The n-Category Café

It’s a shame you couldn’t come, especially as you’d done all the hard work of being on the scientific committee.

You’re asking the million-dollar question, one that I’ve been asked by a few people now. The short answer is “I don’t know”.

One obstruction is that $L^1[0, 1]$ isn’t literally a set of functions, only almost-everywhere equivalence classes of functions. If it were a set of functions, we could hope to construct an evaluation map $$\begin{aligned} ev_x: &L^1[0, 1] &\to &\mathbb{R}\\ &f &\mapsto &f(x) \end{aligned}$$ for each $x \in [0, 1]$ by using the universal property. But it’s not, and we can’t.

I thought about this kind of thing, and although there are some ways to recast the conditions on $F$, it’s not as simple as you suggest. The trouble is that coproducts in $Meas_{emb}$ aren’t disjoint unions (and indeed don’t always exist), for the same kind of reason that coproducts in the category of sets and injections don’t behave as you might at first expect.

I’m feeling pretty slow-witted myself right now, but let me see if I can formulate a thought: it’s somehow not coincidence that your teaching is arranged in an order that makes it awkward to mention this theorem. Pedagogically, it’s natural to progress as follows:

• First study individual functions (such as integrable functions on $[0, 1]$).

• Then study individual spaces of functions (such as the Banach space $L^1[0, 1]$).

• Then study individual systems of spaces of functions (such as the category $Ban$).

…”and so on”, I guess, to ever-higher levels of abstraction and categorical dimensions.

The question on my last slide — can we state precisely why it’s natural to associate a Banach space to a measure space? — belongs on one of those higher levels of abstraction.

In any case, you could tell your class the unique characterization of integration without mentioning the universal characterization of $L^1$.

I can’t help but ask: what on earth are measure spaces for, if not forming their $L^1$? What else would you do with them?

Don’t restrain yourself! Go ahead! Ask!

Indeed, Theorem B is a precise formulation of the gut feeling that you so dramatically express.

Mark asks, rhetorically:

I can’t help but ask: what on earth are measure spaces for, if not forming their $L^1?$ What else would you do with them?

Form their $L^\infty$?

I mention this not just to be a smart-aleck (though that’s always fun), but because my advisor Irving Segal had a concept of ‘integration algebra’ which he advocated as a better way to think about integration. $L^1$ isn’t an algebra, but $L^\infty$ is. It might be fun to revisit his work with a more category-theoretic outlook.

He always hated the points in that big measure space $\Omega$ probability theorists like to have hovering over all their discussions, since everything is done ‘almost everywhere’, not at individual points.

He had an unusually florid style for a mathematician (though maybe that’s not saying much), so I’ll quote the introduction to his paper Algebraic integration theory and then state the main theorem:

Although the theory of the abstract Lebesgue integral has strong intrinsic architectural articulation, the very importance of the theory for a variety of mathematical applications has tended to cloud its basic simplicity and elegance. The large number of approaches and formalisms, each justification by its appropriateness in connection with particular applications, has given the theory as a whole an appearance of heterogeneous complexity which is in fact significantly specious and a hindrance to a deeper understanding of the theory and to its further application. While it would probably be widely granted that integration serves as a focal point for real analysis, the focus has been more virtual than real; the ultimate convergence of the multitude of generally similar but technically distinct theories has been felt rather than proved.

(That reminds me of how certain people invented the concept of ‘natural isomorphism’ to simplify the sprawling mess of different homology theories, back in the 1940’s.

On the other hand, the coherence and algebraic simplicity of the basic theory has been visible and virtually taken for granted for a long time by those working in abstract analysis. The replacement of the sigma-ring of all measurable sets by the abstract Boolean ring of measurable sets modulo null sets is natural from a fundamental statistical viewpoint, and provides probably the earliest approach displaying such features. This Boolean ring approach, which has now been elaborated by many publications, most notably those of Carathéodory and his associates, is, however, an inconvenient one for most applications, inasmuch as these deal mainly with functions, which enter into the Boolean ring approach only in a rather circumlocutory fashion. Around the later thirties there appeared, as a fruit of the spectral and representation theory originating in the well-known work of von Neumann, Stone, and Gelfand, simple abstract characterizations of measure-theoretic function spaces, as ring and/or lattices. However, no comprehensive development of integration theory along such lines took place at that time.

It was the development of a variety of new theories, rather than the desire to embellish old ones, which primarily has led to the development of a complex of results, methods, and ideas here somewhat loosely referred to as ‘algebraic integration theory.’ The introduction of a new term such as this requires some explanation and justification tion, in the light of the rapidly increasing burden which mathematicians must bear. To this end we point out that one almost universal feature of applied work is a tendency to overlook mathematical isomorphisms. In theoretical physics, for example, although analogies are frequently invoked, the concept of a rigorous isomorphism rarely occurs in a non trivial form, although it is relevant in a number of significant ways in contemporary work. The result of this is, naturally, a tendency toward a variety of babel; each special context tends to acquire a distinctive terminology and notation, and close connections or even virtual identities between superficially distinct developments, are rendered quite obscure. This is, of course, not only uneconomical, in the duplication of labor involved, but brings a substantial linguistic encumbrance to bear on the scientific commerce between different areas and the development of interdisciplinary subjects.

Even in probability theory, the most notably successful of originally applied subjects in achieving mathematical clarity and emerging in this century as a distinctive branch of mathematics, cases could be cited in which quite wrong conclusions were reached as to the relation between certain work and earlier extant theories. While the use of analogies is a partial substitute in heuristic work, it has the disadvantage of making it impossible to determine whether the results are merely plausible or actually quite correct; on occasion it even makes it difficult to determine whether the results are meaningful. The obvious corrective for this situation is a closer adherence to a formulation of integration theory which is relatively independent of particularities and revealing of features independent of the notation. This might be called invariant integration theory, but this term might well suggest a subject quite different from this one, namely that of invariant integrals. Since the setting of the theory is naturally algebraic, in its concern with features independent of isomorphisms, the term algebraic integration theory is reasonable—although the subject is distinctly more distant from conventional algebra than is algebraic topology. Such a theory is necessarily abstract, but the term ‘abstract integration theory’ has already a different meaning, signifying usually the theory in which integrals are considered not necessarily over subsets of euclidean space, but over relatively general spaces, and is a more limited and quite distinct notion from that of the theory considered here, whose distinctive description as algebraic seems therefore practical.

This build-up actually goes on for quite a bit longer, but here’s the theorem:

Theorem. Suppose $A$ is a real commutative associative algebra with unit, and let

$$\int : A \to \mathbb{R}$$

be a linear functional such that:

1. $\int a^2 \ge 0$, and equals zero iff $a = 0$.

2. For any $b \in A$ there is a constant $k$ such that $\int a^2 b \le k \int a^2$.

Then $A$ is isomorphic to a dense subalgebra of $L^\infty(X)$ for a finite measure space $X$, with the usual integral as $\int$.

Ugh, typo: I meant 1/2-1/2 versus 1/3-2/3, or perhaps 1/3-1/3-1/3

I agree that 1/2-1/2 is a choice, though I wouldn’t necessarily call it an artificial one. If you chose 1/3-2/3, you’d simply end up with a different measure on $[0, 1]$ (skewed to the right), though I think the two versions of $[0, 1]$ would be isomorphic as measure spaces.

You could equally well use 1/3-1/3-1/3, or 1/4-1/4-1/4-1/4, etc., and they would also produce Lebesgue measure on $[0, 1]$. (Similarly, Freyd’s theorem also holds if you stick together three or more copies of the interval.)

I believe so, yes. The missing ingredient is

Here is the proof of that claim

The first summand is bounded by $K\mu(A)$ and the second summand tends to zero as $K \rightarrow 0$ by the dominated convergence theorem. So for big enough $K$ the right hand side will be small and then for small enough $\mu(A)$ the sum will be small.

This is related to the concept of uniform integrability, in particular a singleton set containing an element of $L^1$ is uniformly integrable.

Great; thanks, Tom. So now I’m optimistic that we can imitate Mark’s example in the more general setting.

Let’s give it a try.

For each (finite, as always) measure space $X = (X, \Sigma_X, \mu_X)$, we have the set $C(\Sigma_X)$ of continuous functions $\Sigma_X \to \mathbb{R}$, where $\Sigma_X$ is metrized as above. This is in fact a Banach space, under the sup norm. It has a closed subspace $C_\ast(\Sigma_X)$ consisting of all those $\phi \in C(\Sigma_X)$ such that $\phi(\emptyset) = 0$; that is itself a Banach space.

Whenever we have an embedding $Y \hookrightarrow X$ of measure spaces, we get a map $C_\ast(\Sigma_Y) \to C_\ast(\Sigma_X)$ defined by $\phi \mapsto \bar{\phi}$, where for any measurable $A \subseteq X$, we put $\bar{\phi}(A) = \phi(A \cap Y)$. This map is linear and of norm $\leq 1$ (that is, $\sup_A |\bar{\phi}(A)| \leq \sup_B |\phi(B)|$). So it’s a map of Banach spaces in the usual sense.

So, we’ve just defined a functor $C_\ast(\Sigma_\bullet): Meas_{emb} \to Ban$.

The next task is to equip $C_\ast(\Sigma_\bullet)$ with enough extra structure that it becomes an object of the category called $\mathcal{B}$ in my slides. That means that for each measure space $X$, we have to pick out a distinguished element $u_X \in C_\ast(\Sigma_X)$. We do this by setting $u_X(A) = \mu_X(A)$ for each measurable $A \subseteq X$ (and checking that $u_X$ is continuous, which is easy). It has the following two properties:

• $\|u_X\| \leq \mu_X(X)$ for each $X$ (again, easy)

• if $Y \rightarrow X \leftarrow Y'$ are embeddings with $Y \amalg Y' = X$ then $\bar{u_Y} + \bar{u_{Y'}} = u_X$ (where I’m using bar notation for the action of the functor $C_*$ on morphisms, as above). That’s just a matter of unwinding the definitions.

So: we’ve defined an object $(C_\ast(\Sigma_\bullet), u)$ of the category $\mathcal{B}$.

By the universal property of $(L^1, I)$, there is a unique map $(L^1, I) \to (C_\ast(\Sigma_\bullet), u)$ in $\mathcal{B}$. We can phrase this fact completely concretely: there is a unique family of linear maps

$$\Bigl( J_X: L^1(X) \to C_\ast(\Sigma_X) \Bigr)_{\text{measure spaces }\,\, X}$$

of norm $\leq 1$ such that:

• $(J_X(\text{constant function 1}))(A) = \mu_X(A)$ for all measure spaces $X$ and measurable $A \subseteq X$

• whenever a measure space $Y$ is embedded into another measure space $X$, and $g \in L^1(Y)$, then $(J_X(g \,\,\text{ extended by }\,\, 0))(A) = (J_Y(g))(Y \cap A)$ for all measurable $A \subseteq X$.

Since these conditions are satisfied by putting $(J_X(f))(A) = \int_A f \,d\mu_X$, that’s what $J$ is.

In other words:

the universal property tells us the integral of any function $f \in L^1(X)$ not only on $X$ itself, but also on every measurable subset of $X$.

(Footnote: I’ve relied here on the assumption that when one measure space $Y$ is embedded into another measure space $X$, and $A$ is a measurable subset of $X$, then $Y \cap A$ is a measurable subset of $Y$. This isn’t actually true with the meaning of “embedding” defined in my slides: there, I only asked that an embedding is injective and that the direct [not inverse] image of a measurable set is measurable. But the theorem still holds if we strengthen the definition of embedding to include the condition that the inverse image of a measurable set is measurable, i.e. that if $A$ is measurable in $X$ then $Y \cap A$ is measurable in $Y$. Perhaps this application suggests that this version of the theorem is superior.)

I don’t think I know such a theorem, but I’m not entirely sure I know what you mean. Can you be any more precise about what you mean by “the existence of Lebesgue measure”?

I believe there’s a theorem (which I once worked out, but only kind of scrappily) characterizing the poset of Lebesgue-measurable subsets of $[0 ,1]$ by some universal property. It’s very similar in spirit to the characterization of $L^1[0, 1]$, but instead of thinking about arbitrary integrable functions, we’re thinking about characteristic functions of measurable subsets.

I don’t think I can suggest anything good to cite. During the talk, I reminded the audience of the thrill they probably felt when they first discovered that various standard mathematical constructions arise as adjoints (perhaps left adjoints to some much simpler functor), or as (co)limits, etc.

One of my favourite examples of “socially important is categorically natural” is how the category $\Delta$ (hence the category of simplicial sets) arises in a canonical way from the forgetful functor $Cat \to Graph$. See e.g. here and here.

It’s probably only my favourite because I discovered it myself. Nevertheless, I like it because I’d never heard a convincing explanation of why simplicial sets are a natural thing to study, from any point of view. (I realize that that sentence may tempt you to have a go; please do!) Geometrically, it’s easy to see why you might want to divide a space into simplices. But why put the vertices in order? Why have the degeneracy maps? Or algebraically, why consider nonempty, finite, totally ordered sets, rather than some other collection of modifiers?

Maybe there’s some good other explanation of simplicial sets that I’m missing. But at the time, I thought the categorical story that I was able to tell was a good, clean explanation for their undoubted social importance.

(I hope it’s more or less clear what I mean by “social importance”. Obviously it doesn’t mean what a non-mathematician would think it means. It’s something like this: if I open a book on algebraic geometry then I might see that the word “divisor” comes up a lot, so I’d be able to deduce that divisors were important to algebraic geometers without even knowing what they are.)

Another example is the Freyd theorem characterizing $[0, 1]$ (slide 6). Undoubtedly, the unit interval is socially important!

Freyd’s algebraic analysis also seems relevant.

Yes. The keystone of that paper, Freyd’s startlingly simple characterization of the set $[0, 1]$, appears on slide 6 of my talk and was also the starting point of all my work on this. (His paper didn’t appear until nearly a decade after he first posted the result on the categories mailing list.)

I find some of the questions and emphases on this blog very strange.

Good! Finding someone else’s approach to something “strange” is often the prelude to learning a new way of thinking, at least if your mind is open.

What do you mean, the definition of Lebesgue integration is too complicated?

No one said it was too complicated. Lebesgue integration in some sense “right”, so the definition is exactly what it ought to be.

But yes, the definition of Lebesgue integration, as usually presented, is moderately complicated. That’s why almost all universities teach Riemann integration before Lebesgue — because even though most people agree that the Lebesgue theory is more satisfactory, Riemann integration is easier to define.

One standard way to present the definition of Lebesgue integration involves a sequence of preliminary definitions, as follows (and as on slide 4 of my talk):

• define null set, hence almost everywhere

• define step function

• define $\mathcal{L}^{inc}$ (the set of functions expressible as an almost-everywhere limit of an increasing sequence of step functions)

• define $\mathcal{L}^1$ (the set of functions expressible as the difference of two functions in $\mathcal{L}^{inc}$)

• define $L^1$ (as $\mathcal{L}^1$ quotiented out by almost everywhere equality)

• define the integral of a step function

• define integration on $\mathcal{L}^1$ by extending by continuity and linearity

• define integration on $L^1$ by showing that two functions in $\mathcal{L}^1$ that are equal almost everywhere have the same integral.

As it says on that slide, that’s not the only way to get to the definition, but it’s a common enough path.

The result I described (“Theorem A”) skips all these preliminary steps, and instead characterizes both $L^1[0, 1]$ and $\int_0^1$ in terms of two concepts: Banach space and mean.

we care about areas under curves and surfaces … The definition of Lebesgue integration makes plain how one calculates such areas

Sure — and everyone knows that. No one disputes the importance of interpreting integration as “area under the curve”, or as the opposite of differentiation.

But it’s often useful to have multiple perspectives on important objects. Theorem A adds a new perspective on $L^1[0, 1]$. It shows that it’s characterized by a simple universal property. Isn’t that interesting? Why wouldn’t you want to know lots of different characterizations of an object you find important?

New results on $L^p$! I hope we can talk about them in person someday.

Have you ever thought about trying to characterize Sobolev spaces? For analysts, those are the next important class of spaces after $L^p$’s. They blend the concepts of integration and differentiation.

I hope so too! And I also hope to put up a new post on this soon.

Mark Meckes asked the same question about Sobolev spaces, and similarly about Orlicz spaces. So the important people seem to agree it’s something worth thinking about :-) I haven’t thought about either class of space, but I hope someone will.

Thanks for the comment. As I see it, there are two levels of result here:

1. The universal characterization of $L^p[0, 1]$. This determines the Banach space $L^p[0, 1]$ (equipped with some extra structure) uniquely up to isomorphism. This theorem would be worth knowing even if it didn’t provide a construction of $L^p[0, 1]$, for all the usual reasons that universal characterizations are worth knowing. For example, the universal property of the tensor product, $$Hom(A \otimes B, C) \cong Hom(A, Hom(B, C)),$$ is extremely useful—fundamental, even—even though it alone doesn’t construct $A \otimes B$.

2. But in fact, in this case the categorical approach does give a construction of $L^p[0, 1]$. This is explained in Remark 2.11. In brief, there is a general categorical result on initial algebras (Adámek’s theorem) which, when applied in this case, constructs the Banach space $L^p[0, 1]$, including its norm. When phrased explicitly, the construction is something completely easy and obvious to any analyst—there’s nothing new about that. It’s the universal property that’s new.