Kevin Ford, Sergei Konyagin, James Maynard, Carl Pomerance, and I have uploaded to the arXiv our paper “Long gaps in sieved sets“, submitted to J. Europ. Math. Soc..

This paper originated from the MSRI program in analytic number theory last year, and was centred around variants of the question of finding large gaps between primes. As discussed for instance in this previous post, it is now known that within the set of primes , one can find infinitely many adjacent elements whose gap obeys a lower bound of the form

where denotes the -fold iterated logarithm. This compares with the trivial bound of that one can obtain from the prime number theorem and the pigeonhole principle. Several years ago, Pomerance posed the question of whether analogous improvements to the trivial bound can be obtained for such sets as

Here there is the obvious initial issue that this set is not even known to be infinite (this is the fourth Landau problem), but let us assume for the sake of discussion that this set is indeed infinite, so that we have an infinite number of gaps to speak of. Standard sieve theory techniques give upper bounds for the density of that is comparable (up to an absolute constant) to the prime number theorem bounds for , so again we can obtain a trivial bound of for the gaps of . In this paper we improve this to

for an absolute constant ; this is not as strong as the corresponding bound for , but still improves over the trivial bound. In fact we can handle more general “sifted sets” than just . Recall from the sieve of Eratosthenes that the elements of in, say, the interval can be obtained by removing from one residue class modulo for each prime up to , namely the class mod . In a similar vein, the elements of in can be obtained by removing for each prime up to zero, one, or two residue classes modulo , depending on whether is a quadratic residue modulo . On the average, one residue class will be removed (this is a very basic case of the Chebotarev density theorem), so this sieving system is “one-dimensional on the average”. Roughly speaking, our arguments apply to any other set of numbers arising from a sieving system that is one-dimensional on average. (One can consider other dimensions also, but unfortunately our methods seem to give results that are worse than a trivial bound when the dimension is less than or greater than one.)

The standard “Erd\H{o}s-Rankin” method for constructing long gaps between primes proceeds by trying to line up some residue classes modulo small primes so that they collectively occupy a long interval. A key tool in doing so are the smooth number estimates of de Bruijn and others, which among other things assert that if one removes from an interval such as all the residue classes mod for between and for some fixed , then the set of survivors has exceptionally small density (roughly of the order of , with the precise density given by the Dickman function), in marked contrast to the situation in which one randomly removes one residue class for each such prime , in which the density is more like . One generally exploits this phenomenon to sieve out almost all the elements of a long interval using some of the primes available, and then using the remaining primes to cover up the remaining elements that have not already been sifted out. In the more recent work on this problem, advanced combinatorial tools such as hypergraph covering lemmas are used for the latter task.

In the case of , there does not appear to be any analogue of smooth numbers, in the sense that there is no obvious way to arrange the residue classes so that they have significantly fewer survivors than a random arrangement. Instead we adopt the following semi-random strategy to cover an interval by residue classes. Firstly, we randomly remove residue classes for primes up to some intermediate threshold (smaller than by a logarithmic factor), leaving behind a preliminary sifted set . Then, for each prime between and another intermediate threshold , we remove a residue class mod that maximises (or nearly maximises) its intersection with . This ends up reducing the number of survivors to be significantly below what one would achieve if one selects residue classes randomly, particularly if one also uses the hypergraph covering lemma from our previous paper. Finally, we cover each the remaining survivors by a residue class from a remaining available prime.

In March, I’ll be talking at Spencer Breiner‘s workshop on Applied Category Theory at the National Institute of Standards and Technology. I’ll be giving a joint talk with John Foley about our work using operads to design networks. This work is part of the Complex Adaptive System Composition and Design Environment project being done by Metron Scientific Solutions and managed by John Paschkewitz at DARPA.

I’ve written about this work before:

• Complex Adaptive Systems: Part 1, Part 2, Part 3, Part 4, Part 5, Part 6.

But we’ve done a lot more, and my blog articles are having trouble keeping up! So I’d like to sketch out the big picture as it stands today.

If I had to summarize, I’d say we’ve developed a formalism for *step-by-step compositional design and tasking, using commitment networks*. But this takes a while to explain.

Here’s a very simple example of a commitment network:

It has four nodes, which represent agents: a port, a helicopter, a UAV (an unmanned aerial vehicle, or drone) and a target. The edges between these notes describe relationships between these agents. Some of these relationships are ‘commitments’. For example, the edges labelled ‘SIR’ say that one agent should ‘search, intervene and rescue’ the other.

Our framework for dealing with commitment networks has some special features. It uses operads, but this isn’t really saying much. An ‘operad’ is a bunch of ways of sticking things together. An ‘algebra’ of the operad gives a particular collection of these things, and says what we get when we stick them together. These concepts are extremely general, so there’s a huge diversity of operads, each with a huge diversity of algebras. To say one is using operads to solve a problem is a bit like saying one is using math. What matters more is the specific *kind* of operad one is using, and how one is using it.

For our work, we needed to develop a new class of operads called **network operads**, which are described here:

• John Baez, John Foley, Joseph Moeller and Blake Pollard, Network models.

In this paper we mainly discuss communication networks. Subsequently we’ve been working on a special class of network operads that describe how to build commitment networks.

Here are some of key ideas:

• Using network operads we can build bigger networks from smaller ones by *overlaying* them. David Spivak’s operad of wiring diagrams only let us ‘wire together’ smaller networks to form bigger ones:

Here networks X_{1}, X_{2} and X_{3} are being wired together to form Y.

Network operads also let us wire together networks, but in addition they let us take one network:

and overlay another:

to create a larger network:

This is a new methodology for designing systems. We’re all used to building systems by wiring together subsystems: anyone who has a home stereo system has done this. But *overlaying* systems lets us do more. For example, we can take two plans of action involving the same collection of agents, and overlay them to get a new plan. We’ve all done this, too: you tell a bunch of people to do things… and then tell the same people, or an overlapping set of people, to do some other things. But lots of problems can arise if you aren’t careful. A mathematically principled approach can avoid some of these problems.

• The nodes of our networks represent agents of various types. The edges represent various relationships between agents. For example, they can represent communication channels. But more interestingly, they can represent *commitments*. For example, we can have an edge from A to B saying that agent A has to go rescue agent B. We call this kind of network a **commitment network**.

• By overlaying commitment networks, we can not only build systems out of smaller pieces but also build complicated plans by overlaying smaller pieces of plans. Since ‘tasking’ means telling a system what to do, we call this **compositional tasking**.

• If one isn’t careful, overlaying commitment networks can produce conflicts. Suppose we have a network with an edge saying that agent A has to rescue agent B. On top of this we overlay a network with an edge saying that A has to rescue agent C. If A can’t do both of these tasks at once, what should A do? There are various choices. We need to build a specific choice into the framework, so we can freely overlay commitment networks and get a well-defined result that doesn’t overburden the agents involved. We call this **automatic deconflicting**.

• Our approach to automatic deconflicting uses an idea developed by the famous category theorist Bill Lawvere: graphic monoids. I’ll explain these later, along with some of their side-benefits.

• Networks operads should let us do **step-by-step compositional tasking.** In other words, they should let us partially automate the process of tasking networks of agents, both

1) **compositionally**: tasking smaller networks and then sticking them together, e.g. by overlaying them, to get larger networks,

and

2) in a **step-by-step** way, starting at a low level of detail and then increasing the amount of detail.

To do this we need not just operads but their algebras.

• Remember, a network operad is a bunch of ways to stick together networks of some kind, e.g. by overlaying them. An *algebra* of this operad specifies a particular collection of networks of this kind, and says what we actually get when we stick them together.

So, a network operad can have one algebra in which things are described in a bare-bones, simplified way, and another algebra in which things are described in more detail. Indeed it will typically have *many* algebras, corresponding to *many* levels of detail, but for simplicity let’s just think about two.

When we have a ‘less detailed’ algebra and a ‘more detailed’ algebra they will typically be related by a map

which ‘forgets the extra details’. This sort of map is called a **homomorphism** of algebras. We give examples in our paper Network models.

But what we usually want to do, when designing a system, is not *forget* extra detail, but rather *add* extra detail to a rough specification. There is not always a systematic way to do this. If there *is*, then we may have a homomorphism

going back the other way. This lets us automate the process of filling in the details. But we can’t usually count on being able to do this. So, often we may have to start with an element of and search for an element of that is mapped to it by And typically we want this element to be optimal, or at least ‘good enough’, according to some chosen criteria. Expressing this idea formally helps us figure out how to automate the search. John Foley, in particular, has been working on this.

That’s an overview of our ideas.

Next, for the mathematically inclined, I want to give a few more details on one of the new elements not mentioned in our Network models paper: ‘graphic monoids’.

In our paper Network models we explain how the ‘overlay’ operation makes the collection of networks involving a given set of agents into a monoid. A **monoid** is a set M with a product that is associative and has an identity element 1:

In our application, this product is overlaying two networks.

A **graphic monoid** is one in which the **graphic identity**

holds for all

To understand this identity, let us think of the elements of the monoid as “commitments”. The product means “first committing to do then committing to do ”. The graphic identity says that if we first commit to do , then , and then again, it’s the same as first committing to do and then Committing to do *again* doesn’t change anything!

In particular, in any graphic monoid we have

so making the same commitment twice is the same as making it once. Mathematically we say every element of a graphic monoid is **idempotent**:

A commutative monoid obeying this law automatically obeys the graphic identity, since then

But for a noncommutative monoid, the graphic identity is stronger than . It says that *after committing to no matter what intervening commitments one might have made, committing to again has no further effect*. In other words: the intervening commitments did not undo the original commitment, so making the original commitment a second time has no effect! This captures the idea of how promises should behave.

As I said, for any network model, the set of all networks involving a fixed set of agents is a monoid. In a **commitment network model**, this monoid is required to be a graphic monoid. Joseph Moeller is writing a paper that shows how to construct a large class of commitment network models. We will follow this with a paper illustrating how to use these in compositional tasking.

For now, let me just mention a side-benefit. In any graphic monoid we can define a relation by

x a = y $ for some

This makes the graphic monoid into a partially ordered set, meaning that these properties hold:

reflexivity:

transitivity:

antisymmetry:

In the context of commitment networks, means that starting from we can reach by making some further commitment : that is, for some . So, as we ‘task’ a collection of agents by giving them more and more commitments, we move up in this partial order.

This is just a short note - a record-keeping, if you like - to report that my long review on "Collider Searches for Diboson Resonances" has now appeared on the online Elsevier site of the journal "Progress of Particle and Nuclear Physics".

I had previously pointed to the preprint version of the same article on this blog, with the aim of getting feedback from experts in the field, and I am happy that this has indeed happened: I was able to integrate some corrections from Robert Shrock, a theorist at SUNY, as well as some integrations to the references list by a couple of other colleagues.

I had previously pointed to the preprint version of the same article on this blog, with the aim of getting feedback from experts in the field, and I am happy that this has indeed happened: I was able to integrate some corrections from Robert Shrock, a theorist at SUNY, as well as some integrations to the references list by a couple of other colleagues.

Jadagul writes:

Got a draft of the course schedule for next year. Looks like I might get to teach real analysis.

I probably need someone to talk me out of trying to do everything in R^n.

A subsequent update indicates that the more standard alternative is teaching one variable analysis.

This is my second go around teaching rigorous multivariable analysis — key points are the multivariate chain rule, the inverse and implicit function theorems, Fubini’s theorem, the multivariate change of variables formula, the definition of manifolds, differential forms, Stokes’ theorem, the degree of a differentiable map and some preview of de Rham cohomology. I wouldn’t say I’m doing a great job, but at least I know why it’s hard to do. I haven’t taught single variable, but I have read over the day-to-day syllabus and problem sets of our most experienced instructor.

Here is the conceptual difference: It is quite doable to start with the axioms of an ordered field and build rigorously to the Fundamental Theorem of Calculus. Doing this gives students a real appreciation for the nontrivial power of mathematical reasoning. I don’t want to say that it is actually impossible to do the same for Stokes’ theorem (according to rumor, Brian Conrad did it), but I never manage — there comes a point where I start waving my hands and saying “Oh yes, and throw in a partition of unity” or “Yes, there is an inverse function theorem for maps between -folds just like the one for maps between open subsets of .” I think most students probably benefit from seeing things done carefully for a term first.

Below the fold, a list of specific topics much harder in more than one variable. If you have found ways not to make them harder, please chime in in the comments!

• No need for linear algebra. Just defining the multivariate derivative uses the concept of a linear map, and stating the chain rule requires you to compose them. If you want your students to ever be able to check the hypotheses of the inverse function theorem, they have to be able to check if matrices are invertible.

• One variable Riemann sums are so nice! If is an increasing bijection, and is a partition of , then is a partition of ; the -substitution formula for integrals follows immediately. If is a smooth bijection, and we have a partition of into rectangles, its image in is quite hard to describe. This is why the change of variables formula is such a pain.

• Regions of integration: In one variable we always integrate over an interval. In many variables, we integrate over complicated regions, so we need to describe the geometry of complicated regions. If you want to give a region up into simpler pieces, you need to introduce some rudimentary notion of "measure zero", to make sure the boundaries you cut along aren't too large.

• Improper integrals: In one variable, we always take limits as the bounds of the integral go somewhere. In many variables, there are uncountably many different limiting processes which could define .

And that's not even getting into manifolds, or multilinear algebra…

*guest post by Daniel Cicala and Jules Hedges*

We continue the Applied Category Theory Seminar with a discussion of Carboni and Walters’ paper *Cartesian Bicategories I*. The star of this paper is the notion of ‘bicategories of relations’. This is an abstraction of relations internal to a category. As such, this paper provides excellent, if technical, examples of internal relations and other internal category theory concepts. In this post, we discuss bicategories of relations while occasionally pausing to enjoy some internal category theory such as relations, adjoints, monads, and the Kleisli construction.

We’d like to thank Brendan Fong and Nina Otter for running such a great seminar. We’d also like to thank Paweł Sobociński and John Baez for helpful discussions.

Shortly after Bénabou introduced bicategories, a program was initiated to study these through profunctor bicategories. Carboni and Walters, however, decided to study bicategories with a more relational flavor. This is not quite as far a departure as one might think. Indeed, relations and profunctors are connected. Let’s recall two facts:

a profunctor from $C$ to $D$ is a functor from $D^{op} \times C$ to $Set$, and

a relation between sets $x$ and $y$ can be described with a $\{0,1\}$-valued matrix of size $x \times y$.

Heuristically, profunctors can be thought of as a generalization of relations when considering profunctors as “$\mathbf{Set}$-valued matrix of size $\text{ob} (C) \times \text{ob} (D)$”. As such, a line between profunctors and relations appears. In *Cartesian Bicategories I*, authors Carboni and Walters walk this line and continue a study of bicategories from a relational viewpoint.

The primary accomplishment of this paper is to characterize ‘bicategories of internal relations’ $\mathbf{Rel}(E)$ and of ‘ordered objects’ $\mathbf{Ord}(E)$ in a regular category $E$. To do this, the authors begin by introducing the notion of Cartesian bicategory, an early example of a *bicategory* with a monoidal product. They then explore **bicategories of relations**, which are Cartesian bicategories whose objects are Frobenius monoids. The name “bicategories of relations” indicates their close relationship with classical relations $\mathbf{Rel}$.

We begin by defining the two most important examples of a bicategory of relations: $\mathbf{Rel}(E)$ and $\mathbf{Ord}(E)$. Knowing these bicategories will ground us as we wade through the theory of Cartesian bicategories. We finish by characterizing $\mathbf{Rel}(E)$ and $\mathbf{Ord}(E)$ in terms of the developed theory.

In set theory, a relation from $x$ to $y$ is a subset of $x \times y$. In category theory, things become more subtle. A relation $r$ from $x$ to $y$ internal to a category $C$ is a ‘jointly monic’ $C$-span $x \xleftarrow{r_0} \hat{r} \xrightarrow{r_1} y$ That is, for any arrows $a , b \colon u \to \hat{r}$ such that $r_0 a = r_0 b$ and $r_1 a = r_1 b$ hold, then $a = b$. In a category with products, this definition simplifies substantially; it is merely a monic arrow $r \colon \hat{r} \to x \times y$.

Given a span $x \xleftarrow{c} w \xrightarrow{d} y$ and the relation $r \coloneqq \langle r_0 , r_1 \rangle$ from above, we say that $c$ is $r$-related to $d$ if there is an $w \to \hat{r}$ so that

commutes. We will write $r \colon c \nrightarrow d$ when $c$ is $r$-related to $d$.

While we can talk about relations internal to $any$ category, we cannot generally assemble them into another category. However, if we start with a regular category $E$, then there is a bicategory $\mathbf{Rel}(E)$ of relations internal to $E$. The objects are those of $E$. The arrows are the relations internal to $E$ with composition given by pullback:

Additionally, we have a unique 2-cell, written $r \leq s$, whenever $s \colon r_0 \nrightarrow r_1$. Diagrammatically, $r \leq s$ if there exists a commuting diagram

We are quite used to the idea of having an order on a set. But what about an order on a category? This is captured by $\mathbf{Ord}(E),$ the bicategory of ordered objects and ideals in a regular category $E$.

The objects of $\mathbf{Ord}(E)$ are ordered objects in $E$. An **ordered object** is a pair $(x,r)$ consisting of an $E$-object $x$ and a reflexive and transitive relation $r : x \to x$ internal to $E$.

*(Puzzle: $r$ is a monic of type $r \colon \hat{r} \to x \times x$. Both reflexivity and transitivity can be defined using morphisms. What are the domains and codomains? What properties should be satisfied?)*

The arrows of $\mathbf{Ord}(E)$ are a sort of ‘order preserving relation’ called an ideal. Precisely, an **ideal** $f \colon (x,r) \to (y,s)$ between ordered objects is a relation $f \colon x \nrightarrow y$ such that given

morphisms $a , a' , b , b'$ with a common domain $z$, and

relations $r \colon a \nrightarrow a'$, $f \colon a' \nrightarrow b'$, and $s \colon b' \nrightarrow b$

then $f \colon a \nrightarrow b$.

In $\mathbf{Set}$, an ordered object is a preordered set and an ideal $f \colon (x,r) \to (y,s)$ is a directed subset of $x \times y$ with the property that if it contains $s$ and $s' \leq s$, then it contains $s'$.

There is at most a single 2-cell between parallel arrows in $\mathbf{Ord}(E)$. Given $f , g \colon (x,r) \to (y,s)$, write $f \leq g$ whenever $g \colon f_0 \nrightarrow f_1$.

Now that we know what bicategories we have the pleasure of working with, we can move forward with the theoretical aspects. As we work through the upcoming definitions, it is helpful to recall our motivating examples $\mathbf{Rel}(E)$ and $\mathbf{Ord}(E)$.

As mentioned above, in the early days of bicategory theory, mathematicians would study bicategories as $V$-enriched profunctor bicategories for some suitable $V$. A shrewd observation was made that when $V$ is Cartesian, a $V$-profunctor bicategory has several important commonalities with $\mathbf{Rel}(E)$ and $\mathbf{Ord}(E)$. Namely, there is the existence of a Cartesian product $\otimes$, plus for each object $x$, a diagonal arrow $\Delta \colon x \to x \otimes x$ and terminal object $\epsilon \colon x \to I$. With this insight, Carboni and Walters decided to take this structute as primitive.

To simplify coherence, we only look at locally posetal bicategories (i.e. $\mathbf{Pos}$-enriched categories). This renders 2-dimensional coherences redundant as all parallel 2-cells manifestly commute. This assumption also endows each hom-poset with 2-cells $\leq$ and, as we will see, local meets $\wedge$. For the remainder of this article, all bicategories will be locally posetal unless otherwise stated.

**Definition.** A locally posetal bicategory $B$ is **Cartesian** when equipped with

a symmetric tensor product $B \otimes B \to B$,

a cocommutative comonoid structure, $\Delta_x \colon x \to x \otimes x$, and $\epsilon_x \colon x \to I$, on every $B$-object $x$

such that

- every 1-arrow $r \colon x \to y$ is a lax comonoid homomorphism, i.e.

$\Delta_y r \leq ( r \otimes r ) \Delta_x \quad \text{and} \quad \epsilon_y r \leq \epsilon_x$

- for all objects $x$, both $\Delta_x$ and $\epsilon_x$ have right adjoints $\Delta^\ast_x$ and $\epsilon^\ast_x$.

Moreover, $(\Delta_x , \epsilon_x)$ is the *only* cocommutative comonoid structure on $x$ admitting right adjoints.

*(Question: This definition contains a slight ambiguity in the authors use of the term “moreover”. Is the uniqueness property of the cocommutative comonoid structure an additional axiom or does it follow from the other axioms?)*

If you’re not accustomed to thinking about adjoints internal to a general bicategory, place yourself for a moment in $\mathbf{Cat}$. Recall that adjoint functors are merely a pair of *arrows* (adjoint functors) together with a pair of *2-cells* (unit and counit) obeying certain equations. But this sort of data can exist in *any* bicategory, not just $\mathbf{Cat}$. It is worth spending a minute to feel comfortable with this concept because, in what follows, adjoints play an important role.

Observe that the right adjoints $\Delta^\ast_x$ and $\epsilon^\ast_x$ turn $x$ into a commutative monoid object, hence a bimonoid. The (co)commutative (co)monoid structure on an object $x$ extend to a tensor product on $x \otimes x$ as seen in this string diagram:

Ultimately, we want to think of arrows in a Cartesian bicategory as generalized relations. What other considerations are required to do this? To answer this, it is helpful to first think about what a generalized function should be.

For the moment, let’s use our $\mathbf{Set}$ based intuition. For a relation to be a function, we ask that every element of the domain is related to an element of the codomain (entireness) and that the relationship is unique (determinism). How do we encode these requirements into this new, general situation? Again, let’s use intuition from relations in $\mathbf{Set}$. Let $r \nrightarrow x \times y$ be a relation and $r^\circ \nrightarrow y \times x$ be the relation defined by $r^{\circ} \colon y \nrightarrow x$ whenever $r \colon x \nrightarrow y$. To say that $r$ is entire is equivalent to saying that the composite relation $r^\circ r$ contains the identity relation on $x$ *(puzzle)*. To say that $r$ is deterministic is to say that the composite relation $rr^\circ$ is contained by the identity *(another puzzle)*. These two containments are concisely expressed by writing $1 \leq r^\circ r$ and $r r^\circ \leq 1$. Hence $r$ and $r^\circ$ form an adjoint pair! This leads us to the following definition.

**Definition.** An arrow of a Cartesian bicategory is a **map** when it has a right adjoint. Maps are closed under identity and composition. Hence, for any Cartesian bicategory $B$, there is the full subbicategory $\mathbf{Map}(B)$ whose arrows are the maps in $B$.

*(Puzzle: There is an equivalences of categories $E \simeq \mathbf{Map}(\mathbf{Rel}(E))$ for a regular category $E$. What does this say for $E := \mathbf{Set}$?)*

We can now state what appears as Theorem 1.6 of the paper. Recall that $(-)^\ast$ refers to the right adjoint.

**Theorem.** Let $B$ be a locally posetal bicategory. If $B$ is Cartesian, then

$\mathbf{Map}(B)$ has finite bicategorical products $\otimes$,

the hom-posets have finite meets $\wedge$ (i.e. categorical products) and the identity arrow in $B(I,I)$ is maximal (i.e. a terminal object), and

bicategorical products and biterminal object in $\mathbf{Map}(B)$ may be chosen so that $r \otimes s = (p^\ast r p) \wedge (p s p^\ast)$,where $p$ denotes the appropriate projection.

Conversely, if the first two properties are satisfied and the third defines a tensor product, then $B$ is Cartesian.

This theorem gives a nice characterisation of Cartesian bicategories. The first two axioms are straightforward enough, but what is the significance of the above tensor product equation?

It’s actually quite painless when you break it down. Note, every bicategorical product $\otimes$ comes with projections $p$ and inclusions $p^\ast$. Now, let $r \colon w \to y$ and $s \colon x \to z$ which gives $r \otimes s \colon w \otimes x \to y \otimes z$. One canonical arrow of type $w \otimes x \to y \otimes z$ is $p^\ast r p$ which first projects to $w$, arrives at $y$ via $r$, which then includes into $y \otimes w$. The other arrow is similar, except we first project to $x$. The above theorem says that by combining these two arrows with a meet $\wedge$, the only available operation, we get our tensor product.

The next stage is to add to Cartesian bicategories the property that each object is a Frobenius monoid. In this section we will study such bicategories and see that Cartesian plus Frobenius provides a reasonable axiomatization of relations.

Recall that an object with monoid and comonoid structures is called a **Frobenius monoid** if the equation

holds. If you’re not familiar with this equation, it has an interesting history as outlined by Walters. Now, if every object in a Cartesian bicategory is a Frobenius monoid, we call it a **bicategory of relations**. This term is a bit overworked as it commonly refers to $\mathbf{Rel}(E)$. Therefore, we will be careful to call the latter a “bicategory of internal relations”.

Why are bicatgories of relations better than simply Cartesian bicategories? For one, they admit a compact closed structure! This appears as Theorem 2.4 in the paper.

**Theorem.** A bicategory of relations has a compact closed structure. Objects are self-dual via the unit

$\Delta \epsilon^\ast_x \colon I \to x \otimes x$

and counit

$\epsilon \Delta^\ast_x \colon x \otimes x \to I.$

Moreover, the dual $r^\circ$ of any arrow $r \colon x \to y$ satisfies

$(r \otimes id) \Delta \leq (1 \otimes r^\circ) \Delta r$

and

$\Delta^\ast (r \otimes id) \leq r \Delta^\ast (1 \otimes r^\circ).$

Or if you prefer string diagrams, the above inequalities are respectively

and

Because a bicategory of relations is Cartesian, maps are still present. In fact, they have a very nice characterization here.

**Lemma.** In a bicategory of relations, an arrow $r$ is a map iff it is a (strict) comonoid homomorphism iff $r \dashv r^\circ$.

As one would hope, the adjoint of a map corresponds with the involution coming from the compact closed structure. The following corollary provides further evidence that maps are well-behaved.

**Corollary.** In a bicategory of relations:

$f$ is a map implies $f^\circ = f^\ast$. In particular, multiplication is adjoint to comultiplication and the unit is adjoint to the counit.

for maps $f$ and $g$, if $f \leq g$ then $f=g$.

But maps don’t merely behave in a nice way. They also contain a lot of the information about a Cartesian bicategory and, when working with bicategories of relations, the local information is quite fruitful too. This is made precise in the following corollary.

**Corollary.** Let $F \colon B \to D$ be a pseudofunctor between bicategories of relations. The following are equivalent:

$F$ strictly preserves the Frobenius structure.

The restriction $F \colon \mathbf{Map}(B) \to \mathbf{Map}(D)$ strictly preserves the comonoid structure.

$F$ preserves local meets and $I$.

The entire point of the theory developed above is to be able to prove things about certain classes of bicategories. In this section, we provide a characterization theorem for bicategories of internal relations. Freyd had already given this characterization using allegories. However, he relied on a proof by contradiction whereas using bicategories of relations allows for a constructive proof.

A bicategory of relations is meant to generalize bicategories of internal relations. Given a bicategory of relations, we’d like to know when an arrow is “like an internal relation”.

**Definition.** An arrow $r \colon x \to y$ is a **tabulation** if there exists maps $f \colon z \to x$ and $g \colon z \to y$ such that $r = g f^\ast$ and $f^\ast f \wedge g^\ast g = 1_z$.

This definition seems bizarre on its face, but it really is analogous to the jointly monic span-definition of an internal relation. That $r = g f^\ast$ is saying that $r$ is like a span $x \xleftarrow{f} z \xrightarrow{g} y$. The equation $f^\ast f \wedge g^\ast g = 1_z$ implies that this span is jointly monic *(puzzle)*.

A bicategory of relations is called **functionally complete** if every arrow $r \colon x \to I$ has a tabulation $i \colon x_r \to x$ and $t \colon x_r \to I$. One can show that the existence of these tabulations together with compact closedness is sufficient to obtain a unique (up to isomorphism) tabulation for every arrow. We now provide the characterization, presented as Theorem 3.5.

**Theorem.** Let $B$ be a functionally complete bicategory of relations. Then:

$\mathbf{Map}(B)$ is a regular category (all 2-arrows are trivial by an above corollary)

There is a biequivalence of bicategories $\mathbf{Rel}(\mathbf{Map}(B)) \simeq B$ obtained by sending the relation $\langle f,g \rangle$ of $\mathbf{Rel}(\mathbf{Map}(B))$ to the arrow $g f^\circ$ of $B$.

So all functionally complete bicategories of relations *are* bicategories of internal relations. An interesting quesion is whether any regular category can be realized as $\mathbf{Map}(B)$ for some functionally complete bicategory of relations. Perhaps a knowledgeable passerby will gift us with an answer in the comments!

From this theorem, we can classify some important types of categories. For instance, bicategories of relations internal to a Heyting category are exactly the functionally complete bicategory of relations having all right Kan extensions. Bicategories of relations internal to an elementary topos are exactly the functionally complete bicategories of relations $B$ such that $B(x,-)$ is representable in $\mathbf{Map}(B)$ for all objects $x$.

The goal of this section is to characterize the bicategory $\mathbf{Ord}(E)$ of ordered objects and ideals. We already introduced $\mathbf{Ord}(E)$ earlier, but that definition isn’t quite abstract enough for our purposes. An equivalent way of defining an ordered object in $E$ is as an $E$-object $x$ together with a relation $r$ on $x$ such that $1 \leq r$ and $r r \leq r$. Does this data look familiar? An ordered object in $E$ is simply a monad in $\mathbf{Rel}(E)$!

*(Puzzle: What is a monad in a general bicategory? Hint: how are adjoints defined in a general bicategory?)*

Quite a bit is known about monads, and we can now apply that knowledge to our study of $\mathbf{Ord}(E)$.

Recall that any monad in $\mathbf{Cat}$ gives rise to a category of adjunctions. The initial object of this category is the Kleisli category. Since the Kleisli category can be defined using a universal property, we can define a Kleisli object in any bicategory. In general, a Kleisli object for a monad $t \colon x \to x$ need not exist but when it does, it is defined as an arrow $k : x \to x_t$ plus a 2-arrow $\theta \colon k t \to k$ such that, given any arrow $f \colon x \to y$ and 2-arrow $\alpha \colon f t \to f$, there exists a unique arrow $h \colon x_t \to y$ such that $h k = f$. The pasting diagrams involved also commute:

As in the case of working inside $\mathbf{Cat}$, we would expect for $k$ to be on the left of an adjoint pair, and indeed it is. We get a right adjoint $k^\ast$ such that the composite $k^\ast k$ is our original monad $t$. The benefit of working in the locally posetal case is we also have that $k k^\ast = 1$. This realizes $t$ as an idempotent:

$t t = k^\ast k k^\ast k = k^\ast k = t.$

It follows that the Kleisli object construction is exactly an idempotent splitting of $t$! This means we can start with an exact category $E$ and construct $\mathbf{Ord}(E)$ by splitting the idempotents of $\mathbf{Rel}(E)$. With this in mind, we move on to the characterization, presented as Theorem 4.6.

**Theorem.** A bicategory $B$ is biequivalent to a bicategory $\mathbf{Ord}(E)$ if and only if

$B$ is Cartesian,

every monad in $B$ has a Kleisli object,

for each object $x$, there is a monad on $x$ and a Frobenius monoid $x_0$ that is isomorphic to the monad’s Kleisli object,

given a Frobenius monoid x and $f \colon x \to x$ with $f \leq 1$, $f$ splits.

The authors go on to look closer at bicategories of relations inside Grothendieck topoi and abelian categories. Both of these are regular categories, and so fit into the picture we’ve just painted. However, each have additional properties and structure that compels further study.

Much of what we have done can be done in greater generality. For instance, we can drop the local posetal requirement. However, this would greatly complicate matters by requiring non-trivial coherence conditions.

It's been three years or so since my last post. It would be cliche to talk of all that has changed, but perhaps the more crucial aspects surround what has remained the same. Ironically, the changes have often been associated with the realisation of themes and patterns which recur again and again...perhaps it is simply the spotting of these patterns which leads to potential growth, though growth often feels like a multi-directional random walk, where over time you get further from where you started, but the directions only proliferate as you find more corners to turn down. The perpendicular nature of Euclidean dimensions is itself probably illusory as here, directions may curve back on themselves and become a lost corridor which you had previously explored and long since left behind, with dust hiding your former footsteps...but the motes still floating reminds you that you've been here before and that there are mistakes still to learn from what may be a narrow and uncomfortable passageway. So perhaps it is all some giant torus, and the trick is counting the closed cycles and trying to find ones which are enjoyable to go around, rather than trying to escape from them all, which is probably futile.

A cycle which recurs is hidden truth, or perhaps, more accurately the hiding of truth...it may be deeper than the hiding of truth from the map as the map-maker himself may be unaware of what is resistance/compromise/expectation/honesty/pain/frustration or joy in a given journey. Google Maps has to go back to the matrix to find any reality hidden within the neurons, and maybe the reality has been erased long ago...'safe passage', or 'dead end' being all that is left...where in fact it may not be a dead end at all, and the paralysis that ensues would be eased with only the simplest of secret code words being divulged...I don't like that...there is a better way...can you do this for me?

This is not to be read as deep or meaningful, but really as a little gush of words, to ease the fingers into writing some more, which I hope to do. This is a deep breath and a stretch after a long long sleep...

A cycle which recurs is hidden truth, or perhaps, more accurately the hiding of truth...it may be deeper than the hiding of truth from the map as the map-maker himself may be unaware of what is resistance/compromise/expectation/honesty/pain/frustration or joy in a given journey. Google Maps has to go back to the matrix to find any reality hidden within the neurons, and maybe the reality has been erased long ago...'safe passage', or 'dead end' being all that is left...where in fact it may not be a dead end at all, and the paralysis that ensues would be eased with only the simplest of secret code words being divulged...I don't like that...there is a better way...can you do this for me?

This is not to be read as deep or meaningful, but really as a little gush of words, to ease the fingers into writing some more, which I hope to do. This is a deep breath and a stretch after a long long sleep...