2.4: Constructions on V-categories

Now that we have a good intuition for what V-categories are, we give three examples of what can be done with V-categories. The first (Section 2.4.1) is known as change of base. This allows us to use a monoidal monotone f : V → W to construct W-categories from V-categories. The second construction (Section 2.4.2), that of V-functors, allows us to complete the analogy: a preorder is to a Bool-category as a monotone map is to what? The third construction (Section 2.4.2) is known as a V-product, and gives us a way of combining two V-categories.

Changing the base of enrichment

Any monoidal monotone V → W between symmetric monoidal preorders lets us convert V-categories into W-categories.
Construction 2.64. Let f : V → W be a monoidal monotone. Given a V-category C, one forms the associated W-category,say \(C_{f}\) as follows.

(i) We take the same objects: Ob(\(C_{f}\)) := Ob(C).

(ii) For any c, d \(\in\) Ob(C), put \(C_{f}\)(c,d) := f(C(c,d)).

This construction \(C_{f}\) does indeed obey the definition of a W-category, as can be seen by applying Definition 2.41 (of monoidal monotone) and Definition 2.46 (of V-category):

(a) for every c \(\in\) C, we have

\(I_{W}\) ≤ f (\(I_{V}\) ) ( f is monoidal monotone)

≤ f (C(c, c)) (C is V-category)

= \(C_{f}\)(c, c) (definition of \(C_{f}\))

(b) for every c, d, e \(\in\) Ob(C) we have

\(C_{f}\)(c,d) \(⊗_{W}\) \(C_{f}\)(d,e) = f(C(c,d)) \(⊗_{W}\) f(C(d,e)) (definition of \(C_{f}\))

≤ f (C(c, d) \(⊗_{V}\) C(d, e)) (f is monoidal monotone)

≤ f (C(c, e)) (C is V-category)

= \(C_{f}\)(c, e) (definition of \(C_{f}\))

Example 2.65.

As an example, consider the function f : [0, ∞] → {true, false} given by

It is easy to check that f is monotonic and that f preserves the monoidal product and monoidal unit; that is, it’s easy to show that f is a monoidal monotone. (Recall Exercise 2.44.)

Thus f lets us convert Lawvere metric spaces into preorders.

Exercise 2.67.

Recall the “regions of the world” Lawvere metric space from Exercise 2.52 and the text above it. We just learned that, using the monoidal monotone f in Eq. (2.66), we can convert it to a preorder. Draw the Hasse diagram for the preorder corresponding to the regions: US, Spain, and Boston. How could you interpret this preorder relation? ♦

Exercise 2.68.

1. Find another monoidal monotone g : Cost → Bool different from the one defined in Eq. (2.66).
2. Using Construction 2.64, both your monoidal monotone g and the monoidal monotone f in Eq. (2.66) can be used to convert a Lawvere metric space into a preorder. Find a Lawvere metric space X on which they give different answers, \(X_{f}\) \(\neq\) \(X_{g}\). ♦

Enriched functors

The notion of functor provides the most important type of relationship between categories.

Definition: 2.69.

Let X and Y be V-categories. A V-functor from X to Y, denoted F : X → Y, consists of one constituent:

(i) a function F : Ob(X) → Ob(Y)

subject to one constraint

(a) for all x1, x2 \(\in\) Ob(X), one has X(\(x_{1}\), \(x_{2}\)) ≤ Y(F(\(x_{1}\)), F(\(x_{2}\))).

Example 2.70.

For example, we have said several times—e.g. in Theorem 2.49—that preorders are Bool-categories, where X(\(x_{1}\),\(x_{2}\)) = true is denoted \(x_{1}\) ≤ \(x_{2}\). One would hope that monotone maps between preorders would correspond exactly to Bool-functors, and that’s true. A monotone map (X, \(≤_{X}\) ) → (Y, \(≤_{Y}\) ) is a function F : X → Y such that for every \(x_{1}\), \(x_{2}\) \(\in\) X, if \(x_{1}\) ≤ \(_{X}\) x2 then F(\(x_{1}\)) \(≤_{Y}\) F(\(x_{2}\)). In other words, we have

X(\(x_{1}\), \(x_{2}\)) ≤ Y(F(\(x_{1}\)), F(\(x_{2}\))),

where the above ≤ takes place in the enriching category V = Bool; this is exactly the condition from Definition 2.69.

Remark 2.71. In fact, we have what is called an equivalence of categories between the category of preorders and the category of Bool-categories. In the next chapter we will develop the ideas necessary to state what this means precisely (Remark 3.59).

Example 2.72.

Lawvere metric spaces are Cost-categories. The definition of Cost-functor should hopefully return a nice notion—a “friend”—from the theory of metric spaces, and it does: it recovers the notion of Lipschitz function. A Lipschitz (or more precisely, 1-Lipschitz) function is one under which the distance between any pair of points does not increase. That is, given Lawvere metric spaces (X, \(d_{X}\) ) and (Y, \(d_{Y}\) ), a Cost- functor between them is a function F: X → Y such that for every \(x_{1}\), \(x_{2}\) \(\in\) X we have \(d_{X}\)(\(x_{1}\), \(x_{2}\)) ≥ \(d_{Y}\)(F(\(x_{1}\)), F(\(x_{2}\))).

Exercise 2.73.

The concepts of opposite, dagger, and skeleton (see Examples 1.58 and 1.72 and Remark 1.35) extend from preorders to V-categories. The opposite of a V-category X is denoted \(X^{op}\) and is defined by

(i) Ob(\(X^{op}\)) := Ob(X), and
(ii) for all x, y \(\in\) X, we have \(X^{op}\)(x,y) := X(y,x).

A V-category X is a dagger V-category if the identity function is a V-functor †: X → \(X^{op}\). And a skeletal V-category is one in which if I ≤ X(x, y) and I ≤ X(y, x), then x = y.

Recall that an extended metric space (X, d) is a Lawvere metric space with two extra properties; see properties (b) and (c) in Definition 2.51.

1. Show that a skeletal dagger Cost-category is an extended metric space.
2. Use Exercise 1.73 to make sense of the following analogy: “preorders are to sets as Lawvere metric spaces are to extended metric spaces.”

Product V-categories

If V = (V, ≤, I, ⊗) is a symmetric monoidal preorder and X and Y are V-categories, then we can define their V-product, which is a new V-category.

Definition: 2.74.

Let X and Y be V-categories. Define their V-product, or simply product, to be the V-category X × Y with

(i) Ob(X × Y) := Ob(X) × Ob(Y),
(ii) (X × Y)((x, y), (x′, y′)) := X(x, x′) ⊗ Y(y, y′),

for two objects (x, y) and (x′, y′) in Ob(X × Y).

Product V-categories are indeed V-categories (Definition 2.46); see Exercise 2.75.

Exercise 2.75.

Let X × Y be the V-product of V-categories as in Definition 2.74.

1. Check that for every object (x, y) \(\in\) Ob(X × Y) we have I ≤ (X × Y)((x, y), (x, y)).
2. Check that for every three objects (\(x_{1}\), \(y_{1}\)), (\(x_{2}\), \(y_{2}\)), and (\(x_{3}\), \(y_{3}\)), we have (X×Y) ((\(x_{1}\),\(y_{1}\)),(\(x_{2}\),\(y_{2}\))) ⊗(X×Y) ((\(x_{2}\),\(y_{2}\)),(\(x_{3}\),\(y_{3}\))) ≤ (X×Y) ((\(x_{1}\),\(y_{1}\)),(\(x_{3}\),\(y_{3}\))).
3. We said at the start of Section 2.3.1 that the symmetry of V (condition (d) of Definition 2.2) would be required here. Point out exactly where that condition is used. ♦

When taking the product of two preorders (P, \(≤_{P}\) ) × (Q , \(≤_{Q}\) ), as first described in Example1.56,wesaythat(\(p_{1}\),\(q_{1}\)) ≤ (\(p_{2}\),\(q_{2}\)) iff both \(p_{1}\) ≤ \(p_{2}\) AND \(q_{1}\) ≤ \(q_{2}\); the AND is the monoidal product ⊗ from of Bool. Thus the product of preorders is an example of a Bool-product.

Example 2.76.

Let X and Y be the Lawvere metric spaces (i.e. Cost-categories) defined by the following weighted graphs:

Their product is defined by taking the product of their sets of objects, so there are six objects in X × Y. And the distance \(d_{X × Y}\)((x, y), (x′, y′)) between any two points is given by the sum dX(x, x′) + dY(y, y′).

Examine the following graph, and make sure you understand how easy it is to derive from the weighted graphs for X and Y in Eq. (2.77):

Exercise 2.78.

Consider \(\mathbb{R}\) as a Lawvere metric space, i.e. as a Cost-category (see Example 2.54). Form the Cost-product \(\mathbb{R}\) × \(\mathbb{R}\). What is the distance from (5, 6) to (−1, 4)?

Hint: apply Definition 2.74; the answer is not \(\sqrt{40}\). ♦

In terms of matrices, V-products are also quite straightforward. They generalize what is known as the Kronecker product of matrices. The matrices for X and Y in Eq. (2.77) are shown below

and their product is as follows:

We have drawn the product matrix as a block matrix, where there is one block shaped like X for every entry of Y. Make sure you can see each block as the X-matrix shifted by an entry in Y. This comes directly from the formula from Definition 2.74 and the fact that the monoidal product in Cost is +.