The n-Category Café

Thank you, Mike.

Can you give a link or reference to the results you mention in your first paragraph?

As for regarding factorization systems as some kind of enrichment, that’s an interesting idea. I should think about it.

And as to whether eventual images are somehow ‘absolute’, this reminds me of the conversation we had after you mentioned the fact that objects with two universal properties need not be unique up to structure-preserving isomorphism. (If $X$ and $Y$ satisfy the same two universal properties then there’s a unique isomorphism $i: X \to Y$ that’s structure-preserving for the first property, and similarly $j: X \to Y$ for the second, but perhaps $i \neq j$.) In the case of the eventual image, it does actually have this uniqueness property. In this sense, it’s ‘robust’. That’s presumably not the same as absoluteness, whatever that might mean, but it feels similar somehow.

Is there some sense in which the presence of a factorization system can be regarded as an “enrichment”?

You know, this is the first surprising question I’ve heard about factorization systems in a while! And you (perhaps) know how I like to think about factorization systems, even if you’re better at it than I am.

Hmm… A factorization system asks for a $\mathcal{P}\{0,1\}$-filtration on all the homsets, and we ask that composition preserves this filtration: a composite of epis is epi, … of monos is mono, etc.

But the versalness of composing epis with monics gives more something; it sounds like a property (a property of factorization systems among $\mathcal{P}\{0,1\}$-filtration-enriched categories), and it can be specified by stuff (an algebraic factorization with properties…), but if it says more about each homset I can’t see yet.

Anyway, there’s a trivial start.

Perhaps a way forward would involve the characterization of factorization systems as distributive laws in spans? (We discussed that back here in another context.)

My vague thought is that we don’t actually need categories-with-factorization-systems to literally be enriched categories; we just need the 2-category of categories-with-factorization-systems to be equipped with proarrows in a well-behaved way. And since the canonical proarrow equipment of profunctors arises from considering categories as monads in spans, perhaps it has a natural extension to distributive laws in spans.

No, I hadn’t heard about inclusion systems — thanks. For some reason my PDF readers are unable to search for text in the PDF file you link to, so in case anyone else has the same problem, I’ll point out: the definition of inclusion system is on page 10.

My first impression is that inclusion systems seem against the geenral categorical spirit of not distinguishing between isomorphic objects. An inclusion system in a category consists of a class $E$ of epics and a class $I$ of maps, called ‘inclusions’, satisfying some axioms. A basic example is the category of sets with $E =$ surjections and $I =$ inclusions — not just injections, but actual subset inclusions. I find this puzzling. I’m not sure the distinction between a subset inclusion and an injection really makes sense to me. But is there a specific reason why you mention them in the context of this post? Do you see a role for them here?

I first encountered this sort of thing in this paper, where the same idea (they call it a “system of inclusions”) is used to describe exactly the extra structure that the category of material sets has. My first reaction was like Tom’s: what the heck? But I got happier when I realized that I could think of it as an $\mathcal{M}$-category, like Todd mentions, rather than an ordinary category with extra (“evil”) structure.

On the other hand, like Tom, I don’t see any connection to the current post, where everything is happening up to isomorphism as usual.

Dear Tom, maybe we should regard the eventual image of an endomorphism f:X–>X as an object Evim(f) equipped with an automorphism (the automorphism induced by f), instead of a naked object. In which case the object Evim(f^n) is obtained from the the object Evim(f) by applying the n-fold power operation on the automorphism (this called an Adams operation). Another property of the eventual image is its invariance under conjugation, if f:X–>Y and g:Y–>X, then Evim(gf) is isomorphic to Evim(fg). Is this also true for Julia sets? It seems that the eventual image behaves like some kind of higher trace. Let me stress that the objects Evim(gf) and Evim(fg) are canonically isomorphic in two ways (at least)! I am confused with the meaning of this.

I definitely agree that we should regard the eventual image as coming equipped with an automorphism. I’m thinking of eventual image as a functor

$$im^\infty: \mathbf{Endo} \to \mathbf{Auto}$$

where $\mathbf{Endo}$ is the category of objects (of our category) equipped with an endomorphism, and $\mathbf{Auto}$ is the category of objects equipped with an automorphism.

Then the statement about $n$th powers (and thanks for the name “Adams operation”) becomes the following. Let $n \geq 0$. Each of the categories $\mathbf{Endo}$ and $\mathbf{Auto}$ has an endofunctor $A_n$ sending $(X, f)$ to $(X, f^n)$. For $n \geq 1$, the square

$$\begin{matrix} \mathbf{Endo} &\stackrel{im^\infty}{\to} &\mathbf{Auto}\\ A_n \downarrow & &\downarrow A_n\\ \mathbf{Endo} &\stackrel{im^\infty}{\to}& \mathbf{Auto} \end{matrix}$$

commutes.

The canonical isomorphism between $im^\infty(g f) = \im^\infty(f g)$ is nice — thanks. I hadn’t consciously realized, though I’d been thinking about very closely related things (more on which below). The way I’d see it is this. Given $f: X \to Y$ and $g: Y \to X$, we get a commutative square

$$\begin{matrix} X &\stackrel{f}{\to} &Y \\ g f \downarrow& &\downarrow f g \\ X &\stackrel{f}{\to} &Y. \end{matrix}$$

This is a map $f: (X, g f) \to (Y, f g)$ in $\mathbf{Endo}$. By functoriality, it induces a map $f: im^\infty(g f) \to im^\infty(f g)$ in $\mathbf{Auto}$. Similarly, $g$ induces a map in the opposite direction:

$$im^\infty(g f) \stackrel{\stackrel{f}{\to}}{\stackrel{\longleftarrow}{g}} im^\infty(f g)$$

The composite map $g f : im^\infty(g f) \to im^\infty(g f)$ is an isomorphism (because $g f$ acts as an automorphism on $im^\infty(g f)$), and similarly on the other side. So $f: im^\infty(g f) \to im^\infty(f g)$ is an isomorphism.

Similarly, $g: im^\infty(f g) \to im^\infty(g f)$ is an isomorphism. But it’s not the inverse to $f$, so $g^{-1}$ and $f$ are different canonical isomorphisms from $im^\infty(g f)$ to $im^\infty(f g)$.

OK, maybe I’ve recreated your thought now…

There must actually be a $\mathbb{Z}$-indexed family of isomorphisms $im^\infty(g f) \to im^\infty(f g)$, namely

$$f(g f)^k = (f g)^k f : im^\infty(g f) \to im^\infty(f g)$$

for each $k \in \mathbb{Z}$. When $k = 0$, this is $f$; when $k = -1$, this is $g^{-1}$.

(I don’t know whether the Julia set of $f g$ is equal to that of $g f$. It’s true in trivial cases, but I don’t know how to calculate the simplest nontrivial case. Julia sets do have another property that I find quite startling. Let $f$ and $g$ be complex rational functions, both of degree $\geq 2$. If $f g = g f$ then $J(f) = J(g)$.)

Speaking of eventual image as higher trace (as André was), there are some things that can be said in the vector space case.

Let $T$ be a linear operator on a finite-dimensional vector space $X$ over an algebraically closed field $k$. Write $ker^\infty(T)$ for its eventual kernel (the union of $ker(T^n)$ over all positive integers $n$). Then

$$X = ker^\infty(T) \oplus im^\infty(T),$$

but also

$$X = \bigoplus_{\lambda \in k} ker^\infty(T - \lambda).$$

In fact the decompositions are compatible, in the sense that

$$im^\infty(T) = \bigoplus_{\lambda \neq 0} ker^\infty(T - \lambda).$$

The dimensions of the subspaces $ker^\infty(T - \lambda)$ are the algebraic multiplicities of the eigenvalues. So the restricted operator $T$ on $im^\infty(T)$ has the same spectrum, with the same algebraic multiplicities, as $T$, except possibly at $0$. In particular, it has the same trace.

It’s also a fact (which some of us recently discussed) that if $f: X \to Y$ and $g: Y \to X$ are linear maps then

$$ker^\infty(g f - \lambda) \cong ker^\infty(f g - \lambda)$$

for each $\lambda \neq 0$. Taking the direct sum over all $\lambda \neq 0$ gives

$$im^\infty(g f) \cong \im^\infty(f g).$$

In fact, the proof of the isomorphism for the eventual kernels is similar to the one for eventual images. Whether $\lambda$ is zero or not, $f$ and $g$ restrict to maps

$$ker^\infty(g f - \lambda) \stackrel{\stackrel{f}{\to}}{\stackrel{\longleftarrow}{g}} ker^\infty(f g - \lambda).$$

The composite map $g f$ is an automorphism of $ker^\infty(g f - \lambda)$ unless $\lambda = 0$, and similarly on the other side. So, assuming that $\lambda \neq 0$, this restricted map $f$ is an isomorphism. So too is the inverse of the restricted map $g$, and more generally, so is $f (g f)^r = (g f)^r f$ for any $r \in \mathbb{Z}$.

Your comment made me realize that I’d missed a whole class of examples of objects with two universal properties. An object of a category $C$ is, as we know, initial iff it’s the colimit of the empty diagram in $C$. But it’s also initial iff it’s the limit of the identity functor on $C$. Since every universal property can be rephrased in terms of initial objects, every object with a universal property has two universal properties.

On the other hand, they’re not dual universal properties in the sense I’ve been using the term. One’s a limit and one’s a colimit, but they’re limits and colimits over different diagrams.

That’s an interesting point.

The coinciding limits and colimits that I mentioned earlier are limits and colimits of the same diagram, but the weights might be different for the limit and the colimit. In fact, in general they have to be different: since weights for limits and weights for colimits have different variances, asking whether they’re the same would be a type error. But it happens that in many cases, the diagram shape is self-dual and both weights turn out to be “conical”, so the colimit can be regarded as the dual of the limit.

Actually this result is not much more than Tom’s observation above that if the initial object exists then it’s the limit of the identity functor. It happens to be remarkably useful in this context though.

Thanks Benjamin for the extra informations on the properties of the one-sided shift map. It is very interesting. I must absolutly get a copy the book of Hindmann and Strauss. I am delighted to learn that the theory of compact semi-group has applications to number theory! The connection between locally compact groups and number theory is an important theme in the work of André Weil. His book “Basic Number theory” is really about the classification of all locally compact (non-discrete) fields, commutative or skew. The adèles of a number field are forming a nice locally compact ring. The work of Langlands is rooted in the representation theory of semi-simple (locally compact) Lie groups. I would love to understand Langlands program… I have been studying it with a few collegues during last year. Our strategy is to study simple examples. A kind of bottom-up approach. We shall keep digging until we can see the light at the end of the tunnel. It may take us years.

Ben, I found this comment of yours rather thought-provoking. Below are three thoughts it provoked.

First let me say back to you what I think you said, for the good of my soul. Any monoid homomorphism $f: M \to N$ induces a functor

$$f^* = -\circ f: Set^{N^{op}} \to Set^{M^{op}}$$

which has a left adjoint $f_!$ and a right adjoint $f_*$. (That much is true just because $f$ is a functor between small categories.) Now, given a monoid $M$ and a central idempotent $e \in M$, we get a new monoid $M e$ and a monoid homomorphism $f: M \to M e$. And in that case, as you explained, $f_! = f_*$.

My three thoughts:

1. You say this “explains the eventual range case”. What do you mean? What exactly is the connection between eventual images and this fact about monoids?
2. A special case of the fact you explained is this. Let $M$ be the two-element monoid $\{1, e\}$ consisting of the identity and an idempotent. Then $M e$ is the trivial monoid. Now $Set^{M^{op}}$ is the category in which an object is a set equipped with an idempotent, $Set^{(M e)^{op}} = Set$, and $f^*$ is the ‘diagonal’ functor equipping a set with its identity. The simultaneous left-and-right adjoint $f_! = f_*$ splits the idempotent. This is the double universal property mentioned at the beginning of my post.
3. In general, for which functors $F: \mathbf{A} \to \mathbf{B}$ is it the case that $$F_! = F_*: Set^{\mathbf{A}^{op}} \to Set^{\mathbf{B}^{op}}?$$ Is an explicit condition known? Any such functor is certainly final — that’s what you get by testing for equality at the presheaf with constant value $1$. And any such functor preserves all limits in $\mathbf{A}$ that exist. But those conditions can’t be enough. The answer must have something to do with absolute flatness.