May 3, 2023

Metric Spaces as Enriched Categories II

Posted by Simon Willerton

In the previous post I set the scene a little for enriched category theory by implying that by working ‘over’ the category of sets is a bit like working ‘over’ the integers in algebra and sometimes it is more appropriate to have a different base category just as it is sometimes more appropriate to have a different base ring. Below, we’ll see with the case of metric spaces that changing the base category can seemingly change the flavour quite a lot.

An example which I was using for illustration in the last post was that whilst, on the one hand, you can encapsulate the group actions of a group $G$ via the functor category $[{\mathbf{\mathcal{B}}} G, \mathbf{\mathcal{S}et}]$ where ${\mathbf{\mathcal{B}}} G$ is the one object category with $G$ as its set of morphisms, on the other hand, you cannot in ordinary category theory encapsulate the category of representations of $A$, an algebra over the complex numbers, as a category of functors into the category of vector spaces as you might hope. Indeed in ordinary category theory you can’t really see the structure of a vector space lurking in the one-object category ${\mathbf{\mathcal{B}}} A$.

In this post I’ll explain what an enriched category is and how enriched category theory can, for example, allow a natural expression for a representatation category as a functor category. I’ll go on to show, following Lawvere’s insight, how metric spaces and much metric space theory can be seen to live within the realm of enriched category theory.

I’ll finish with an afterword on my experiences and thoughts on why enriched categories should be more appreciated but aren’t!

Enriched categories

We get the notion of an enriched category by first considering the following definition of category. (Actually, it’s a definition of locally small category as it requires that the morphisms between a pair of objects form a set rather than anything bigger.) Here we will switch notation for hom-sets, moving from $\operatorname{Hom}_{\mathbf{\mathcal{C}}}(c, c')$ to the category theorists’ ${\mathbf{\mathcal{C}}}(c, c')$.

Definition 1. A category consists of a collection of objects, $\operatorname{Ob}{\mathbf{\mathcal{C}}}$, together with the following data which is required to satisfy the unit and associativity axioms:

(i) for all $c, c'\in \operatorname{Ob}{\mathbf{\mathcal{C}}}$ there is a specified set ${\mathbf{\mathcal{C}}}(c, c') \in \operatorname{Ob}{\mathbf{\mathcal{S}et}};$

(ii) for all $c, c', c''\in \operatorname{Ob}{\mathbf{\mathcal{C}}}$ there is a specified function between sets ${\mathbf{\mathcal{C}}}(c, c') \times {\mathbf{\mathcal{C}}}(c', c'') \to {\mathbf{\mathcal{C}}}(c, c'');$

(iii) for all $c \in \operatorname{Ob}{\mathbf{\mathcal{C}}}$ there is a specified element $\operatorname{id}_c\in {\mathbf{\mathcal{C}}}(c, c)$ which we can consider as a specified function $\operatorname{id}_c\colon \{\ast\} \to {\mathbf{\mathcal{C}}}(c, c).$

We obtain the defintion of enriched category by simply replacing the references to the monoidal category $({\mathbf{\mathcal{S}et}}, \times, \{\ast\})$ with references to the monoidal category $({\mathbf{\mathcal{V}}}, \otimes, \mathbb{1})$ that we are using as our base category.

Definition 2. A category enriched over ${\mathbf{\mathcal{V}}}$, or simply a ${\mathbf{\mathcal{V}}}$-category, consists of a collection of objects, $\operatorname{Ob}{\mathbf{\mathcal{C}}}$, together with the following data which is required to satisfy unit and associativity axioms:

(i) for all $c, c'\in \operatorname{Ob}{\mathbf{\mathcal{C}}}$ there is a specified object ${\mathbf{\mathcal{C}}}(c, c') \in \operatorname{Ob}{\mathbf{\mathcal{V}}};$

(ii) for all $c, c', c''\in \operatorname{Ob}{\mathbf{\mathcal{C}}}$ there is a specified morphism in ${\mathbf{\mathcal{V}}}$ ${\mathbf{\mathcal{C}}}(c, c') \otimes {\mathbf{\mathcal{C}}}(c', c'') \to {\mathbf{\mathcal{C}}}(c, c'');$

(iii) for all $c \in \operatorname{Ob}{\mathbf{\mathcal{C}}}$ there is a specified morphism in ${\mathbf{\mathcal{V}}}$ $\operatorname{id}_c\colon \mathbb{1}\to {\mathbf{\mathcal{C}}}(c, c).$

If you take ${\mathbf{\mathcal{V}}}$ to be the category of vector spaces ${\mathbf{\mathcal{V}ect}}$ then you see that a ${\mathbf{\mathcal{V}ect}}$-category is like a category except the ‘hom-sets’ are actually vector spaces and the composition maps are bilinear. The identity morphism in ${\mathbf{\mathcal{C}}}(c, c)$ is recovered by taking $\operatorname{id}_c(1)$ where $1\in \mathbb{C}$ means the unit complex number. For instance, for an algebra $A$, there is the one-object ${\mathbf{\mathcal{V}ect}}$-category ${\mathbf{\mathcal{B}}}A$ and the ${\mathbf{\mathcal{V}ect}}$-category ${\mathbf{\mathcal{R}ep}}(A)$ of $A$-representations.

There is an accompanying notion of ${\mathbf{\mathcal{V}}}$-functor between ${\mathbf{\mathcal{V}}}$-categories.

Definition 3. For ${\mathbf{\mathcal{V}}}$-categories ${\mathbf{\mathcal{C}}}$ and ${\mathbf{\mathcal{D}}}$, a ${\mathbf{\mathcal{V}}}$-functor $F\colon {\mathbf{\mathcal{C}}}\to {\mathbf{\mathcal{D}}}$ consists of a function $F\colon \operatorname{Ob}{\mathbf{\mathcal{C}}}\to \operatorname{Ob}{\mathbf{\mathcal{D}}}$ and for each pair $c,c'\in \operatorname{Ob}{\mathbf{\mathcal{C}}}$ a specified morphism ${\mathbf{\mathcal{C}}}(c, c') \to {\mathbf{\mathcal{D}}}(F(c), F(c'))$ in ${\mathbf{\mathcal{V}}}$. These are required to satisfy composition and unit axioms.

In particular, a ${\mathbf{\mathcal{V}ect}}$-functor is required to be given by linear maps between the hom-spaces.

The nicer that the enriching category ${\mathbf{\mathcal{V}}}$ is, the nicer that the theory of ${\mathbf{\mathcal{V}}}$-categories is. I won’t go into detail here, but in particular, we have the following.

• If ${\mathbf{\mathcal{V}}}$ is so-called closed monoidal then you can make ${\mathbf{\mathcal{V}}}$ into a ${\mathbf{\mathcal{V}}}$-category itself, analogous to how you can make a ring into a module over itself.

• If ${\mathbf{\mathcal{V}}}$ is braided (in particular if it is symmetric) so that there are natural and coherent isomorphisms $v_1 \otimes v_2 \cong \v_2 \otimes v_1$ for $v_1, v_2 \in {\mathbf{\mathcal{V}}}$, then for a pair of ${\mathbf{\mathcal{V}}}$-categories ${\mathbf{\mathcal{C}}}$ and ${\mathbf{\mathcal{D}}}$ you can form the tensor product ${\mathbf{\mathcal{V}}}$-category ${\mathbf{\mathcal{C}}}\otimes {\mathbf{\mathcal{D}}}$, analogous to how you can take tensor product of modules for commutative rings.

• If ${\mathbf{\mathcal{V}}}$ has sufficiently many limits then for a pair of ${\mathbf{\mathcal{V}}}$-categories ${\mathbf{\mathcal{C}}}$ and ${\mathbf{\mathcal{D}}}$ you can form the functor ${\mathbf{\mathcal{V}}}$-category $[{\mathbf{\mathcal{C}}}, {\mathbf{\mathcal{D}}}]$ where the objects are ${\mathbf{\mathcal{V}}}$-functors of the form $F\colon {\mathbf{\mathcal{C}}}\to {\mathbf{\mathcal{D}}}$. (There can be size issues here if $C$ is not sufficiently small.)

In many cases of interest the enriching category ${\mathbf{\mathcal{V}}}$ is braided closed monoidal with sufficiently many limits, so a small ${\mathbf{\mathcal{V}}}$-category ${\mathbf{\mathcal{C}}}$ has a ${\mathbf{\mathcal{V}}}$-category $[{\mathbf{\mathcal{C}}}, {\mathbf{\mathcal{V}}}]$ of scalar-valued functors, or presheafs, on ${\mathbf{\mathcal{C}}}$.

For instance, for an algebra $A$, the ${\mathbf{\mathcal{V}ect}}$-category ${\mathbf{\mathcal{R}ep}}(A)$ of representations of $A$ can be given as the functor category on ${\mathbf{\mathcal{B}}}A$:

${\mathbf{\mathcal{R}ep}}_A = [{\mathbf{\mathcal{B}}}A, {\mathbf{\mathcal{V}ect}}].$

This is what we were hoping for above.

Metric spaces as $\overline{\mathbb{R}}_{+}$-categories

If we enrich over the monoidal category $\overline{\mathbb{R}}_{+}$ of extended non-negative real numbers then we can simplify the general definition a little and get the following.

Definition 4. An $\overline{\mathbb{R}}_{+}$-category, consists of a collection of objects, $\operatorname{Ob}X$, together with the following data:

(i) for all $x, x'\in \operatorname{Ob}X$ there is a specified number $X(x, x') \in [0,\infty];$

(ii) for all $x, x', x''\in \operatorname{Ob}X$ there is an inequality $X(x, x') + X(x', x'') \ge X(x, x'');$

(iii) for all $x \in \operatorname{Ob}X$ there is an equality $0 = X(x, x).$

There are several things to note here. Firstly, because $\overline{\mathbb{R}}_{+}$ is a thin category, all diagrams commute so the associativity and unitality conditions are automatically satisfied and so do not need to be specified. Secondly, the “composition morphism” in (ii) is clearly a “triangle inequality”, so we can think of the hom-object $X(x, x')$ as a “distance” from $x$ to $x'$. Thirdly, from the definition of enriched category the “zero self-distance” in (iii) would be written as $0 \ge X(x, x)$ but as we know that the hom-object is non-negative, we can conclude the inequality is actually an equality.

If, as indicated above, we think of the hom-objects as being distances then we can think of a $\overline{\mathbb{R}}_{+}$-category as a kind of generalized metric space. The traditional notion of metric space was due to Fréchet: by contrast, $\overline{\mathbb{R}}_{+}$-categories are sometimes known as Lawvere metric spaces; these differ from Fréchet metric spaces in the following three ways.

1. The distance is not necessarily symmetric, so we allow $X(x, x') \ne X(x', x)$.

2. The distance between two points can be infinite.

3. The distance from one point to a different point can be zero.

We’ll see some examples illustrating these below.

From this generalized metric space perspective, an $\overline{\mathbb{R}}_{+}$-functor $f\colon X \to Y$ is viewed as a short map or distance non-increasing function, so it is a function $f\colon \operatorname{Ob}X \to \operatorname{Ob}Y$ such that $X(x, x') \ge Y\big(f(x), f(x')\big)$ for all $x, x' \in \operatorname{Ob}X$.

Here are the promised examples.

• We can equip $\overline{\mathbb{R}}_{+}$ with the structure of an $\overline{\mathbb{R}}_{+}$-category. This is basically the fact that $\overline{\mathbb{R}}_{+}$ is a closed monoidal category. It is a slight abuse of notation to refer to the category and the $\overline{\mathbb{R}}_{+}$-category as $\overline{\mathbb{R}}_{+}$ but it doesn’t usually cause confusion. Anyway we define the generalized metric as follows. $\overline{\mathbb{R}}_{+}(a, b) = b  \stackrel{.}{-} a \coloneqq \max(b - a, 0).$ The operation $ \stackrel{.}{-} $ is sometimes referred to as truncated subtraction. If we think of the extended non-negative real numbers as standing going upwards, then for $b \ge a$ we can think of it being ‘free’ to descend from $b$ to $a$, but having a ‘cost’ of $b-a$ associated with ascending from $a$ to $b$.

• As $\overline{\mathbb{R}}_{+}$ is sufficiently nice – it has all limits, which are given by suprema – if $X$ and $Y$ are $\overline{\mathbb{R}}_{+}$-categories then there is the functor $\overline{\mathbb{R}}_{+}$-category $[X, Y]$ consisting of short maps from $X$ to $Y$ and the generalized metric given as follows:

$[X, Y](f, g) \coloneqq\sup_{x\in X} Y(f(x), g(x)).$

This is a measurement of the furthest apart that $f$ and $g$ get. In particular, if we take $Y= \overline{\mathbb{R}}_{+}$ then we can get the following generalized metric on scalar-valued short maps: $[X, \overline{\mathbb{R}}_{+}](f, g) \coloneqq\sup_{x\in X} \left(g(x)  \stackrel{.}{-} f(x)\right).$

The Yoneda embedding is an distance preserving (ie. isometric) function: $X \hookrightarrow [X^{\mathrm{op}}, \overline{\mathbb{R}}_{+}]; \quad x \mapsto X({-}, x).$ In the case of a classical metric space we have $X = X^\mathrm{op}$ as the distance is symmetric and the embedding in sometimes known as the Kuratowski embedding.

We can see that this notion of generalized metric on function spaces is a refinement of the standard notion of ‘sup-metric’ on function spaces. If we look at all functions from the unit interval $I = \{x \mid 0 \le x \le 1\}$ to the set $\mathbb{R}_{\ge 0}$ then we can consider these as short maps from $I_\delta$ to $\overline{\mathbb{R}}_{+}$ where $I_\delta$ is the interval equipped with the discrete metric so that the distance $I(x, x')$ is $0$ if $x=x'$ and is $\infty$ otherwise. Then for functions $f, g\colon I\to \mathbb{R}_{\ge 0}$ we have that the sup-metric between them is a symmetrization of the enriched category metric: \begin{aligned}\textstyle\sup_x \left| f(x) - g(x) \right| &= \max\Bigl( \sup_x (g(x)  \stackrel{.}{-} f(x)), \sup_x (f(x)  \stackrel{.}{-} g(x))\Bigr)\\ &=\max\Bigl([I_\delta, \overline{\mathbb{R}}_{+}](f, g), [I_\delta, \overline{\mathbb{R}}_{+}](g, f)\Bigr). \end{aligned}

• Now we can see an example where even if we are just interested in classical metric spaces we naturally end up with an asymmetric metric, although it is usually symmetrized to obtain the Hausdorff metric! Given a classical metric space $M$ we can take the set of compact, non-empty subsets, $S_M \coloneqq\{A \subseteq M \mid A\ \text{compact},A\ne \emptyset\}$. We can then define a generalized metric on this by $S_M(A, B) \coloneqq \sup_{a\in A}\inf_{b\in B} M(a, b).$

One way of thinking about $S_M(A, B)$ is that it is measuring the furthest you would have to go if you were dropped at a random point in $A$ and wanted to take the shortest route to $B$. Because we are considering compact sets we have the following simple characterization of zero-distance, which is telling us that the generalized metric is encoding the usual order on subsets: $S_M(A, B) = 0 \quad \Longleftrightarrow \quad A \subseteq B.$ (If we hadn’t used compact subsets then we wouldn’t have had such a nice characterization.) This leads us to see that this generalized metric is naturally asymmetric even though we started with a symmetric metric on $M$. The usual Hausdorff metric $\operatorname{d}_{\mathrm{H}}$ is obtained by symmetrizing this generalized metric: $\operatorname{d}_{\mathrm{H}}(A, B) = \max\big(S_M(A, B), S_M(B, A)\big).$ Clearly, however, the Hausdorff metric is losing information, such as the partial order, that was contained in the generalized metric.

This concludes the quick introduction to metric spaces as enriched categories. However, this perspective does allow other categorical techniques to be used with metric spaces, leading to some fruitful mathematics.

Afterword

It’s probably worth mentioning my experiences a little. As a Part III student at Cambridge, like many students before and after me, I attended Peter Johnstone’s Category Theory course, so I had familiarity with category theory, but, as I didn’t take the exam, I didn’t have a deep understanding. I then went on to do my PhD and later work on things related to algebraic topology and topological quantum field theory which both have strong categorical underpinnings. Coming back to the basics of category theory when I was involved in the Catsters with Eugenia Cheng I realised that various things didn’t chime with my experience of using categories and certain things seemed almost arcane to me, for example the importance put on things like monoids and functors into the category of sets.

It wasn’t until many, many years later that I started to understand that this cognitive dissonance was because in areas like algebraic topology, topological field theory, representation theory and algebraic geometry it is much more useful to consider that you’re not working over the category of sets but over some other category like vector space or abelian groups or chain complexes. As a ‘working mathematician’, rather than monoids and functors into the category of sets, I would have been much more familiar with algebras or rings and with functors into the category of vector spaces or abelian groups.

Although I was aware of the notion of enriched category theory, I didn’t get a thorough appreciation of quite what it meant until I’d spent a lot of time thinking about metric spaces in this context in particular via Lawvere’s rather wonderful paper Metric spaces, generalized logic and closed categories. I think one reason that enriched category is not more appreciated as a perspective is that the main text, Kelly’s Basic Concepts of Enriched Category Theory, is not particularly welcoming and only entered into by the brave! I could probably try to make the area more welcoming by polishing up my first attempt at a draft book on enriched categories, but with all of the other things I want to do, unfortunately I don’t see that happening too soon.

Posted at May 3, 2023 2:16 PM UTC

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Re: Metric Spaces as Enriched Categories II

You liken enriched categories to working with k-algebras for k not necessarily the integers. Is there a notion of “base change” in the enriched category context?

Typos btw: definiton, cateogry.

Posted by: Allen Knutson on May 4, 2023 3:16 AM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

[TL;DR: A lax monoidal functor $\mathcal{V}\to \mathcal{W}$ gives rise to a base-change functor $\mathcal{V}\text{-cat}\to \mathcal{W}\operatorname{-cat}$.]

I guess that “base change” can mean a few different things, but in the enriched category context there is the following.

Suppose that $(\mathcal{V}, \otimes, \mathbb{1}_{\mathcal{V}})$ and $(\mathcal{W}, \odot, \mathbb{1}_{\mathcal{W}})$ are two monoidal categories to be used as base categories for enrichment. Then there are the associated (ordinary) categories $\mathcal{V}$-cat and $\mathcal{W}$-cat, where $\mathcal{V}$-cat is the category of $\mathcal{V}$-categories and $\mathcal{V}$-functors, and similarly for $\mathcal{W}$-cat.

[You can also consider $\mathcal{V}$-natural transformations and have an associated $2$-category of $\mathcal{V}$-categories, $\mathcal{V}$-functors and $\mathcal{V}$-natural transformations, but I’ll keep things simpler here.]

The correct notion of map between the base categories here is that of lax-monoidal functor, and that means a functor $F\colon \mathcal{V} \to \mathcal{W}$ which laxly preserves the monoidal structure in the sense that there are comparison maps

$F(v_1) \odot F(v_2) \to F(v_1 \otimes v_2) \quad for all v_1, v_2$

and

$\mathbb{1}_{\mathcal{W}} \to F(\mathbb{1}_{\mathcal{V}})$

which are natural and appropriately coherent, but which are not necessarily isomorphisms.

For instance, every monoidal category $(\mathcal{V}, \otimes, \mathbb{1}_{\mathcal{V}})$ has a lax monoidal functor to the category of sets, $\Gamma \colon \mathcal{V} \to \mathbf{\mathcal{S}et}$, given by

$\Gamma(v) \coloneqq \mathcal{V}(\mathbb{1}_\mathcal{V}, v).$

This goes by many names, such as the “underlying set” functor or the “set of elements” functor or “set of generalized elements” functor. You can think about what that functor is when $\mathcal{V}$ is $\mathbf{\mathcal{V}ect}$ or $\bar{\mathbb{R}}_+$.

Another example of a lax monoidal functor is the “free vector space” functor $({\mathbf{\mathcal{S}et}}, \times, \{\ast\}) \to (\mathbf{\mathcal{V}ect}, \otimes, \mathbb{C})$ given by $S\mapsto \mathbb{C}^S$.

Anyway, given such a lax monoidal functor $F\colon \mathcal{V}\to \mathcal{W}$ you get a “base-change” functor $F_\ast \colon\mathcal{V}\text{-cat}\to \mathcal{W}\text{-cat}$. If $\mathcal{C}$ is a $\mathcal{V}$-category then $F_\ast(\mathcal{C})$ has the same objects as $\mathcal{C}$ but the hom-objects are given by applying $F$:

$F_\ast(\mathcal{C})(c_1, c_2) \coloneqq F\bigl(\mathcal{C}(c_1, c_2)\bigr).$

From the “set of generalized elements” functor $\Gamma \colon \mathcal{V} \to \mathbf{\mathcal{S}et}$ above, we get, for any base category $\mathcal{V}$ an “underlying category” functor $\Gamma_\ast \colon \mathcal{V}\text{-cat} \to \mathbf{\mathcal{S}et}\text{-cat}$.

Exercises (for anyone)

1. What is the “underlying category” of a $\mathbf{\mathcal{V}ect}$-category?

2. What is the “underlying category” of an $\bar{\mathbb{R}}_+$-category?

3. Given a classical metric spaces $M$, what is the “underlying category” $\Gamma_\ast (S_M)$ of $S_M$, the generalized metric space of non-empty, compact subsets of $M$?

Posted by: Simon Willerton on May 4, 2023 12:37 PM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

Thanks for a great post. Attempt at the exercises:

1. I think it will be the “actual” underlying category (since a Vect-enriched category is in particular a category).

2. I think it will be the poset (thought of as a category) where $x \lt y$ if and only if $Hom(x,y) \in (0, \infty)$.

3. I think it will be the poset (thought of as a category) of non-empty compact subspaces of $M$, ordered by inclusion.

Posted by: Bruce Bartlett on May 4, 2023 6:59 PM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

If I’ve untangled the definitions correctly, then I think exercise 2 “is” the weighted digraph representation of the generalized metric space.

So exercise 3, as a special case of that, is the digraph whose nodes are nonempty compact subsets of $M$, and the unique arrow $X \to Y$ has as weight the maximum distance from a point in $X$ to $Y$ (or maybe the other way around).

I agree with Bruce on exercise 1.

Posted by: Mark Meckes on May 4, 2023 11:10 PM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

1. What is the “underlying category” of a $\mathbf{\mathcal{V}ect}$-category?

Bruce said:

I think it will be the “actual” underlying category (since a Vect-enriched category is in particular a category).

Well, the answer given is kind of right, but the reason given is not great and will lead to errors, as I’ll try to illustrate with an example.

[From a didactic point of view it’s actually a great answer as it illustrates a common line of reasoning. Knowing Bruce, I could almost think that he gifted it to me. :-) ]

Suppose we take ${\mathbf{\mathcal{V}}}$ to be ${\mathbf{\mathcal{R}ep}}(G)$ the category of finite dimensional complex representations of a finite group $G$. The objects are representations of $G$: I’ll write a representation as $(V, \tau)$ for $\tau \colon G\to \operatorname{End}(V))$. The hom set ${\mathbf{\mathcal{R}ep}}(G)\bigl((V, \tau), (W, \rho)\bigr)$ is the set of equivariant or intertwining maps from $(V, \tau)$ to $(W, \rho)$. The category ${\mathbf{\mathcal{R}ep}}(G)$ is a monoidal category using the tensor product of representations and the monoidal unit is the complex numbers with the trivial action.

Define the ${\mathbf{\mathcal{V}}}$-category (i.e., ${\mathbf{\mathcal{R}ep}}(G)$-category) ${\mathbf{\mathcal{C}}}$ to have, again, representations of $G$ as objects, but now the homs are going to be ${\mathbf{\mathcal{V}}}$ objects, i.e., $G$-representations, and we take ${\mathbf{\mathcal{C}}}\bigl((V, \tau), (W, \rho)\bigr)$ to be $\operatorname{Lin}(V, W)$ the vector space of all linear maps from $V$ to $W$ equipped with the action defined for $\varphi\in \operatorname{Lin}(V, W)$ by $g \cdot \varphi= \rho(g)\circ \varphi\circ \tau(g^{-1})$. This is the usual action you would expect.

What is $\Gamma_\ast({\mathbf{\mathcal{C}}})$, the “underlying category” of ${\mathbf{\mathcal{C}}}$?

If we apply the reasoning that the hom-objects are already sets (sets of linear maps) and so we already have a category there then we obtain the answer that it is the category where the objects are representations of $G$ and the hom-sets are the sets of linear maps between the underlying vector spaces. This is the wrong answer!

Exercise. 4. Why is this the wrong answer? What is the right answer?

Hint: You should first figure out the “set of generalized elements” functor $\Gamma \colon {\mathbf{\mathcal{R}ep}}(G) \to {\mathbf{\mathcal{S}et}}$.

Posted by: Simon Willerton on May 5, 2023 10:53 AM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

If $(V,\tau)$ is a $G$-representation, then $\Gamma(V,\tau)$ is the set of $G$-intertwining maps from the trivial representation to $(V,\tau)$. That is, $F \in \Gamma(V,\tau)$ is a linear map $F:\mathbb{C} \to V$ such that $\tau(g) \circ F = F$ for every $g \in G$, i.e., the one-dimensional (assuming non-zero) image of $F$ is a $\tau$-invariant subspace.

So, up to a canonical identification of $F \in Lin(\mathbb{C},V)$ with $F(1)$, $\Gamma(V,\tau)$ is the subspace of $V$ on which $G$ acts trivially via $\tau$. (My knowledge of representation theory is quite rusty and I’m forgetting what that would usually be called.)

Then in your category $\mathcal{C}$, $\Gamma_*((V,\tau),(W,\rho))$ is the subspace of $Lin(V,W)$ consisting of linear maps $\varphi$ such that $\rho(g) \circ \varphi \circ \tau(g^{-1}) = \varphi$. In other words, the space of intertwiners. (Except that in $\mathcal{S}et$ we’re forgetting that it’s a vector space and just treating it as a set.)

In the case of $\mathcal{V}ect$-categories, none of the intertwining considerations come up and you end up with all of $Lin(V,W)$, as Bruce said.

If we asked for the “underlying $\mathcal{V}ect$-category of a $\mathcal{R}ep(G)$-category”, all this would be the same except at the end we would still remember that the homs are vector spaces, right?

Posted by: Mark Meckes on May 5, 2023 2:21 PM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

In working through that I was so focused on the individual steps that I never turned back to see where I was coming from, so I missed the punchline: the underlying category of the $\mathcal{R}ep(G)$-category $\mathcal{C}$ is the category $\mathcal{R}ep(G)$ itself.

And my last sentence above is the observation that likewise, the underlying $\mathcal{V}ect$-category of the $\mathcal{R}ep(G)$-category $\mathcal{C}$ is the $\mathcal{V}ect$-category $\mathcal{R}ep(G)$ itself.

Posted by: Mark Meckes on May 5, 2023 6:52 PM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

(Wrong answer)… Knowing Bruce, I could almost think that he gifted it to me.

Sadly it was a genuine conceptual error :-)

Exercise. 4. Why is this the wrong answer? What is the right answer?

The right answer is the one Mark gave.

It’s a nice example. Is there a “derived” version of this enriched category theory where the action of G on the hom-sets is “up to homotopy”… in that case the “underlying derived category” of the derived version of $\mathcal{Rep}(G)$ would be the derived quotient or something.

Posted by: Bruce Bartlett on May 9, 2023 9:12 AM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

Bruce said

Sadly it was a genuine conceptual error :-)

But still it was a good way to get the conversation started!

As for your question about derived things, I don’t know at this point.

I can say two things that might or might not be relevant to how you are thinking.

Firstly, consider the category of chain complexes of $R$-modules for a commutative ring $R$: this is closed monoidal.

The “underlying set functor” takes a chain complex to its set of zero-cycles (which is just its set of zero-chains if the complex is non-negatively graded). Naively I might have thought that it gave zeroth homology, but that’s not the case. You can then see what the “underlying category” of a chain-complex-enriched category is.

Secondly, taking homology groups gives a lax monoidal functor from the category of chain complexes to the category of graded $R$-modules. For nice $R$, the Künneth Theorem tells us how far the functor is from being strong monoidal: for example, for $R$ a field, the functor is strong monoidal, and for $R$ a PID, such as $\mathbb{Z}$, the failure to be strong is measured by $\operatorname{Tor}$ groups.

Anyway, this tells us that taking homology of hom-complexes gives a functor from chain-complex-enriched categories to graded-module-enriched categories.

Posted by: Simon Willerton on May 9, 2023 1:40 PM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

The underlying set functor takes a chain complex to its set of zero-cycles (which is just its set of zero-chains if the complex is non-negatively graded).

Ok, that’s interesting.

Secondly, taking homology groups gives a lax monoidal functor from the category of chain complexes to the category of graded modules… Anyway, this tells us that taking homology of hom-complexes gives a functor from chain-complex-enriched categories to graded-module-enriched categories.

That’s nice - it’s a nice example of “lax monoidal” being very useful. In practice, people will just write down this functor and check by hand that it is a functor. But one thing the enriched category theory has done is isolate the crucial thing which makes it a functor (the lax monoidal-ness).

Posted by: Bruce Bartlett on May 18, 2023 11:55 AM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

2${}$. What is the “underlying category” of an $\bar{\mathbb{R}}_+$-category?

Neither Bruce nor Mark have quite nailed this. So let me go back a step.

Exercise. 2a. What is the “set of generalized elements” functor $\Gamma\colon \bar{\mathbb{R}}_+ \to \mathbf{\mathcal{S}et}$?

Posted by: Simon Willerton on May 5, 2023 11:07 AM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

Oops, I think I had my arrows reversed in $\overline{\mathbb{R}}_+$. So now I think the underlying category of a classical metric space as an $\overline{\mathbb{R}}_+$-category is just the discrete category on the same set of objects.

But for a generalized (Lawvere) metric space, there is also a unique morphism $x \to y$ whenever $d(x,y) = 0$.

Posted by: Mark Meckes on May 5, 2023 1:18 PM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

Hit “Post” sooner than I meant to.

So the underlying category of an $\overline{\mathbb{R}}_+$-category $X$ is a poset where the partial order on objects is given by $x \le y$ iff $X(x,y) = 0$.

For the case of $S_M$, the partial order on compact nonempty subsets of $M$ says that $A \le B$ iff $\sup_{A\in A} \inf_{b \in B} d(a,b) = 0$, hence iff $\inf_{b \in B} d(a,b) = 0$ for every $a \in A$, hence (by compactness) iff $A \subseteq B$. Which is to say, I now believe Bruce’s answer to exercise 3 was correct.

Posted by: Mark Meckes on May 5, 2023 1:30 PM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

Full marks to Mark on all the questions now.

I will just fill in some details.

Exercise. 2a. What is the “set of generalized elements” functor $\Gamma\colon \bar{\mathbb{R}}_+ \to \mathbf{\mathcal{S}et}$.

Solution. If $x \in \bar{\mathbb{R}}_+$ then $\Gamma(x) \coloneqq \bar{\mathbb{R}}_+(0, x)$ and in $\bar{\mathbb{R}}_+$ there is a single morphism $a\to b$ precisely when $a\ge b$ and no morphism otherwise. Thus

$\Gamma(x) = \begin{cases} \{0\xrightarrow{!} 0 \}&\text{if}\quad x=0;\\ \emptyset&\text{if}\quad x\mathop{>}0.\end{cases}$

Posted by: Simon Willerton on May 5, 2023 5:15 PM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

Allen said:

Typos btw: definiton, cateogry.

Thanks; fixed.

Posted by: Simon Willerton on May 5, 2023 10:58 AM | Permalink | Reply to this

Mozibur Rahman

I had a similar experience with Kelly! But I found a chapter in Borceaux’s second volume of Categorical Algebra a bit more enlightening. But as I’m more of an interested bystander rather than a professional mathematician I promptly forgot most of it.

Recently though I dipped into it again as I got curious as to why it’s not possible to enrich over topological spaces. According to Borceaux it’s enough for a category to be monoidal to enrich over it. And as Top has a Cartesian monoidal structure, we can enrich over it. And because it’s symmetric monoidal, we can even construct its opposite. But because it’s not closed monoidal, we can’t show that Top itself is an enriched category. And this means we can’t have hom categories and hence no Yoneda lemma. So although we can enrich over Top, it doesn’t have enough nice properties that we have a nice healthy enriched category. I’m guessing that this is what is meant by saying we can’t enrich over Top.

However, diffeology is nice category - which I think - and I’m no expert, so take this with a pinch of salt - that is a kind of blending between topology and smooth spaces, taking the best of both. As it is a closed Cartesian category, we ought to be able to get a nice enriched category over it which I think ought to be a nice replacement for Top enriched categories.

But can we do better?

This leads me onto a question which I hope isn’t too naive. As diffeology is a certain construction over smooth manifolds, it seems to me we ought to be able to do the same over topological manifolds and which for the sake of this post, I’ll christen as “topeology”. And this ought to also be Cartesian closed and so we ought to be able to get “topeologically enriched categories” which should be closer replacement.

Probably this is all well known and I just don’t happen to know it. And even more probably I got it all wrong. Like I said I’m an amateur at this. So feel free to correct me.

Posted by: Mozibur Rahman Ullah on May 4, 2023 11:43 AM | Permalink | Reply to this

Re: Mozibur Rahman

I don’t know much about this, I’m afraid, but hopefully there are people around here who can say more.

I certainly know that the category of topological spaces is not a nice category for working with, definitely not a nice category of spaces in the sense of the nLab; and I know that people tend to enrich over other categories of “spaces” such as compactly generated spaces or simplicial sets which are closed monoidal.

Posted by: Simon Willerton on May 4, 2023 1:19 PM | Permalink | Reply to this

Re: Mozibur Rahman

Mozibur wrote:

However, diffeology is nice category - which I think - and I’m no expert, so take this with a pinch of salt - that is a kind of blending between topology and smooth spaces, taking the best of both.

Most people I know regard diffeological spaces not as a blend of topology and smooth spaces, but as a nice concept of smooth spaces. A diffeological space is a generalization of smooth manifold where we weaken the definition of ‘chart’ enough to get a cartesian closed category of diffeological spaces. By the way, Alex Hoffnung and I have a paper on this:

One thing we point out is that diffeological spaces form something better than a cartesian closed category: they have all limits and colimits, and indeed form a ‘quasitopos’—but not a topos. A slight further generalization (and simplification!) produces an actual topos.

As it is a closed Cartesian category, we ought to be able to get a nice enriched category over it which I think ought to be a nice replacement for Top enriched categories.

Yes, it’s nice to enrich over diffeological spaces. Personally I would say that categories enriched over diffeological spaces form a nice replacement for categories enriched over the category of smooth manifolds (which is not cartesian closed). It allows you to do differential geometry on the hom-sets.

I got interested in diffeological spaces because I was doing things in geometry where I wanted to look at categories internal to diffeological spaces. But categories enriched in diffeological spaces can be seen as a special case of these.

As diffeology is a certain construction over smooth manifolds, it seems to me we ought to be able to do the same over topological manifolds and which for the sake of this post, I’ll christen as “topeology”.

Yes, I see a way to do this: just take the definition of diffeological space and replace the word ‘smooth’ with the word ‘continuous’ everywhere. This gives a quasitopos which you can think of as an improved version of the category of topological manifolds.

Posted by: John Baez on May 5, 2023 7:35 AM | Permalink | Reply to this

Mozibur Rahman

Thanks for the clarification.

It was actually your paper with Alex Hoffnung on diffeological spaces that got me interested in them! I thought it very readable and a great introduction to the area.

I also want to thank Simon Willerton for his Catsters videos on category theory with Eugenia Cheng. I thought they were fab too.

As far as I understand quasitoposes - which isn’t very far - they have a classifier for strong subobjects instead of ordinary objects.

I’m curious as to how one turns the category of diffeological spaces into a topos? Presumably we have to get rid of the points in some way. Actually, doesn’t that also turn each topological space into Cartesian closed category too? I mean by considering it simply as a locale of opens? Am I then being too hopeful in expecting the category of all such locales - I think they’re called sober locales - form a closed Cartesian category?

Just one last thought here given that this cafe is devoted to philosophy. I recall reading somewhere that Aristotle was against infinite division in the continuum. This was a bit of a puzzle to me until I realised the other day it was likely he was against this because he was against the notion that the continuum was made up of points.

Posted by: Mozibur Rahman Ullah on May 6, 2023 3:02 AM | Permalink | Reply to this

Mozibur Rahman

Well, I’ve discovered that Loc isn’t closed Cartesian and that the exponentiable locales are the locally compact ones. So it was too much to hope for.

Posted by: Mozibur Rahman Ullah on May 6, 2023 9:38 AM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

Just two quick thoughts.

Truncated subtraction is like “ReLU” in machine learning, i.e. $f(x) = x$ if $x \geq 1$ and $f(x) = 0$ otherwise. Machine learning has been on my mind :-)

It is also like put and call options in finance. If the share price today is dollars, and today you are given a call option on the share 1 year from now at a strike price $K=20$ dollars (i.e. you will have the option, but not the obligation, to buy the share at 20 dollars 1 year from now), then the payout 1 year from now will be

$P = max(S - K, 0)$

where $S$ is the price of the share 1 year from now. So the payout of a call option is the truncated difference between the share price and the strike price (and conversely for a put option).

Posted by: Bruce Bartlett on May 4, 2023 7:11 PM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

It reminds me of valves and diodes, (ideally) physical one-way streets.

Posted by: unekdoud on May 8, 2023 1:21 AM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

Simon wrote:

I could probably try to make the area more welcoming by polishing up my first attempt at a draft book on enriched categories, but with all of the other things I want to do, unfortunately I don’t see that happening too soon.

Maybe it’d help, and be a lot less work, to turn this series of blog posts into an expository article that you could put on the arXiv. The perfect can be the enemy of the good!

Posted by: John Baez on May 5, 2023 2:53 PM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

You might like to add some comments to the MathOverflow question “Mention of Bernoulli principle by Bill Lawvere” to widen the audience.

Posted by: Tom Copeland on May 14, 2023 10:20 PM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

Found another typo: expcept

Thanks for the interesting post!

Posted by: Stéphane Desarzens on May 6, 2023 10:56 PM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II

Fixed, thanks.

Posted by: Simon Willerton on May 9, 2023 2:24 PM | Permalink | Reply to this

Re: Metric Spaces as Enriched Categories II: Related MO-Q

I suspect you could provide some informative comments to the recently posted MathOverflow question Mention of Bernoulli principle by Bill Lawvere.

Posted by: Tom Copeland on May 14, 2023 10:15 PM | Permalink | Reply to this

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