( \text{Vect}_k \) if and only if it is bijective. Since \( U \) maps \( f \) to its underlying function, \( U(f) \) is an isomorphism in \( \text{Set} \) (i.e., a bijection) if and only if \( f \) is an isomorphism in \( \text{Vect}_k \). Thus, \( U \) reflects isomorphisms.

For $(3)$, a \( U \)-split pair in \( \text{Vect}_k \) consists of linear maps \( f, g: V \to W \) such that there exist set-theoretic maps \( h: U(W) \to U(V) \) and \( k: U(W) \to U(W) \) making \( U(f), U(g) \) a split pair in \( \text{Set} \). The coequalizer of \( f, g \) in \( \text{Vect}_k \) is the quotient space \( W / \text{Im}(f - g) \), and \( U \) maps this to the set-theoretic coequalizer of \( U(f), U(g) \) in \( \text{Set} \). Since \( U \) is a right adjoint, it preserves coequalizers of \( U \)-split pairs. Moreover, because \( U \) reflects isomorphisms and the splitting conditions ensure that the coequalizer in \( \text{Set} \) lifts uniquely to \( \text{Vect}_k \), \( U \) also reflects coequalizers of \( U \)-split pairs. Thus, \( U \) preserves and reflects such coequalizers.

In particular, in \(\text{Vect}_k\), let \(V\) and \(W\) be vector spaces, and let \(f, g: V \to W\) be linear maps. The coequalizer of \(f\) and \(g\) is the quotient space \(W / \text{Im}(f - g)\), equipped with the canonical projection map \(\pi: W \to W / \text{Im}(f - g)\). For any vector space \(Z\) and linear map \(\pi': W \to Z\) satisfying \(\pi' \circ f = \pi' \circ g\), there exists a unique linear map \(h: W / \text{Im}(f - g) \to Z\) induced by the universal property of the coequalizer, such that \(\pi' = h \circ \pi\). This universal property characterizes the coequalizer in \(\text{Vect}_k\).

As the forgetful functor \( U: \text{Vect}_k \to \text{Set} \) satisfies all the conditions of Beck's Monadicity Theorem, it is monadic. This particularly means that \( \text{Vect}_k \) is equivalent to the category of algebras over the monad \( T = UF \) on \( \text{Set} \).

Now, the nice thing is that $\text{Set}$ has all limits by design. Since the canoncial forgetful functor from $\text{Vect}_k$ is monadic, we see $\text{Vect}_k$ also has all limits. It remains to show that the category of chain complexes $ \text{CH}(k) $ has all limits. We can do this analogously to how we showed $\text{Vect}_k$ has all limits given $\text{Set}$ has all limits.

To prove that the category of chain complexes $ \text{CH}(k) $ is a reflective subcategory of a category of functors into $ \text{Vect}_k $, we need to show that the inclusion functor $ i: \text{CH}(k) \to \text{Fun}(\mathcal{C}, \text{Vect}_k) $ has a left adjoint. This left adjoint is called the reflector, and it provides a way to "reflect" objects from the larger category (functors into $ \text{Vect}_k $) back into the subcategory $ \text{CH}(k) $.

Let $ \text{Vect}_k $ be the category of vector spaces over a field $ k $.

Let $ \text{CH}(k) $ be the category of chain complexes of $ k $-vector spaces. Objects are sequences of vector spaces $ (C_n)_{n \in \mathbb{Z}} $ with differentials $ d_n: C_n \to C_{n-1} $ such that $ d_{n-1} \circ d_n = 0 $, and morphisms are chain maps.

Let $ \mathcal{C} $ be a small category, and let $ \text{Fun}(\mathcal{C}, \text{Vect}_k) $ be the category of functors from $ \mathcal{C} $ to $ \text{Vect}_k $. Objects are functors $ F: \mathcal{C} \to \text{Vect}_k $, and morphisms are natural transformations.

Consider the category $ \mathcal{C} = \mathbb{Z} $, where $ \mathbb{Z} $ is viewed as a poset category (with objects integers and a unique morphism $ n \to m $ iff $ n \geq m $).

A functor $ F: \mathbb{Z} \to \text{Vect}_k $ assigns a vector space $ F(n) $ to each integer $ n $ and a linear map $ F(n \to m): F(n) \to F(m) $ for each $ n \geq m $.

A chain complex $ (C_n, d_n) $ can be viewed as a functor $ F: \mathbb{Z} \to \text{Vect}_k $, where $ F(n) = C_n $ and $ F(n \to n-1) = d_n $, with the condition $ d_{n-1} \circ d_n = 0 $.

Thus, $ \text{CH}(k) $ is a subcategory of $ \text{Fun}(\mathbb{Z}, \text{Vect}_k) $.

To show that $ \text{CH}(k) $ is a reflective subcategory, we need to construct a left adjoint $ L: \text{Fun}(\mathbb{Z}, \text{Vect}_k) \to \text{CH}(k) $ to the inclusion functor $ i: \text{CH}(k) \to \text{Fun}(\mathbb{Z}, \text{Vect}_k) $.

For a functor $ F \in \text{Fun}(\mathbb{Z}, \text{Vect}_k) $, define $ L(F) $ as the chain complex obtained by "forcing" the condition $ d_{n-1} \circ d_n = 0 $.

To construct the chain complex $ L(F) $ from a functor $ F: \mathbb{Z} \to \text{Vect}_k $, we start by defining $ L(F)_n = F(n) $ and the differential $ d_n = F(n \to n-1) $. However, this does not automatically satisfy the chain complex condition $ d_{n-1} \circ d_n = 0 $. To enforce this condition, we modify $ L(F)_n $ by taking the quotient of $ F(n) $ by the image of $ d_{n+1}: F(n+1) \to F(n) $, particularly submodule generated by this image. This quotient ensures that the composition $ d_{n-1} \circ d_n $ becomes zero, as the image of $ d_{n+1} $ is factored out. The resulting object $ L(F) $ is now a chain complex, as the differentials satisfy the required condition $ d_{n-1} \circ d_n = 0 $. This process of quotienting effectively "reflects" the functor $ F $ into the subcategory of chain complexes.

The reflector $ L $ is left adjoint to the inclusion functor $ i $. We see there is a natural isomorphism:

$$\text{Hom}_{\text{CH}(k)}(L(F), C) \simeq \text{Hom}_{\text{Fun}(\mathbb{Z}, \text{Vect}_k)}(F, i(C))$$

for all $ F \in \text{Fun}(\mathbb{Z}, \text{Vect}_k) $ and $ C \in \text{CH}(k) $.

This adjunction arises because $ L(F) $ is the "best approximation" of $ F $ as a chain complex, and morphisms from $ L(F) $ to $ C $ correspond naturally to morphisms from $ F $ to $ i(C) $.

Since the inclusion functor $ i: \text{CH}(k) \to \text{Fun}(\mathbb{Z}, \text{Vect}_k) $ has a left adjoint $ L $, the category $ \text{CH}(k) $ is a reflective subcategory of $ \text{Fun}(\mathbb{Z}, \text{Vect}_k) $.

A category where all limits exist is called a "complete category".

These notes were inspired by a remark in this video.