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Stream: learning: questions

Topic: Spaces vs/and Categories

John Onstead (Mar 14 2024 at 14:36):

Recently, I read an article here that specified a number of points that will be well familiar to anyone on this server. For instance, the author discusses how "mathematical objects are like people: we understand them better when we study them in families with other objects of the same type". In addition, they later discuss mappings between classes of objects and how they allow you to use your understanding of a simpler class to understand a more complicated one, and how maps will take an object of one type to a similar one of another type. These sound exactly like categories and functors, right? Well, not quite- the author is talking about intersection theory and moduli spaces, not category theory and categories, but it really is odd just how similar these talking points are! There are many similar examples of families of similar objects being able to form either categories or spaces with which they can be studied together, but I am having difficulty reconciling this...

John Onstead (Mar 14 2024 at 14:41):

One way is by seeing spaces like moduli spaces as certain enriched categories, notably Lawvere metric spaces, but this isn't fully satisfying to me because the information about the mathematical objects is conveyed in different ways from spaces (and Lawvere metric spaces) to categories. After a recent discussion on "chimera categories" (which may or may not be relevant here), it got me thinking about if there was such a thing as a "category with distances". This could potentially be a "chimera category" with an object of objects that is a space like a metric space of some form and an object of morphisms which is a set. It's basically a 1-category with the extra structure of having a notion of "distance" between the objects, which is meant to convey some level of "similarity" between the math objects the category's objects represent. In any case, this construction would allow us to see how mathematical objects can form a single structure that is BOTH a category AND a space at the same time! However, I am not exactly sure how to construct such a category, and if such a category has been described already in the literature, or what uses it could have. Any thoughts on these, and also on any personal thoughts about spaces vs categories in terms of settings by which we can study families of math objects, are very welcome!

David Egolf (Mar 14 2024 at 15:24):

I was recently reading this blog post. The things you are saying about having a structure that can be viewed as a category and as a space reminds me of an idea discussed in that blog post, where I learned that Grothendieck defined sheaves on any category equipped with a 'coverage' or 'Grothendieck topology'. And I also learned that sheaves on a topological space contain a lot of information about that topological space. So maybe one can consider the category of sheaves on a site (which is a category equipped with a coverage) as a sort of topological thing that also contains a lot of information about the original category?

David Egolf (Mar 14 2024 at 15:30):

A less fancy idea occurs to me, which involves considering a category enriched over metric spaces. Then each hom-set is also a metric space. Using that, we could define the distance from an object $A$ to an object $B$ as the minimum distance between the identity morphism on $A$ and any endomorphism of $A$ that factors through $B$.

My intuition here is to measure how much of $A$ you irreparably lose when you change $A$ in to (part of) $B$, e.g. as one does in categories where the morphisms are functions that respect certain "internal structure" of the objects - like the category of groups and group homomorphisms.

David Egolf (Mar 14 2024 at 15:31):

(I haven't checked to see if this satisfies much in the way of properties that we'd like of a notion of distance; it's just a very rough idea! Although at least if $A$ and $B$ are isomorphic the distance from $A$ to $B$ and the distance from $B$ to $A$ should both be zero!)

David Egolf (Mar 14 2024 at 15:45):

One last idea - if you have a functor $F: \mathsf{C} \to \mathsf{D}$ and you have some notion of distance between objects in $\mathsf{D}$, then you can maybe "pull back" this notion of distance to $\mathsf{C}$. If $d$ is used to compute distance between objects in $\mathsf{D}$, we could try computing a distance in $\mathsf{C}$ from an object $A$ to an object $B$ as $d(F(A), F(B))$.

My intuition is that this involves measuring the similarity of two objects by measuring the similarity of some particular property of those objects.

Mike Shulman (Mar 14 2024 at 15:51):

I haven't looked at the links, but another thing that connects categories to spaces is groupoids. A groupoid is a kind of category, and every category has an underlying groupoid of objects and invertible morphisms. But a groupoid is also a kind of space: the 2-category of groupoids is equivalent to the homotopy 2-category of spaces that are homotopy 1-types. Moduli spaces are often a kind of groupoid, or perhaps a sheaf of groupoids.

Eric M Downes (Mar 14 2024 at 16:40):

If you have something like a Hahn measure from your category $|..|:\mathsf{C}\to\mathbb{R}_+$ (like cardinality etc), you can use a "homutator" (my made-up term) to measure distance $d$ (in a Lawvere metric space sense) between morphisms in $\mathsf{C}$ and the hom-endofunctor in $\mathsf{D}$ given a functor $\mathcal{F}:\mathsf{D}\to\mathsf{C}$. Essentially, if you can construct a $\mathsf{D}$-object in $\mathsf{C}$, you're good to go.

These tend to be obvious if you're familiar with the categories. Like, lets say you have a set function between groups $f:G\to K$, and you want to know how close it is to a group homomorphism; then there are a couple measures readily available given the images of inverses $\zeta$ and group laws $\mu$ in $\mathsf{Set}$ under $\mathcal{F}$; here is one in which you invert after multiplying and mapping

$\mathcal{H}f:GG\to K; \mathcal{H}f= \mu_K\circ\big(f\circ\mu_G\times \zeta_K\circ\mu_K\circ(f\times f)\big)\circ(23)\circ\Delta_{4\leftarrow 2}$
$d(f,hom.\mathsf{Grp})=\log|\mathcal{H}f^*(G G)|$
(where $(23)$ is a transposition of arguments in cycle notation, $\Delta$ dups args etc.)

You can realize the cross ratio from projective geometry as a homutator vs inclusion-exclusion between two sets this way as well. I suspect this all should follow from May's papers on categorifying traces, but I don't know yet... still working through the latter slowly; @Spencer Breiner might be interested in this.
https://www.math.uchicago.edu/~may/PAPERS/AddJan01.pdf

John Onstead (Mar 14 2024 at 21:11):

@David Egolf Interesting perspectives! I think the site idea is very interesting, though it may not work fully for my purposes as the objects in sites are interpreted to be like the open sets of a topological space as opposed to the points of a topological space (though maybe I'm overlooking something!). The metric space idea is also interesting because in some cases, you can consider a category to be enriched over two other categories. For instance, the category Ab of abelian groups can be viewed as enriched over itself, but also enriched over Set (if we take the underlying set of the hom abelian groups between the objects in Ab). In addition, Cat is self-enriched in multiple ways, giving things such as the Gray tensor product for the category of 2-categories. Perhaps there exists certain (2-)categories that can be viewed as enriched over both metric spaces and categories?
@Mike Shulman This is an interesting connection. I believe I read somewhere that there is some form of equivalence between topological spaces and infinity-groupoids through the "fundamental infinity-groupoid". Is this true or did I misread/misremember something?
@Eric M Downes This is a very interesting analysis, but it does go a little above what I have learned so far. Though I wonder if one can assign a Hahn measure to the product category C x C to get a value C x C -> R. In a way this resembles the definition of a metric spaces in terms of sets, since those are given by a function S x S -> R that satisfies the axioms of a metric space. It could be like an "internal metric space" internal to Cat rather than Set?

John Onstead (Mar 14 2024 at 21:18):

As a related question, I was also wondering about the difference between a Set-enriched category and a "discrete Lawvere metric space". A discrete Lawvere metric space is one that assigns a discrete natural number to every pair of objects in a category, rather than a real number as the usual Lawvere metric space does. I am wondering about the connection between these two kinds of category because you might be able to express a normal 1-category as a discrete Lawvere metric space where the number assigned between two objects is the cardinality of that Hom set. So if I have objects A and B in some normal category C and have 4 morphisms from A to B, then a corresponding discrete Lawvere metric space would assign the number 4 to Hom(A, B). Is this a valid construction or thing to consider?

John Baez (Mar 14 2024 at 21:30):

John Onstead said:

These sound exactly like categories and functors, right? Well, not quite- the author is talking about intersection theory and moduli spaces, not category theory and categories, but it really is odd just how similar these talking points are!

I think the reason is that most moduli spaces in nature are watered-down versions of moduli stacks. A moduli space is a set with extra structure - like a topological space, or algebraic variety. A moduli stack is a groupoid with analogous extra structure.

This may sound mysterious, but the basic idea is simple.

To get a moduli space, we start with a category $C$ of some sort. Then we take the set of isomorphism classes of objects. Then we notice that this set has extra structure: for example, we can tell when one isomorphism class of objects is 'close' to another. So, we put this extra structure on the set of isomorphism classes... and that's our moduli space.

To get a moduli stack, we start with $C$ again. Then we form the groupoid $\mathrm{core}(C)$ of objects and isomorphisms between them. Then we notice that this has extra structure: for example, we can tell when one object is 'close' to being isomorphic to another. Then we put this structure on $\mathrm{core}(C)$... and that's our moduli stack.

John Baez (Mar 14 2024 at 21:32):

The details get a bit technical, and they depend a lot on what sort of extra structure we're dealing with. So it's good to look at examples. Here's a nice example:

• John Baez, The moduli space of acute triangles, The n-Category Cafe.

John Baez (Mar 14 2024 at 21:35):

I only discuss the moduli space, not the stack. But the fact that some acute triangles have extra symmetry (isosceles and right triangles) manifest itself in the moduli space: these triangles give points at the edges or corners of the moduli space. And the moduli stack actually keeps track of the extra symmetries.

John Baez (Mar 14 2024 at 21:35):

All this is very typical of moduli spaces and moduli stacks.

John Baez (Mar 14 2024 at 21:36):

Anyone wanting to learn about stacks using concrete examples can try this:

• Kai Behrend, An introduction to algebraic stacks, in Moduli Spaces, eds. L. Brambila-Paz, P. Newstead, R. P. Thomas and O. García-Prada, Cambridge U. Press, Cambridge 2014, pp. 1–131.

He uses moduli stacks of triangles as his examples!

John Baez (Mar 14 2024 at 21:37):

First he talks about topological stacks, and then algebraic stacks. These are two choices of 'extra structure'.

John Onstead (Mar 14 2024 at 22:03):

@John Baez Ah! This kind of extra structure on categories/groupoids is exactly what I was looking for. Thanks!

Mike Shulman (Mar 14 2024 at 22:27):

John Onstead said:

you might be able to express a normal 1-category as a discrete Lawvere metric space where the number assigned between two objects is the cardinality of that Hom set. So if I have objects A and B in some normal category C and have 4 morphisms from A to B, then a corresponding discrete Lawvere metric space would assign the number 4 to Hom(A, B).

The result won't (in general) satisfy the triangle inequality or (ever) the reflexivity property $d(x,x)=0$, so it won't be a metric space.

I think it would be closer to sensible to define the distances to be the logarithms of the cardinalities of homsets, since the monoidal structure of Set corresponds to multiplication of cardinalities and the logarithm would take that to addition. Of course the distances would no longer be discrete, but at least you could get $d(x,x)=0$ in the case when $x$ has no nonidentity endomorphisms. But the triangle identity is still going to fail.

John Onstead (Mar 14 2024 at 22:35):

@Mike Shulman Ah thanks, I can't believe I totally forgot about the triangle inequality!

David Egolf (Mar 14 2024 at 23:03):

Just throwing another idea out there, with regards to equipping an arbitrary category with a notion of distance between objects:

• An object can be "observed" by morphisms from other objects. These morphisms tell us about that object. For example, the isomorphism classes of the monomorphisms of an object $A$ are the subobjects of $A$.
• It seems plausible to consider two objects to be similar or "close" if a lot of their subobjects are closely related.
• For example, we could fix a "witness" object $B$, and consider all subobjects from $B$ to $A$ and from $B$ to $A'$. Given a subobject $f: B \to A$, let's say that $f$ is "linked to $A'$" if there is a subobject $f':B \to A'$ and a morphism $g:A' \to A$ such that $g \circ f' = f$. The idea is that in this case $A'$ contains the information to produce the subobject of $A$ in question.
• Then we consider what fraction of the subobjects of $A$ are linked to $A'$, as the witnessing object varies. If $A \cong A'$, all subobjects of $A$ should be linked to $A'$, so in that case this fraction is 1.
• One could maybe define a "distance" from $A$ to $A'$ as the negative logarithm of the fraction of subobjects of $A$ that are linked to $A'$. So the distance from $A$ and $A'$ is zero if $A \cong A'$.
• Disclaimer: I did not check if this satisfies any other properties we'd want for a notion of distance!

David Egolf (Mar 14 2024 at 23:09):

It's probably good to note that this "distance" I sketched above is not symmetric (the distance from $A$ to $A'$ is in general different from the distance from $A'$ to $A$). So maybe one would wish to symmetrize this by computing the distance from $A$ to $A'$ and the distance from $A'$ to $A$ (in the sense above) and then computing a symmetrized distance as their sum.

(I have no idea if the triangle inequality is remotely close to holding!)

David Egolf (Mar 14 2024 at 23:40):

To give an example: I wonder what this symmetrized "distance" between a set with two elements (call it "2") and a set with one element (call it "1") is in $\mathsf{Set}$. All the subobjects of 1 are linked to 2, I think. Working with subobjects of 2:

• the empty subobject is linked to 1
• each subobject picking out a particular element of 2 is linked to 1 (there are two of these, I think)
• the subobject that is all of 2 is not linked to 1

So, 3/4 of the subobjects of 2 are linked to 1. Then our total distance between 1 and 2 is -log(3/4)+-log(1), which is about 0.29.

(Hopefully I did that right!)

Eric M Downes (Mar 15 2024 at 02:16):

In the context of measuring the distance between objects in categories with all limits and colimits, I like categorizing this metric:

$d:\mathbb{N}_+^2\to\mathbb{R}_{\geq0};~~d(m,n)=\log\frac{lcm(m,n)}{gcd(m,n)}$

The uniqueness of the prime power expansion proves the triangle inequality, and in $\mathsf{Set}$ each $p:1\to N$ plays the role of a prime in a squarefree expansion; they span the object $N$. Of course gcd and lcm have their natural categorizations, and if you have some sense of both quotient and a log functor, as with for instance Lie groups, perhaps you just stay in the category.

However for a strict metric, one then need a measure $|..|$ of the "weight" of the fibers weaving through these objects. Cardinality works for $\mathsf{FinSet}$, I'm not sure if there is a natural way to choose this fiber-measure in general, but using the sum-product algorithm in a category enriched over a ring seems to work. When two fibers meet, you add their weights, with each composition you multiply the weight assigned to that morphism.

Maybe its obvious to others, but here things get a little wobbly for me. I agree with you @David Egolf to think in terms of monos. Generally it seems one must determine what kind of module the fibers of monos form; we like modules where the limits and colimits always have unique expansions relative to one another... some kind of PBW theorem.

In $\mathsf{Grp}$ if we restrict to finitely-generated groups, again things are nice and maps from the generators of $N\vee M$ play the role of primes and we can just use cardinality. Something more interesting happens when we think of the image of each group homomorphism acting on the rest of the fibers incident upon a group... there should be a Haar measure consistent with these "action bundles".

Eric M Downes (Mar 15 2024 at 02:55):

John Onstead said:

Eric M Downes Though I wonder if one can assign a Hahn measure to the product category C x C to get a value C x C -> R. In a way this resembles the definition of a metric spaces in terms of sets, since those are given by a function S x S -> R that satisfies the axioms of a metric space. It could be like an "internal metric space" internal to Cat rather than Set?

First -- yeah! Do it.

Second. See above, though also I will note that a morphism $\phi:X\to Y$ is already very close to an object in $obj.\mathsf{C}\times obj.\mathsf{C}$... the homset of the category is like a set relation++, extra data (direction, composition data with other morphisms), and multiple morphisms per (co)domain pair, but adjoint to the product of objects. MacLane just defines a "product" $\times_\mathsf{C}$ that returns exactly $hom.\mathsf{C}$! :)

No guarantees this is always good idea, but sometimes I find it useful to think of a category algebra. Same way as you construct a group algebra over a ring $R[\mathsf{C}]$; I imagine a cayley table describing products of unit vectors, with entry in row $X$ and column $Y$: $hom(X,Y)$. The sum of unit vectors is just the free abelian group (though in a symmetric monoidal category there is an internal choice), and the algebraic "product" of vectors is the composition of homs distributing over the sum. When $\nexists f:X\to Y$, we just default to $0_R$. (I think I got the direction right there, often screw that up.) This is nearly a category enriched over a ring though it has a different construction.

All that is to say, I don't think looking at a metric over objects in $\mathsf{C}\times\mathsf{C}$ is fundamentally that far from looking at a metric over morphisms in $\mathsf{C}$, though obviously the details can differ substantially. If you buy my category algebra construction (admittedly vague, but I swear it's legit; other people do this too! :) then you might see that if we can calculate distances of morphisms from some platonic ideal (like what I'm trying to do), then we can immediately compute the distances between objects using a sum-product approach. And if we first assign distances to objects, these are either inconsistent with any assignment of weights to morphisms, or they constrain said assignments.