7. Constructions

7.1. Set and class

In Section 3.2 we saw that we could start with a category \(\mathcal C\) and obtain a (possibly different) category \(\mathcal C^{\mathrm{op}}\). There are many such constructions which produce new examples of categories.

7.1.1. Small and locally small category

As mentioned briefly in Section 2.2, there are set-theoretic difficulties that can arise when one attempts to handle some categories “all at once”. Namely, the objects and morphisms of a category often form proper classes. In order to prevent these issues in some of our constructions we introduce the following terminology.

A category is called small if both its objects and morphisms form sets.

A category \(\mathcal C\) is locally small if for every pair \(A\), \(B\) of objects in \(\mathcal C\) the collection of morphisms from \(A\) to \(B\) is a set. In this case, the set of morphisms from \(A\) to \(B\), called a “hom set,” is denoted by \(\mathrm{Hom}_{\mathcal C}(A, B)\).

Let’s pause to consider an example of a small category.

Example 7.1

The category Ord \(_{\mathrm{fin}}\) of finite ordinals (also called the simplex category \(\Delta\)) has

• objects. \(\underline{n} = \{0, 1, \dots, n-1\}\) for each \(n \in \mathbb N\);
• morphisms. \(f \colon \underline{n} \to \underline{m}\) monotone functions (\(i\leq j \; \Rightarrow f i \leq f j\)).

Note that all morphisms of \(\Delta\) are generated by (i.e., can be written as finite compositions of) face maps and degeneracy maps.

Given a positive natural number \(n > 0\) and a natural number \(i\) such that \(0\leq i < n\), the face map \(\delta^n_i \colon \underline{n-1} \to \underline{n}\) is define as follows:

\[\begin{split}\delta^n_i (j) = \begin{cases} j, & j < i,\\ j+1, & j \geq i. \end{cases}\end{split}\]

The degeneracy map \(\sigma^n_i \colon \underline{n+1} \to \underline{n}\) is define as follows:

\[\begin{split}\sigma^n_i (j) = \begin{cases} j, & j \leq i,\\ j-1, & j > i. \end{cases}\end{split}\]

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Exercise 7.1.1: Verify that Ord \(_{\mathrm{fin}}\) is a category.

Exercise 7.1.2: Verify that the faces and degeneracies satisfy the following so-called simplicial identities: for \(0 ≤ i ≤ j ≤ n\),

\[δ^{n+1}_j ∘ δ^n_i = δ^{n+1}_i ∘ δ^n_{j-1}, \qquad σ^{n}_j ∘ σ^{n+1}_i = σ^{n}_i ∘ σ^{n+1}_{j+1},\]

\[\begin{split}σ^{n}_j ∘ δ^{n+1}_i = \left\{ \begin{array}{ll}δ^n_i ∘ σ^{n-1}_{j-1}, & 0 ≤ i < j < n,\\ \mathrm{id}_n, & 0 ≤ j < n \text{ and either } i = j \text{ or } i = j+1,\\ δ^n_{i-1} ∘ σ^{n-1}_{j}, & 0 ≤ j \text{ and } j + 1 < i ≤ n. \end{array} \right.\end{split}\]

Exercise 7.1.3: Show that any morphism in \(Δ\) can be written as a composition of face and degeneracy maps. That is, show that the face and degeneracy maps generate all the morphisms in \(Δ\).

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The intuition is that the elements of \(\Delta\) are \(n\)-simplices, i.e., \(n\)-dimensional triangles.

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An example of a large category is the category of small categories, denoted \(\mathbf{Cat}\).

Cat:

The (large) category of small categories has

• objects. small categories;
• morphisms. functors \(F\colon \mathcal C \to \mathcal D\).

Naturally, the usual functor composition is morphism composition and the identity functors are the identity morphisms.

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Exercise 7.1.4: Verify that Cat is a category.

Exercise 7.1.5: We call a category large iff it is not locally small. Prove or disprove: Cat is large.

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7.2. Functor category

Given categories \(\mathcal C\) and \(\mathcal D\), the functor category from \(\mathcal C\) to \(\mathcal D\) has

• objects. functors from \(F \colon \mathcal C \to \mathcal D\);
• morphisms. natural transformations \(\alpha \colon F \Rightarrow G\).

Some common notations for this category are \(\mathrm{Fun}(\mathcal C, \mathcal D)\) and \(\mathcal D^{\mathcal C}\) and \((\mathcal C, \mathcal D)\).

To define composition of morphisms in \(\mathcal D^{\mathcal C}\), suppose \(F, G, H\) are functors from \(\mathcal C\) to \(\mathcal D\), and suppose \(\alpha \colon F \Rightarrow G\) and \(\beta \colon G \Rightarrow H\) are natural transformations.

The composition of \(\alpha\) and \(\beta\) is the natural transformation \(\beta\alpha\colon F \Rightarrow H\) with components \((\beta \alpha)_A = \beta_A \alpha_A\).

A special case is the category \(\mathbf{Set}^{\mathcal C}\) of functors from a category \(\mathcal C\) into the category of sets.

Example 7.2

If \(\mathcal C\) is a category, then \(\mathbf{Set}^{\mathcal C}\) is the category with

• objects. functors \(F \colon \mathcal C \to \mathbf{Set}\);
• morphisms. natural trasformations \(\alpha \colon F \Rightarrow D\).

7.2.1. Evaluation functor

With \(\mathbf{Set}^{\mathcal C}\) in hand, let us now revisit the evaluation functor introduced above. This is the functor \(Ev\colon \mathcal C \times \mathbf{Set}^{\mathcal C}\to \mathbf{Set}\), which takes each pair \((A, F) \in \mathcal C_0 \times (\mathbf{Set}^{\mathcal C})_0\) of objects to the set \(Ev(A, F) = FA\), and takes each pair \((g, \mu) \in \mathcal C_1 \times (\mathbf{Set}^{\mathcal C})_1\) of morphisms to a function on sets, namely

\[Ev(g, \mu) = \mu_{A'} \circ Fg = F'g \circ \mu_A,\]

where \(g\in \mathcal C(A, A')\) and \(\mu \colon F \Rightarrow F'\).

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7.3. Morphism category

We can create categories whose objects are the morphisms from a given category. Three standard cases are described below, but before describing them we recall some notational conventions appearing in these examples.

If \(\mathcal C\) is a category, then \(\mathcal C_0\) denotes the collection of objects of \(\mathcal C\).

If \(A, B \in \mathcal C_0\) then \(\mathcal C(A,B)\) denotes the collection of morphisms from \(A\) to \(B\) in \(\mathcal C\).

Arrow:

Given a category \(\mathcal C\), the arrow category \(\mathcal C^\rightarrow\) has

• objects. triples \((A, B, f)\) such that \(A, B \in \mathcal C_0\) and \(f\in \mathcal C(A,B)\);
• morphisms. pairs \((h_1, h_2) \colon (A, B, f) \to (C, D, g)\) such that \(h_1\in \mathcal C(A,C)\), \(h_2 \in \mathcal C(B, D)\) and \(g \circ h_1 = h_2 \circ f\), so that the diagram below commutes.

Slice:

Given a category \(\mathcal C\) and an object \(C\) from \(\mathcal C\), the slice category \(\mathcal C/C\) has

• objects. pairs \((A, f)\) such that \(f\in \mathcal C(A, C)\);
• morphisms. \(h\colon (A, f)\to (B, g)\) where \(h\in \mathcal C(A, B)\) and \(g\circ h=f\), so that the diagram below commutes.

Comma:

Given categories \(\mathcal C\) and \(\mathcal D\) and functors \(F \colon \mathcal C \to \mathcal D\) and \(G\colon \mathcal C' \to \mathcal D\) (with a common codomain), the comma category \((F \downarrow G)\) has

• objects. triples \((A, f, A')\), where \(A\in \mathcal C_0\), \(A' \in \mathcal C'_0\), and \(f\in \mathcal D(FA, GA')\).
• morphisms. pairs \((\varphi, \psi) \colon (A, f, A') \to (B, g, B')\), where \(\varphi \in \mathcal C(A,B)\), \(\psi \in \mathcal C'(A',B')\) and \(G\psi \circ f = g \circ F\varphi\), so that the square in the figure below commutes.

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Exercise 7.3.1: Define composition of morphisms in \(\mathcal C^{→}\) in the natural way and show that \(\mathcal C^{→}\) is a category.

Exercise 7.3.2: Define composition of morphisms in \(\mathcal C/C\) in the natural way and show that \(\mathcal C/C\) is a category.

Exercise 7.3.3: Show that the slice category construction is a functor \(\mathcal C/\_ \colon \mathcal C \to \mathbf{Cat}\).

Exercise 7.3.4: Consider the claim, "the identity map \(\mathrm{id}_{B^A × A}\) is a perfectly good substitute for the composition \((λ π_1 × \mathrm{id}_A) ∘ h\)." Do you agree with the claim? Why (not)?

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7.4. Presheaf

Another important special case of a functor category is the category of contravariant functors with values in sets. This is denoted by \(\mathbf{Set}^{\mathcal C^{\mathrm{op}}}\) and is called the category of presheaves on \(\mathcal C\).

In particular, the objects of \(\mathbf{Set}^{\Delta^{\mathrm{op}}}\) are called simplicial sets.

A simplicial set \(X \colon \Delta^{\mathrm{op}} \to \mathbf{Set}\) is a geometric object constructed by glueing together simplices, i.e., \(n\)-dimensional triangles.

\(X\underline{n}\) is the set of simplices in \(X\) of dimension \(n\).

The images of the face maps are used to specify how the simplices are glued together.

\(X\delta^n_i : X\underline{n} \to X\underline{(n-1)}\) specifies which \((n-1)\)-simplices are the \(i\)-th face of the \(n\)-simplices.

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7.5. Solutions

Exercise 7.1.1

Solution (to do).

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Exercise 7.1.2

Solution (to do).

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Exercise 7.1.3

Solution (to do).

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Exercise 7.1.4

Solution (to do).

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Exercise 7.1.5

Solution (to do).

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Exercise 7.3.1

Solution (to do).

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Exercise 7.3.2

Solution (to do).

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Exercise 7.3.3

Solution (to do).

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Solution to Exercise 7.3.4.

The claim, “the identity map \(\mathrm{id}_{B^A × A}\) is a perfectly good substitute for the composition \((λ π_1 × \mathrm{id}_A) ∘ h\)” is false.

To see this, take an element \((f, a) ∈ B^A × A\) and trace around the commuting diagram in the definitions of product and exponential object. We arrive at \(f(a) = eval^A_B(f, a) = π_1 (h(f,a))\) and \(a = π_2 (h(f, a))\), from which it follows that \(h(f,a) = (f(a), a)\).

The function \(λ π_1: B → B^A\) takes \(b ∈ B\) to the constant map \(λ x . b\). Therefore,

\[((λ π_1 × \mathrm{id}_A) ∘ h) (f, a) = (λ π_1 × \mathrm{id}_A) (f(a), a) = (λ x . f(a), a),\]

where \(λ x . f(a)\) denotes the constant function that takes every \(x ∈ A\) to \(f(a)\).

If \((λ π_1 × \mathrm{id}_A) ∘ h\) were really the identity map, \(\mathrm{id}_{B^A × A}\), then we should have ended up with \((f, a)\) instead of \((λ x . f(a), a)\).