In category theory, as in life, you spend half of your time trying to forget things, and half of the time trying to recover them. A morphism, the basic building block of every category, is like a defective isomorphism. It maps the initial state to the final state, but it provides no guarantees that you can recover the original. But it seems like this lossiness is what makes morphisms useful.

There are people who can memorize mathematical formulas perfectly but have no idea what they mean. And there are those who get just the gist of it, but can derive the rest when needed. Somehow understanding is related to lossy compression.

We can’t recover lost information. Once it’s gone, it’s gone. All we can do is to try to figure out what the original might have been like. In fact, knowing how the information was lost, we might be able to generate all possible inputs that could have led to a given output. It’s the closest we can get to inverting the uninvertible. This is the main idea behind fibrations.

Let me illustrate this concept with an example. Consider the function isEven:

isEven :: Integer -> Bool
isEven n = n `mod` 2 == 0

This function is definitely not invertible. If I only told you that the output was True, you couldn’t tell me what the input was. You could, however, give me the set of all inputs that could have produced this output: it’s the set of even numbers. We often call this set, which is the inverse image of True, a fiber over True. Similarly, the fiber over False is the set of odd numbers. In this case we only have only two fibers and they happen to be isomorphic.

Here’s a more interesting example. Consider a set of all lists of integers and a function that returns the length of a list: a natural number:

length :: [Integer] -> Nat

This function is not invertible, but it defines fibers over natural numbers. The fiber over zero is a one-element set that contains only the empty list. The fiber over 1 is the set of lists of length one (which is isomorphic to the set of integers). The fiber over 2 is a set of 2-element lists, or pairs of integers, and so on. You may recognize these fibers as length-indexed lists, or vectors. You can find them, for instance, in the Haskell Vec library or as Vect in Idris. These are not your typical data types, though. They are examples of dependent types–types that depend on values (here, natural numbers).

Notice that the name “length-indexed lists” suggests a slightly different interpretation of these types. You may think of them as families of types parameterized by natural numbers. This would suggest a mapping from elements of a set (natural numbers) to types. These two views are equivalent, but in category theory we try to avoid, if possible, talking about sets (and set elements in particular). The interpretation of dependent types as fibrations is more general, so let’s dig into fibrations.

As a generalization of functions like isEven or length, we’ll consider a morphism \pi \colon e \to b, and call it a projection, since it projects each fiber down to one element. Our goal, though, is to define a fiber as the pre-image of an element in b. But what’s an element? The closest we can get to defining an element in category theory is to consider a morphism x from the terminal object 1. Such morphism is called a global element and, in Set it really picks a single element from a (non-empty) set. Now we have two morphisms converging on b: \pi and x. Conceptually, a fiber is a subobject e_x of e, which means that there must be a morphism s that embeds e_x in e. Moreover, when this embedding is followed by the projection \pi, it must produce the same element as x. The best such object is given by a universal construction which, in this case, is a pullback.

Fig. 1.

(The exclamation mark stands for the unique morphism to the terminal object.) Incidentally, this is why a pullback is sometimes called a fiber product.

If fibers over all elements x are isomorphic, the pair (e, \pi) is, quite fittingly, called a fiber bundle. The object b, from which the fibers sprout, is called the base. (Notice that lenght-indexed lists don’t form a bundle.)

Anything you can do with functions, you can do with functors, only better. So we can have a category E, another category B, and a functor p \colon E \to B. Since a functor acts as a function on objects (modulo size issues), we can define a fiber as a set of objects in E that are mapped to a single object in B. The big question is, what do we do with morphisms? We have potentially lots of morphisms in E that go between any two fibers, and which get projected down to a single hom-set in B. If we want to invert p, we have to design a procedure for lifting morphisms from B to E.

Here’s the idea: We would like each fiber to form a subcategory of E, and we’d like to pick morphisms between fibers in such a way that p^{-1} becomes a functor from B to Cat. In other words, we want p^{-1} to map objects of B to categories (the fibers), and morphisms of B to functors between those categories. If this is too much to ask (which it often is), we’ll settle for p^{-1} to be a pseudo-functor, which is a functor that preserves unit and composition only up to isomorphism. In fact the original construction (attributed to Grothendieck) produces a contravariant pseudo-functor. In this post I’ll describe the covariant version of this construction, which is called opfibration, and which is easier to explain.

The starting point of Grothendieck fibration is the recipe for lifting morphisms from the base category B to the total category E. There is a universal construction for doing that. The resulting morphisms are called opcartesian.

Let’s start with a morphism f \colon a \to b in the base category and pick an arbitrary object s (source) in the fiber over a (hence a = p s). This will be the source of our opcartesian morphism. We have a lot of choices for the target. Strictly speaking, the target should be one of the objects over b, and that’s what we are aiming for. However, a universal construction should look at a much larger pool of candidates, some of them with targets in other fibers. This pool enlargement helps narrow down the final choice with greater accuracy. (Remember, universal constructions are unique only up to isomorphism.)

The opcartesian morphism over f, with the specified source s, is a morphism g \colon s \to t, such that p g = f.

Fig. 2.

It must satisfy a universal property that I’m about to describe.

First, we pick an arbitrary object x and a morphism h \colon s \to x. This is supposed to be the competition for g. When projected down to B, it becomes p h \colon a \to p x.

Fig. 3.

We are interested in the case when p h factorizes through f, that is, there is a morphism u \colon b \to p x such that p h = u \circ f. Whenever such factorization is possible in B, we demand that there be a unique lifting of it to E.

Fig. 4.

In other words, there exists a unique \nu \colon t \to x such that h = \nu \circ g and p \nu = u.

If you find this definition a little confusing, you’re in good company. So let’s try a slightly different imagery that has more to do with the original ideas from algebraic topology. Think of objects as shapes. A morphism f \colon a \to b is a proof that b is a proper subset of a, or that a contains b. A functor between two categories of shapes must map shapes to shapes in a way that preserves inclusion. It may map many shapes to one, so imagine that the shapes in the category E are three-dimensional, and their projections using the functor p are their flat shadows. Functoriality means that, if s contains t, then its shadow a = p s contains b = p t. Next, we introduce a new object x that is contained inside s, and the proof of that is h \colon s \to x. It follows from functoriality that a contains the shadow of x.

Now suppose that this shadow falls inside the smaller b (with the proof u \colon b \to p x).

Fig. 5.

Normally, this would not imply that x is inside t. It’s possible that (parts of) x are sticking out below or above t. Our universal condition demands that this cannot happen. There can be no room above or below t— it’s a cylinder carved into s. Universality guarantees that we get the absolutely optimal shape.

Now that we know what an opcartesian morphism is, we might ask the question, does it always exist? Given an arbitrary morphism f \colon a \to b in B and an object s over a, can we always find an opcartesian morphism g \colon s \to t such that t is over b? If we can, then we call the pair (E, p \colon E \to B) an opfibration.

Here’s an interesting observation. You might wonder whether the definition of an opcartesian morphism isn’t overly complicated. Wouldn’t it be enough to restrict the pool of possible candidates to those morphisms whose target, the x in our picture, was over b, the target of f? This was, in fact, the original idea in the Grothendieck construction. The problem was that, with such definition, there was no guarantee that a composition of two opcartesian morphisms would be again opcartesian. The current definition makes that automatic.

Given an opfibration, we now face the opposite problem: there may be too many opcartesian morphisms. Remember, we wanted to (a) make fibers into subcategories of E and (b) use opcartesian morphisms to define functors between them. The first part is relatively easy: a fiber E_a has, as objects, those objects of E whose projection is a. We select as morphisms in E_a those morphisms that project down to identity, id_a (notice that we ignore other endomorphism a \to a). These are called vertical morphisms. But to define functors between fibers we need to map each object of one fiber to exactly one object in the other fiber (and the same for vertical morphisms). Think of this as transporting objects between fibers. In a fibered category, we could use opcartesian morphisms for transport. Any time two objects are connected in the base by a morphism, we have a bunch of opcartesian morphisms over it starting from every single object in the source fiber. We could use them to try to define a functor between fibers.

But in general we have more than one opcartesian morphism between a source object in one fiber and candidate target objects in the other fiber. But we can design a procedure to pick one (if you’re into set theory, you’ll notice that we have to use the Axiom of Choice). Such choice is called an opcleavage, and the resulting construction is called cloven opfibration.

Formally, an opcleavage is described by a function \kappa (f, s)

Fig. 6.

It takes a morphism f \colon a \to b and an object s (such that p s = a), and produces an object t (such that p t = b), which is the target of some opcartesian morphism s \to t. This is exactly the morphism selected by opcleavage.

The universal construction of opcartesian morphisms can then be used to define the mapping of vertical morphisms thus completing the definition of a functor between fibers.

A geometric intuition is that an opcleavage provides a way of transporting objects in the horizontal direction. Vertical morphisms transport objects vertically, and the functors defined by the opcleavage transport them horizontally, in such a way that their shadows follow the arrows in the base. The origin of this intuition goes back to differential geometry, where one is able to define continuous paths in the base manifold and use them to transport objects, such as vectors, between fibers. Category theory lets us abstract away continuity (and differentiability) from this picture. You might also see transport used in homotopy type theory, with paths standing for equality proofs.

Now, remember what I said about the composition of opcartesian morphisms resulting in an opcartesian morphism? Unfortunately, once we start picking individual morphisms to construct an opcleavage, this compositionality might be lost. The composition of any two morphisms from the selected pool is still opcartesian, but it’s not necessarily part of the opcleveage. This is why we might have to relax compositionality and embrace pseudofunctors.

But sometimes an opcleavage preserves compositionality. We call this situation split opfibration. It must satisfy these two conditions:

\kappa (id_a, s) = id_s

\kappa (f', s') \circ \kappa(f, s) = \kappa(f' \circ f, s)

for any f \colon a \to b and f' \colon b \to c.

A split opfibration defines a functor B \to Cat, which maps objects from the base category to fibers seen as categories; and morphisms from the base category to functors between those fibers. So defined functor may be interpreted as an attempt at inverting the original projection p \colon E \to B.

If the splitting conditions are satisfied only up to isomorphism, we get a pseudofunctor B \to Cat. This makes things more complicated but also more interesting. It means that horizontal transport depends on the path. In particular, transport along a closed path–a chain of morphisms in the base that compose to identity–may produce an object that’s different from (albeit isomorphic to) the starting object. In differential geometry we would say that the space has non-zero curvature.

Interestingly, this procedure of generating split opfibrations has its inverse. Given a functor B \to Cat it’s possible to reconstruct the total category E and a projection p \colon E \to B. This is called the Grothendieck construction.

Since our new slogan is “lenses are everywhere,” it should come as no big surprise that a split opfibration may be seen as a type of a lens. The projection corresponds to view or get. It extracts a, the focus of the lens, out of s. The opcleavage part of opfibration, \kappa(f, s) corresponds to put or, more precisely to over. It takes a morphism that modifies the focus from a to b and it takes the object s, and produces the new object t. In programming, get and put are just functions between sets, here they are object mappings of two functors, but the similarity is hard to ignore.

Acknowledment

I’m grateful to Bryce Clarke for reading the draft and helpful comments.

Papers to read

  1. Johnson, Rosebrugh, and Wood, Lenses, fibrations and universal translations
  2. Johnson and Rosebrugh, Delta lenses and opfibrations
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In my previous blog post, Programming with Universal Constructions, I mentioned in passing that one-to-one mappings between sets of morphisms are often a manifestation of adjunctions between functors. In fact, an adjunction just extends a universal construction over the whole category (or two categories, in general). It combines the mapping-in with the mapping-out conditions. One functor (traditionally called the left adjoint) prepares the input for mapping out, and the other (the right adjoint) prepares the output for mapping in. The trick is to find a pair of functors that complement each other: there are as many mapping-outs from one functor as there are mapping-ins to the other functor.

To gain some insight, let’s dig deeper into the examples from my previous post.

The defining property of a product was the universal mapping-in condition. For every object c equipped with a pair of morphisms going to, respectively, a and b, there was a unique morphism h mapping c to the product a \times b. The commuting condition ensured that the correspondence went both ways, that is, given an h, the two morphisms were uniquely determined.

A pair of morphisms can be seen as a single morphism in a product category C\times C. So, really, we have an isomorphism between hom-sets in two categories, one in C\times C and the other in C. We can also define two functors going between these categories. An arbitrary object c in C is mapped by the diagonal functor \Delta to a pair \langle c, c\rangle in C\times C. That’s our left functor. It prepares the source for mapping out. The right functor maps an arbitrary pair \langle a, b\rangle to the product a \times b in C. That’s our target for mapping in.

The adjunction is the (natural) isomorphism of the two hom-sets:

(C\times C)(\Delta c, \langle a, b\rangle) \cong C(c, a \times b)

Let’s develop this intuition. As usual in category theory, an object is defined by its morphisms. A product is defined by the mapping-in property, the totality of morphisms incoming from all other objects. Hence we are interested in the hom-set between an arbitrary object c and our product a \times b. This is the right hand side of the picture. On the left, we are considering the mapping-out morphism from the object \langle c, c \rangle, which is the result of applying the functor \Delta to c. Thus an adjunction relates objects that are defined by their mapping-in property and objects defined by their mapping-out property.

Another way of looking at the pair of adjoint functors is to see them as being approximately the inverse of each other. Of course, they can’t, because the two categories in question are not isomorphic. Intuitively, C\times C is “much bigger” than C. The functor that assigns the product a \times b to every pair \langle a, b \rangle  cannot be injective. It must map many different pairs to the same (up to isomorphism) product. In the process, it “forgets” some of the information, just like the number 12 forgets whether it was obtained by multiplying 2 and 6 or 3 and 4. Common examples of this forgetfulness are isomorphisms such as

a \times b \cong b \times a

or

(a \times b) \times c \cong a \times (b \times c)

Since the product functor loses some information, its left adjoint must somehow compensate for it, essentially by making stuff up. Because the adjunction is a natural transformation, it must do it uniformly across the whole category. Given a generic object c, the only way it can produce a pair of objects is to duplicate c. Hence the diagonal functor \Delta. You might say that \Delta “freely” generates a pair. In almost every adjunction you can observe this interplay of “freeness” and “forgetfulness.” I’m using these therm loosely, but I can be excused, because there is no formal definition of forgetful (and therefore free or cofree) functors.

Left adjoints often create free stuff. The mnemonic is that “the left” is “liberal.” Right adjoints, on the other hand, are “conservative.” They only use as much data as is strictly necessary and not an iota more (they also preserve limits, which the left adjoints are free to ignore). This is all relative and, as we’ll see later, the same functor may be the left adjoint to one functor and the right adjoint to another.

Because of this lossiness, a round trip using both functors doesn’t produce an identity. It is however “related” to the identity functor. The combination left-after-right produces an object that can be mapped back to the original object. Conversely, right-after-left has a mapping from the identity functor. These two give rise to natural transformations that are called, respectively, the counit \varepsilon and the unit \eta.

Here, the combination diagonal functor after the product functor takes a pair \langle a, b\rangle to the pair \langle a \times b, a \times b\rangle. The counit \varepsilon then maps it back to \langle a, b\rangle using a pair of projections \langle \pi_1, \pi_2\rangle (which is a single morphism in C \times C). It’s easy to see that the family of such morphisms defines a natural transformation.

If we think for a moment in terms of set elements, then for every element of the target object, the counit extracts a pair of elements of the source object (the objects here are pairs of sets). Note that this mapping is not injective and, therefore, not invertible.

The other composition–the product functor after the diagonal functor–maps an object c to c \times c. The component of the unit natural transformation, \eta_c \colon c \to c \times c, is implemented using the universal property of the product. Indeed, such a morphism is uniquely determined by a pair of identity morphsims \langle id_c, id_c \rangle. Again, when we vary c, these morphisms combine to form a natural transformation.

Thinking in terms of set elements, the unit inserts an element of the set c in the target set. And again, this is not an injective map, so it cannot be inverted.

Although in an arbitrary category we cannot talk about elements, a lot of intuitions from Set carry over to a more general setting. In a category with a terminal object, for instance, we can talk about global elements as mappings from the terminal object. In the absence of the terminal object, we may use other objects to define generalized elements. This is all in the true spirit of category theory, which defines all properties of objects in terms of morphisms.

Every construction in category theory has its dual, and the product is no exception.

A coproduct is defined by a mapping out property. For every pair of morphisms from, respectively, a and b to the common target c there is a unique mapping out from the coproduct a + b to c. In programming, this is called case analysis: a function from a sum type is implemented using two functions corresponding to two cases. Conversely, given a mapping out of a coproduct, the two functions are uniquely determined due to the commuting conditions (this was all discussed in the previous post).

As before, this one-to-one correspondence can be neatly encapsulated as an adjunction. This time, however, the coproduct functor is the left adjoint of the diagonal functor.

The coproduct is still the “forgetful” part of the duo, but now the diagonal functor plays the role of the cofree funtor, relative to the coproduct. Again, I’m using these terms loosely.

The counit now works in the category C and  it “extracts a value” from the symmetric coproduct of  c with c. It does it by “pattern matching” and applying the identity morphism.

The unit is more interesting. It’s built from two injections, or two constructors, as we call them in programming.

I find it fascinating that the simple diagonal functor can be used to define both products and coproducts. Moreover, using terse categorical notation, this whole blog post up to this point can be summarized by a single formula.

That’s the power of adjunctions.

There is one more very important adjunction that every programmer should know: the exponential, or the currying adjunction. The exponential, a.k.a. the function type, is the right adjoint to the product functor. What’s the product functor? Product is a bifunctor, or a functor from C \times C to C. But if you fix one of the arguments, it just becomes a regular functor. We’re interested in the functor (-) \times b or, more explicitly:

(-) \times b : a \to a \times b

It’s a functor that multiplies its argument by some fixed object b. We are using this functor to define the exponential. The exponential a^b is defined by the mapping-in property.  The mappings out of the product c \times b to a are in one to one correspondence with morphisms from an arbitrary object c to the exponential a^b .

C(c \times b, a) \cong C(c, a^b)

The exponential a^b is an object representing the set of morphisms from b to a, and the two directions of the isomorphism above are called curry and uncurry.

This is exactly the meaning of the universal property of the exponential I discussed in my previous post.

The counit for this adjunction extracts a value from the product of the function type (the exponential) and its argument. It’s just the evaluation morphism: it applies a function to its argument.

The unit injects a value of type c into a function type b \to c \times b. The unit is just the curried version of the product constructor.

I want you to look closely at this formula through your programming glasses. The target of the unit is the type:

b -> (c, b)

You may recognize it as the state monad, with b representing the state. The unit is nothing else but the natural transformation whose component we call return. Coincidence? Not really! Look at the component of the counit:

(b -> a, b) -> a

It’s the extract of the Store comonad.

It turns out that every adjunction gives rise to both a monad and a comonad. Not only that, every monad and every comonad give rise to adjunctions.

It seems like, in category theory, if you dig deep enough, everything is related to everything in some meaningful way. And every time you revisit a topic, you discover new insights. That’s what makes category theory so exciting.


Previously we were exploring universal constructions for products, coproducts, and exponentials. In particular, we were able to prove the distributive law:

(a + b) \times c \cong a \times c + b \times c

The power of this law is that it relates the mapping-in universal construction (product on the left) with the mapping-out one (coproduct on the right). If you take into account that products and coproducts are just special cases of limits and colimits, you may ask a more general question: under what conditions limits commute with colimits. In a cartesian closed category a product of sums is not equal to the sum of products:

(a + b) \times (c + d) \ncong a \times c + b \times d

So, in general, products don’t commute with coproducts. But if you replace coproducts with a special kind of colimits, then it can be shown that:
Theorem.
In Set, filtered colimits commute with finite limits.

In this post I’ll try to explain these terms and provide some intuition why it works and how filtered colimits are related to the more traditional notion of limits that we know from calculus.

Limits

Let’s start with limits. They are like products, except that, instead of just two objects at the bottom, you have any number of objects plus a bunch of morphisms between them. That’s called a diagram. Then you have an apex with arrows going down to all the objects in the diagram; and you get what is called a cone. If you have morphisms in your diagram, they form triangles. These triangles must commute. For instance, in Fig 1, we have:

g \circ \pi_1 = \pi_3

Fig. 1. A cone

This means that not all projections are independent–that you may obtain one projection from another by post-composing it with a morphism from the diagram. In Fig 1, for instance, you may extract a value of c either directly using \pi_3 or by applying g to the result of \pi_1.

A limit is defined as the universal cone with the apex Lim. It means that, if you have any other cone with some apex c, built over the same diagram, there is a unique morphism h from c to Lim that makes all the triangles commute. For instance, in Fig 2, one of the commuting conditions is:

\pi_1 \circ h = f_1

and so on. We’ve seen similar commuting conditions in the definition of the product.

Fig. 2. A universal cone

If you think of Lim in this example as a data structure, you would implement it as a product of a_1, a_2, and a_3, together with two functions:

g_3 : a_1 \to a_3

g_2 : a_1 \to a_2

But because of the commuting conditions, the three values stored in Lim cannot be independent. If you pick a value for a_1, then the values for a_2 and a_3 are uniquely determined.

A limit, just like a product, is defined by a mapping-in property. If you want to define a morphism from some c to Lim, you need to provide three morphisms f_1, f_2, and f_3. However, unlike in the case of a product, these morphisms must satisfy some commuting conditions. Here, f_3 must be equal to g_3 \circ f_1 and f_2 = g_2 \circ f_1. So, really, you only need to define f_1, and that uniquely determines h. This is why the cones in Fig 2 can be simplified, as shown in Fig 3.

Fig. 3. A simplified universal cone

Notice that the diagram essentially forms a subcategory inside the category C, even if we don’t explicitly draw all the identity morphisms or all the compositions. This is because triangles built by composing commuting triangles are again commuting. It therefore makes sense to define a diagram as a functor F from an (often much smaller) index category J to C. In our case it would be a category with just three objects, j_1, j_2, j_3, and two non-identity morphisms. (The diagram category for the product is even simpler: just two objects, no non-trivial morphisms.)

The properties of the diagram category determine the nature of cones and the nature of the limits. For instance, functors from a finite category will produce finite limits.

Fig. 4. Diagram category J

The diagram category J in our example has a very peculiar property: it has a cone for every pair of objects (it’s a cone inside J, not to be confused with the cone in C). For instance, the pair j_2, j_3 is part of the cone with the apex j_1. This is also the apex for the (somewhat degenerate) cone based on j_1 and j_2 (with or without the connecting morphism). A category in which there is a cone for every finite subdiagram is called cofiltered. Limits defined by functors from cofiltered categories are called cofiltered limits.

The intuition is that cofiltered categories exhibit some kind of ordering. You may think of j_1 as a lower bound of j_2 and j_3. Following these bounds, you might eventually get to some kind of roots–here it’s the object j_1–and these roots will dictate the behavior of cones and the behavior of limits. Things get really interesting when the diagram category is infinite, because then there is no guarantee that you’ll ever reach a root. There is, for instance, no smallest (negative) integer, even though integers are ordered. You can begin to see parallels with traditional limits, like:

\lim_{j \to -\infty} a_j

That’s where these ideas originally came from.

Limits in the category of sets have a particularly simple interpretation. In Set, we can use functions from the terminal object–the singleton set–to pick individual set elements.

Fig. 5. Elements of the limit

For every selection in Fig 5. of x_1, x_2, x_3 there is a unique h(x_1, x_2, x_3) that picks an element in Lim. But a selection of x_1, x_2, x_3 is nothing but a cone with the apex 1. So there is a one-to-one correspondence between elements of Lim and such cones. In other words, Lim is a set of apex-1 cones.

Colimits

Colimits are dual to limits–you get them by inverting all the arrows. So, instead of projections, you get injections, and the universal condition defines a mapping out of a colimit (see Fig 6).

Fig. 6. A universal cocone

If you look at the colimit as a data structure, it is similar to a coproduct, except that not all the injections are independent. In the example in Fig 6, i_3 and i_2 are determined by pre-composing i_1 with g_3 and g_2, respectively. It’s not clear how to implement a colimit in Haskell, so here’s a pseudo-Haskell attempt using imaginary dependent-type syntax:

  data Colim a1 a2 a3 (g2 :: a2 -> a1) (g3 :: a3 -> a1) =
           = A1 a1 | A2 a2 | A3 a3

To deconstruct this colimit, you only need to provide one function f_1 : a_1 \to c.

  h :: (a1 -> c) -> Colim a1 a2 a3 g2 g3 -> c
  h f1 (A1 a1)    = f1 a1
  h f1 (A2 a2) = f1 (g2 a2)
  h f1 (A3 a3) = f1 (g3 a3)

Granted, in a lazy language like Haskell, this would be an overkill way to store essentially just one value.

A colimit in the category of sets simplifies to a disjoint union of sets, in which some elements are identified. Suppose that the colimit Colim_J F is defined by some diagram category J and a functor F : J \to Set. Each object j in J produces a set F j.

Fig. 7. Colimit in Set. On the left, the diagram category J.

The disjoint union of all these sets is a set whose elements are the pairs (x, j) where x \in F j. (Notice that the sets may overlap, but each element from the overlap will be counted as many times as the number of sets it belongs to.) Coproduct injections are then functions that take an element x \in F j and map it into an element (x, j) \in Colim_J F. But that doesn’t take into account the presence of morphisms in the diagram. These morphisms are mapped to functions between corresponding sets. For instance, in Fig 7, we can take an element x \in F j_2. It is injected, using i_2, as an element (x, j_2) \in Colim_J F. But there is another path from F j_2 that uses F g_2 followed by i_1. That produces ((F g_2) x, j_1). If the triangle is to commute, these two must be equal. So in the actual colimit, they must be identified. In general, any two elements of the disjoint union that satisfy this relation:

(x, j) \rightsquigarrow (x', j')\;\; \text{if}\;\; \exists_{g : j \to j'} (F g) x = x'

must be identified. This is not an equivalence relation, but it can be extended to one (by first symmetrizing it, and then making it transitive again). A colimit is then a quotient of the disjoint union by this equivalence.

As before, I chose this example to illustrate a special type of a diagram. This is a diagram that can be obtained using a functor from a filtered category. A filtered category has this property that for any finite subdiagram, there is a cocone under it. Here, for instance, the subdiagram formed by j_2 and j_3 has a cocone with the apex j_1. Again, you may think of j_1 as a kind of upper bound of j_2 and j_3. If the filtered category is finite, following upper bounds will eventually lead you to some roots. And in Set, the equivalence relation will allow you shift all the elements down to those roots. But in an infinite case (think natural numbers) there may be no largest element–no root. And that brings filtered colimits closer to the intuition we have for limits in calculus. In fact, all the interesting filtered colimits are based on infinite diagrams.

Commuting Limits and Colimits

What does it mean for a limit to commute with a colimit? A single colimit is generated by a functor from some index category I \to C. What we need is a bunch of such colimits so that we can take a limit over those. Therefore we need a bunch of functors I \to C. Moreover, those colimits have to form a diagram. So we need another index category J to parameterize those functors. Altogether, we need a functor of two arguments:

F : I \times J \to C

It follows that, for any given j in J we have a functor F(-, j) : I \to C. We can take a colimit of that. Then we gather those colimits into a diagram whose shape is defined by J, and then take its limit. We get:

Lim_J (Colim_I F)

Alternatively, when we fix some i in I, we get a functor F(i, -) : J \to C. We can take a limit of that. Then we can gather all those limits and form a diagram whose shape is defined by I. Finally we can take a colimit of that:

Colim_I (Lim_J F)

Fig. 8. Commuting limits (red diagram of shape J) and colimits (black diagram of shape I)

It’s not difficult to construct the mapping:

Colim_I (Lim_J F) \to Lim_J (Colim_I F)

using the universal property, since the colimit has the mapping-out property. It’s the other way around that’s tricky. But it always works in the special case when I is filtered, J is finite, and C is Set.

Here’s the sketch of this amazing proof, which you can find in Saunders Mac Lane’s Categories for the Working Mathematician.

Since the target of the functor is Set, it might help to visualize its image as a rectangular array of sets. A fixed j picks up a row of such sets, whereas a fixed i picks up a column. Because we are dealing with sets, we can try to define the mapping:

Lim_J (Colim_I F) \to Colim_I (Lim_J F)

pointwise. Let’s pick an element of the limit on the left. As we’ve established earlier, a limit in Set is a set of apex-1 cones. So let’s pick one such cone. It’s just a selection of elements from a bunch of colimits.

As we’ve seen before, a colimit in Set is a discriminated union with some identifications. So our apex-1 cone will pick a set of representatives, one per colimit, say (x_n, i_n). Any time there is a morphism g : i_n \to i'_n, we can replace one representative with another (g (x_n), i'_n). The intuition is that we can slide the representatives horizontally within each row along morphisms.

If I is a filtered category, then for any finite number of objects i_n, we can always find a common root (it will be the apex i of a cocone formed by i_n in I). So we can slide all the representatives to a single column. In other words, our cone can be brought to a set of representatives (y_n, i), with a common i.

Fig. 9. A single cone after shifting representatives from all colimits to a common column

But that’s just a cone over J. It’s an element of Lim_J F. And we can inject it into a colimit over I to get an element of Colim_I (Lim_J F). We have thus defined our mapping.

Conclusion

If you didn’t get the proof the first time, don’t get discouraged. Take a break, sleep over it, and then read it slowly again. Make sure you have internalized all the definitions. Draw your own pictures. The two major tricks are: (1) visualizing an element of a limit as a cone originating from the singleton set, and (2) the idea of sliding the elements of multiple colimits to a common column.

The importance of this theorem is that it tells you when and how you can define mappings out of limits. For instance, how to define functions from a product or from an end.

Acknowledgment

I’m grateful to Derek Elkins for correcting mistakes in the original version of this post.


As functional programmers we are interested in functions. Category theorists are similarly interested in morphisms. There is a slight difference in approach, though. A programmer must implement a function, whereas a mathematician is often satisfied with the proof of existence of a morphism (unless said mathematician is a constructivist). Category theory if full of such proofs. It turns out that many of these proofs can be converted to code, often resulting in quite unexpected encodings.

A lot of objects in category theory are defined using universal constructions and universality is used all over the place to show the existence (as a rule: unique, up to unique isomorphism) of morphisms between objects.

There are two major types of universal constructions: the ones asserting the mapping-in property, and the ones asserting the mapping-out property. For instance, the product has the mapping-in property.

Product

Recall that a product of two objects a and b is an object a \times b together with two projections:

\pi_1 : a \times b \to a

\pi_2 : a \times b \to b

This object must satisfy the universal property: for any other object c with a pair of morphisms:

f : c \to a

g : c \to b

there exists a unique morphism h : c \to a \times b such that:

f = \pi_1 \circ h

g = \pi_2 \circ h

In other words, the two triangles in Fig 1 commute.

Fig. 1. Universality of the product

This universal property can be used any time you need to find a morphism that’s mapping into the product, and it can actually produce code.

For instance, let’s say you want to find a morphism from the terminal object 1 to a \times b. All you need is to define two morphisms x : 1 \to a and y : 1 \to b. This is not always possible, but if it is, you are guaranteed the existence of a morphism h : 1 \to a \times b (Fig 2).

Fig. 2. Global element of a product

Morphisms from the terminal object are called global elements, so we have just shown that, as long as a and b have global elements, say x and y, their product has a global element too. Moreover the projection \pi_1 of this global element is the same as x, and \pi_2 is the same as y. In other words, an element of a product is a pair of elements. But you probably knew that.

The universal construction of the product is implemented as an operator in Haskell:

  (&&&) :: (c->a) -> (c->b) -> (c -> (a, b))

We can also go the other way: given a mapping-in h : c \to a \times b, we can always extract a pair of morphisms:

f = \pi_1 \circ h

g = \pi_2 \circ h

This bijection between h and a pair of morphisms (f, g) is in fact an adjunction.

You might think this kind of reasoning is very different from what programmers do, but it’s not. Here’s one possible definition of a product in Haskell (besides the built-in one, (,)):

  data Product a b = MkProduct { fst :: a
                               , snd :: b }

It is in one-to-one correspondence with what I’ve just explained. The two functions fst and snd are \pi_1 and \pi_2, and MkProduct corresponds to our h : 1 \to a \times b. The categorical definition is just a different, much more general, way of saying the same thing.

Here’s another application of universality: Show that product is functorial. Suppose that you have a pair of morphisms:

f : a \to a'

g : b \to b'

and you want to lift them to a morphism:

h : a \times b \to a' \times b'

Since we are dealing with products, we should use the mapping-in property. So we draw the universality diagram for the target a' \times b', and put the source a \times b at the top. The pair of functions that fits the bill is (f \circ \pi_1, g \circ \pi_2) (Fig 3).

Fig. 3. Functoriality of the product

The universal property gives us, uniquely, the h, which is usually written simply as f \times g.

Exercise for the reader: Show, using universality, that categorical product is symmetric.

Coproduct

The coproduct, being the dual of the product, is defined by the universal mapping-out property, see Fig 4.

Fig. 4. Universality of the coproduct

So if you need a morphism from a coproduct a + b to some c, it’s enough to define two morphisms:

f : a \to c

g : b \to c

This universal property may also be restated as the isomorphism between pairs of morphisms (f, g) and morphisms of the type a+b \to c (so there is, in fact, a corresponding adjunction).

This is easily illustrated in Haskell:

  h :: Either a b -> c
  h (Left a)  = f a
  h (Right b) = g b

Here Left and Right correspond to the two injections i_1 and i_2. There is a convenient function in Haskell that encapsulates this universal construction:

  either :: (a->c) -> (b->c) -> (Either a b -> c)

Exercise for the reader: Show that coproduct is functorial.

So next time you ask yourself, what can I do with a universal construction? the answer is: use it to define a morphism, either mapping in or mapping out of your construct. Why is it useful? Because it decomposes a problem into smaller problems. In the examples above, the problem of constructing one morphism h was nicely decomposed into defining f and g separately.

The flip side of this is that there is no simple way of defining a mapping out of a product or a mapping into a coproduct.

Distributive Law

For instance, you might wonder if the familiar distributive law:

(a + b) \times c \cong a \times c + b \times c

holds in an arbitrary category that defines products and coproducts (so called bicartesian category). You can immediately see that defining a morphism from right to left is easy, because it involves the mapping out of a coproduct. All we need is to define a pair of morphisms leading to the common target (Fig 5):

f : a \times c \to (a + b) \times c

g : b \times c \to (a + b) \times c

Fig. 5. Right to left proof

The trick is to take advantage of the functoriality of the product, which we have already established, and use it to implement f and g as:

f = i_1 \times id_c

g = i_2 \times id_c

But if you try to construct a proof in the other direction, from left to right, you’re stuck, because it would require the mapping out of a product. So the distributive property does not hold in general.

“Wait a moment!” I hear you say, “I can easily implement it in Haskell.”

  f :: (Either a b, c) -> Either (a, c) (b, c)
  f (Left a, c)  = Left  (a, c)
  f (Right b, c) = Right (b, c)

Exponential

That’s correct, but Haskell does a little cheating behind the scenes. You can see it clearly when you convert this code to point free notation (I’ll explain later how I figured it out):

  f = uncurry (either (curry Left) (curry Right))

I want to direct your attention to the use of curry and uncurry. Currying is the application of another universal construction, namely that of the exponential object c^b, representing the function type b -> c. This is exactly the construction that provides the missing mapping out of a product, (a, b) -> c. Here we go:

  uncurry :: (c -> (a -> b)) -> ((c, a) -> b)

Categorically, we have the bijection between morphisms (again, a sign of an adjunction):

h : c \to b^a

f : c \times a \to b

Universality tells us that for every c and f there is a unique h in Fig 6 (and vice versa). The arrow h \times id_a is the lifting of the pair (h, id_a) by the product functor (we’ve established the functoriality of the product earlier).

Fig. 6. Universality of the exponential

Not every category has exponentials–the ones that do are called cartesian closed (cartesian, because they must also have products).

So how does the fact that we have exponentials in Haskell help us here? We are trying to define a mapping out of a product:

f : (a + b) \times c \to a \times c + b \times c

Here’s where the exponential saves the day. This mapping exists if we can define another mapping:

h : (a + b) \to (a \times c + b \times c)^c

see Fig 7.

Fig. 7. Uncurrying

This morphism, in turn, is easy to define, because it involves a mapping out of a sum. We just need a pair of morphisms:

h_1 : a \to (a \times c + b \times c)^c

h_2 : b \to (a \times c + b \times c)^c

We can define the first morphism using the universal property of the exponential, picking the injection i_1:

Fig. 8. Defining h_1

This translates to Haskell as h1 = curry Left. Similarly for h_2 we get curry Right.

We can now combine all these diagrams into a single point-free definition, and that’s exactly how I came up with the original code:

  f = uncurry (either (curry Left) (curry Right))

Notice that curry is used to get from f to h, and uncurry from h to f in the original diagram.

Products and coproducts are examples of more general constructions called limits and colimits. Importantly, the universal property of limits can be used to define the mapping-in morphisms, whereas the universal property of colimits allows us to define the mapping-out morphisms. I’ll talk more about it in the upcoming post.


I’ve been working with profunctors lately. They are interesting beasts, both in category theory and in programming. In Haskell, they form the basis of profunctor optics–in particular the lens library.

Profunctor Recap

The categorical definition of a profunctor doesn’t even begin to describe its richness. You might say that it’s just a functor from a product category \mathbb{C}^{op}\times \mathbb{D} to Set (I’ll stick to Set for simplicity, but there are generalizations to other categories as well).

A profunctor P (a.k.a., a distributor, or bimodule) maps a pair of objects, c from \mathbb{C} and d from \mathbb{D}, to a set P(c, d). Being a functor, it also maps any pair of morphisms in \mathbb{C}^{op}\times \mathbb{D}:

f\colon c' \to c
g\colon d \to d'

to a function between those sets:

P(f, g) \colon P(c, d) \to P(c', d')

Notice that the first morphism f goes in the opposite direction to what we normally expect for functors. We say that the profunctor is contravariant in its first argument and covariant in the second.

But what’s so special about this particular combination of source and target categories?

Hom-Profunctor

The key point is to realize that a profunctor generalizes the idea of a hom-functor. Like a profunctor, a hom-functor maps pairs of objects to sets. Indeed, for any two objects in \mathbb{C} we have the set of morphisms between them, C(a, b).

Also, any pair of morphisms in \mathbb{C}:

f\colon a' \to a
g\colon b \to b'

can be lifted to a function, which we will denote by C(f, g), between hom-sets:

C(f, g) \colon C(a, b) \to C(a', b')

Indeed, for any h \in C(a, b) we have:

C(f, g) h = g \circ h \circ f \in C(a', b')

This (plus functorial laws) completes the definition of a functor from \mathbb{C}^{op}\times \mathbb{C} to Set. So a hom-functor is a special case of an endo-profunctor (where \mathbb{D} is the same as \mathbb{C}). It’s contravariant in the first argument and covariant in the second.

For Haskell programmers, here’s the definition of a profunctor from Edward Kmett’s Data.Profunctor library:

class Profunctor p where
  dimap :: (a' -> a) -> (b -> b') -> p a b -> p a' b'

The function dimap does the lifting of a pair of morphisms.

Here’s the proof that the hom-functor which, in Haskell, is represented by the arrow ->, is a profunctor:

instance Profunctor (->) where
  dimap ab cd bc = cd . bc . ab

Not only that: a general profunctor can be considered an extension of a hom-functor that forms a bridge between two categories. Consider a profunctor P spanning two categories \mathbb{C} and \mathbb{D}:

P \colon \mathbb{C}^{op}\times \mathbb{D} \to Set

For any two objects from one of the categories we have a regular hom-set. But if we take one object c from \mathbb{C} and another object d from \mathbb{D}, we can generate a set P(c, d). This set works just like a hom-set. Its elements are called heteromorphisms, because they can be thought of as representing morphism between two different categories. What makes them similar to morphisms is that they can be composed with regular morphisms. Suppose you have a morphism in \mathbb{C}:

f\colon c' \to c

and a heteromorphism h \in P(c, d). Their composition is another heteromorphism obtained by lifting the pair (f, id_d). Indeed:

P(f, id_d) \colon P(c, d) \to P(c', d)

so its action on h produces a heteromorphism from c' to d, which we can call the composition h \circ f of a heteromorphism h with a morphism f. Similarly, a morphism in \mathbb{D}:

g\colon d \to d'

can be composed with h by lifting (id_c, g).

In Haskell, this new composition would be implemented by applying dimap f id to precompose p c d with

f :: c' -> c

and dimap id g to postcompose it with

g :: d -> d'

This is how we can use a profunctor to glue together two categories. Two categories connected by a profunctor form a new category known as their collage.

A given profunctor provides unidirectional flow of heteromorphisms from \mathbb{C} to \mathbb{D}, so there is no opportunity to compose two heteromorphisms.

Profunctors As Relations

The opportunity to compose heteromorphisms arises when we decide to glue more than two categories. The clue as how to proceed comes from yet another interpretation of profunctors: as proof-relevant relations. In classical logic, a relation between sets assigns a Boolean true or false to each pair of elements. The elements are either related or not, period. In proof-relevant logic, we are not only interested in whether something is true, but also in gathering witnesses to the proofs. So, instead of assigning a single Boolean to each pair of elements, we assign a whole set. If the set is empty, the elements are unrelated. If it’s non-empty, each element is a separate witness to the relation.

This definition of a relation can be generalized to any category. In fact there is already a natural relation between objects in a category–the one defined by hom-sets. Two objects a and b are related this way if the hom-set C(a, b) is non-empty. Each morphism in C(a, b) serves as a witness to this relation.

With profunctors, we can define proof-relevant relations between objects that are taken from different categories. Object c in \mathbb{C} is related to object d in \mathbb{D} if P(c, d) is a non-empty set. Moreover, each element of this set serves as a witness for the relation. Because of functoriality of P, this relation is compatible with the categorical structure, that is, it composes nicely with the relation defined by hom-sets.

In general, a composition of two relations P and Q, denoted by P \circ Q is defined as a path between objects. Objects a and c are related if there is a go-between object b such that both P(a, b) and Q(b, c) are non-empty. As a witness of this relation we can pick any pair of elements, one from P(a, b) and one from Q(b, c).

By convention, a profunctor P(a, b) is drawn as an arrow (often crossed) from b to a, a \nleftarrow b.

Composition of profunctors/relations

Profunctor Composition

To create a set of all witnesses of P \circ Q we have to sum over all possible intermediate objects and all pairs of witnesses. Roughly speaking, such a sum (modulo some identifications) is expressed categorically as a coend:

(P \circ Q)(a, c) = \int^b P(a, b) \times Q(b, c)

As a refresher, a coend of a profunctor P is a set \int^a P(a, a) equipped with a family of injections

i_x \colon P(x, x) \to \int^a P(a, a)

that is universal in the sense that, for any other set s and a family:

\alpha_x \colon P(x, x) \to s

there is a unique function h that factorizes them all:

\alpha_x = h \circ i_x

Universal property of a coend

Profunctor composition can be translated into pseudo-Haskell as:

type Procompose q p a c = exists b. (p a b, q b c)

where the coend is encoded as an existential data type. The actual implementation (again, see Edward Kmett’s Data.Profunctor.Composition) is:

data Procompose q p a c where
  Procompose :: q b c -> p a b -> Procompose q p a c

The existential quantifier is expressed in terms of a GADT (Generalized Algebraic Data Type), with the free occurrence of b inside the data constructor.

Einstein’s Convention

By now you might be getting lost juggling the variances of objects appearing in those formulas. The coend variable, for instance, must appear under the integral sign once in the covariant and once in the contravariant position, and the variances on the right must match the variances on the left. Fortunately, there is a precedent in a different branch of mathematics, tensor calculus in vector spaces, with the kind of notation that takes care of variances. Einstein coopted and expanded this notation in his theory of relativity. Let’s see if we can adapt this technique to the calculus of profunctors.

The trick is to write contravariant indices as superscripts and the covariant ones as subscripts. So, from now on, we’ll write the components of a profunctor p (we’ll switch to lower case to be compatible with Haskell) as p^c\,_d. Einstein also came up with a clever convention: implicit summation over a repeated index. In the case of profunctors, the summation corresponds to taking a coend. In this notation, a coend over a profunctor p looks like a trace of a tensor:

p^a\,_a = \int^a p(a, a)

The composition of two profunctors becomes:

(p \circ q)^a\, _c = p^a\,_b \, q^b\,_c = \int^b p(a, b) \times q(b, c)

The summation convention applies only to adjacent indices. When they are separated by an explicit product sign (or any other operator), the coend is not assumed, as in:

p^a\,_b \times q^b\,_c

(no summation).

The hom-functor in a category \mathbb{C} is also a profunctor, so it can be notated appropriately:

C^a\,_b = C(a, b)

The co-Yoneda lemma (see Ninja Yoneda) becomes:

C^c\,_{c'}\,p^{c'}\,_d \cong p^c\,_d \cong p^c\,_{d'}\,D^{d'}\,_d

suggesting that the hom-functors C^c\,_{c'} and D^{d'}\,_d behave like Kronecker deltas (in tensor-speak) or unit matrices. Here, the profunctor p spans two categories

p \colon \mathbb{C}^{op}\times \mathbb{D} \to Set

The lifting of morphisms:

f\colon c' \to c
g\colon d \to d'

can be written as:

p^f\,_g \colon p^c\,_d \to p^{c'}\,_{d'}

There is one more useful identity that deals with mapping out from a coend. It’s the consequence of the fact that the hom-functor is continuous. It means that it maps (co-) limits to limits. More precisely, since the hom-functor is contravariant in the first variable, when we fix the target object, it maps colimits in the first variable to limits. (It also maps limits to limits in the second variable). Since a coend is a colimit, and an end is a limit, continuity leads to the following identity:

Set(\int^c p(c, c), s) \cong \int_c Set(p(c, c), s)

for any set s. Programmers know this identity as a generalization of case analysis: a function from a sum type is a product of functions (one function per case). If we interpret the coend as an existential quantifier, the end is equivalent to a universal quantifier.

Let’s apply this identity to the mapping out from a composition of two profunctors:

p^a\,_b \, q^b\,_c \to s = Set\big(\int^b p(a, b) \times q(b, c), s\big)

This is isomorphic to:

\int_b Set\Big(p(a,b) \times q(b, c), s\Big)

or, after currying (using the product/exponential adjunction),

\int_b Set\Big(p(a, b), q(b, c) \to s\Big)

This gives us the mapping out formula:

p^a\,_b \, q^b\,_c \to s \cong p^a\,_b \to q^b\,_c \to s

with the right hand side natural in b. Again, we don’t perform implicit summation on the right, where the repeated indices are separated by an arrow. There, the repeated index b is universally quantified (through the end), giving rise to a natural transformation.

Bicategory Prof

Since profunctors can be composed using the coend formula, it’s natural to ask if there is a category in which they work as morphisms. The only problem is that profunctor composition satisfies the associativity and unit laws (see the co-Yoneda lemma above) only up to isomorphism. Not to worry, there is a name for that: a bicategory. In a bicategory we have objects, which are called 0-cells; morphisms, which are called 1-cells; and morphisms between morphisms, which are called 2-cells. When we say that categorical laws are satisfied up to isomorphism, it means that there is an invertible 2-cell that maps one side of the law to another.

The bicategory Prof has categories as 0-cells, profunctors as 1-cells, and natural transformations as 2-cells. A natural transformation \alpha between profunctors p and q

\alpha \colon p \Rightarrow q

has components that are functions:

\alpha^c\,_d \colon p^c\,_d \to q^c\,_d

satisfying the usual naturality conditions. Natural transformations between profunctors can be composed as functions (this is called vertical composition). In fact 2-cells in any bicategory are composable, and there always is a unit 2-cell. It follows that 1-cells between any two 0-cells form a category called the hom-category.

But there is another way of composing 2-cells that’s called horizontal composition. In Prof, this horizontal composition is not the usual horizontal composition of natural transformations, because composition of profunctors is not the usual composition of functors. We have to construct a natural transformation between one composition of profuntors, say p^a\,_b \, q^b\,_c and another, r^a\,_b \, s^b\,_c, having at our disposal two natural transformations:

\alpha \colon p \Rightarrow r

\beta \colon q \Rightarrow s

The construction is a little technical, so I’m moving it to the appendix. We will denote such horizontal composition as:

(\alpha \circ \beta)^a\,_c \colon p^a\,_b \, q^b\,_c \to r^a\,_b \, s^b\,_c

If one of the natural transformations is an identity natural transformation, say, from p^a\,_b to p^a\,_b, horizontal composition is called whiskering and can be written as:

(p \circ \beta)^a\,_c \colon p^a\,_b \, q^b\,_c \to p^a\,_b \, s^b\,_c

Promonads

The fact that a monad is a monoid in the category of endofunctors is a lucky accident. That’s because, in general, a monad can be defined in any bicategory, and Cat just happens to be a (strict) bicategory. It has (small) categories as 0-cells, functors as 1-cells, and natural transformations as 2-cells. A monad is defined as a combination of a 0-cell (you need a category to define a monad), an endo-1-cell (that would be an endofunctor in that category), and two 2-cells. These 2-cells are variably called multiplication and unit, \mu and \eta, or join and return.

Since Prof is a bicategory, we can define a monad in it, and call it a promonad. A promonad consists of a 0-cell C, which is a category; an endo-1-cell p, which is a profunctor in that category; and two 2-cells, which are natural transformations:

\mu^a\,_b \colon p^a\,_c \, p^c\,_b \to p^a\,_b

\eta^a\,_b \colon C^a\,_b \to p^a\,_b

Remember that C^a\,_b is the hom-profunctor in the category C which, due to co-Yoneda, happens to be the unit of profunctor composition.

Programmers might recognize elements of the Haskell Arrow in it (see my blog post on monoids).

We can apply the mapping-out identity to the definition of multiplication and get:

\mu^a\,_b \colon p^a\,_c \to p^c\,_b \to p^a\,_b

Notice that this looks very much like composition of heteromorphisms. Moreover, the monadic unit \eta maps regular morphisms to heteromorphisms. We can then construct a new category, whose objects are the same as the objects of \mathbb{C}, with hom-sets given by the profunctor p. That is, a hom set from a to b is the set p^a\,_b. We can define an identity-on-object functor J from \mathbb{C} to that category, whose action on hom-sets is given by \eta.

Interestingly, this construction also works in the opposite direction (as was brought to my attention by Alex Campbell). Any indentity-on-objects functor defines a promonad. Indeed, given a functor J, we can always turn it into a profunctor:

p(c, d) = D(J\, c, J\, d)

In Einstein notation, this reads:

p^c\,_d = D^{J\, c}\,_{J\, d}

Since J is identity on objects, the composition of morphisms in D can be used to define the composition of heteromorphisms. This, in turn, can be used to define \mu, thus showing that p is a promonad on \mathbb{C}.

Conclusion

I realize that I have touched upon some pretty advanced topics in category theory, like bicategories and promonads, so it’s a little surprising that these concepts can be illustrated in Haskell, some of them being present in popular libraries, like the Arrow library, which has applications in functional reactive programming.

I’ve been experimenting with applying Einstein’s summation convention to profunctors, admittedly with mixed results. This is definitely work in progress and I welcome suggestions to improve it. The main problem is that we sometimes need to apply the sum (coend), and at other times the product (end) to repeated indices. This is in particular awkward in the formulation of the mapping out property. I suggest separating the non-summed indices with product signs or arrows but I’m not sure how well this will work.

Appendix: Horizontal Composition in Prof

We have at our disposal two natural transformations:

\alpha \colon p \Rightarrow r

\beta \colon q \Rightarrow s

and the following coend, which is the composition of the profunctors p and q:

\int^b p(a, b) \times q(b, c)

Our goal is to construct an element of the target coend:

\int^b r(a, b) \times s(b, c)

Horizontal composition of 2-cells

To construct an element of a coend, we need to provide just one element of r(a, b') \times s(b', c) for some b'. We’ll look for a function that would construct such an element in the following hom-set:

Set\Big(\int^b p(a, b) \times q(b, c), r(a, b') \times s(b', c)\Big)

Using Einstein notation, we can write it as:

p^a\,_b \, q^b\,_c \to r^a\,_{b'} \times s^{b'}\,_c

and then use the mapping out property:

p^a\,_b \to q^b\,_c \to r^a\,_{b'} \times s^{b'}\,_c

We can pick b' equal to b and implement the function using the components of the two natural transformations, \alpha^a\,_{b} \times \beta^{b}\,_c.

Of course, this is how a programmer might think of it. A mathematician will use the universal property of the coend (p \circ q)^a\,_c, as in the diagram below (courtesy Alex Campbell).

Horizontal composition using the universal property of a coend

In Haskell, we can define a natural transformation between two (endo-) profunctors as a polymorphic function:

newtype PNat p q = PNat (forall a b. p a b -> q a b)

Horizontal composition is then given by:

horPNat :: PNat p r -> PNat q s -> Procompose p q a c
        -> Procompose r s a c
horPNat (PNat alpha) (PNat beta) (Procompose pbc qdb) = 
  Procompose (alpha pbc) (beta qdb)

Acknowledgment

I’m grateful to Alex Campbell from Macquarie University in Sydney for extensive help with this blog post.

Further Reading


Oxford, UK.    2019 July 22 – 26

Dear scientists, mathematicians, linguists, philosophers, and hackers,

We are writing to let you know about a fantastic opportunity to learn about the emerging interdisciplinary field of applied category theory from some of its leading researchers at the ACT2019 School.   It will begin in January 2019 and culminate in a meeting in Oxford, July 22-26.

Applied category theory is a topic of interest for a growing community of researchers, interested in studying systems of all sorts using category-theoretic tools.  These systems are found in the natural sciences and social sciences, as well as in computer science, linguistics, and engineering. The background and experience of our community’s members are as varied as the systems being studied.

The goal of the ACT2019 School is to help grow this community by pairing ambitious young researchers together with established researchers in order to work on questions, problems, and conjectures in applied category theory.

Who should apply?

Anyone from anywhere who is interested in applying category-theoretic methods to problems outside of pure mathematics. This is emphatically not restricted to math students, but one should be comfortable working with mathematics. Knowledge of basic category-theoretic language—the definition of monoidal category for example— is encouraged.

We will consider advanced undergraduates, Ph.D. students, and post-docs. We ask that you commit to the full program as laid out below.

Instructions on how to apply can be found below the research topic descriptions.

Senior research mentors and their topics

Below is a list of the senior researchers, each of whom describes a research project that their team will pursue, as well as the background reading that will be studied between now and July 2019.

Miriam Backens

Title: Simplifying quantum circuits using the ZX-calculus

Description: The ZX-calculus is a graphical calculus based on the category-theoretical formulation of quantum mechanics.  A complete set of graphical rewrite rules is known for the ZX-calculus, but not for quantum circuits over any universal gate set.  In this project, we aim to develop new strategies for using the ZX-calculus to simplify quantum circuits.

Background reading:

  1. Matthes Amy, Jianxin Chen, Neil Ross. A finite presentation of CNOT-Dihedral operators. arXiv:1701.00140
  2. Miriam Backens. The ZX-calculus is complete for stabiliser quantum mechanics. arXiv:1307.7025

Tobias Fritz

Title: Partial evaluations, the bar construction, and second-order stochastic dominance

Description: We all know that 2+2+1+1 evaluates to 6. A less familiar notion is that it can partially evaluate to 5+1.  In this project, we aim to study the compositional structure of partial evaluation in terms of monads and the bar construction and see what this has to do with financial risk via second-order stochastic dominance.

Background reading:

  1. Tobias Fritz, Paolo Perrone. Monads, partial evaluations, and rewriting. arXiv:1810.06037
  2. Maria Manuel Clementino, Dirk Hofmann, George Janelidze. The monads of classical algebra are seldom weakly cartesian. Available here.
  3. Todd Trimble. On the bar construction. Available here.

Pieter Hofstra

Title: Complexity classes, computation, and Turing categories

Description: Turing categories form a categorical setting for studying computability without bias towards any particular model of computation. It is not currently clear, however, that Turing categories are useful to study practical aspects of computation such as complexity. This project revolves around the systematic study of step-based computation in the form of stack-machines, the resulting Turing categories, and complexity classes.  This will involve a study of the interplay between traced monoidal structure and computation. We will explore the idea of stack machines qua programming languages, investigate the expressive power, and tie this to complexity theory. We will also consider questions such as the following: can we characterize Turing categories arising from stack machines? Is there an initial such category? How does this structure relate to other categorical structures associated with computability?

Background reading:

  1. J.R.B. Cockett, P.J.W. Hofstra. Introduction to Turing categories. APAL, Vol 156, pp 183-209, 2008.  Available here .
  2. J.R.B. Cockett, P.J.W. Hofstra, P. Hrubes. Total maps of Turing categories. ENTCS (Proc. of MFPS XXX), pp 129-146, 2014.  Available here.
  3. A. Joyal, R. Street, D. Verity. Traced monoidal categories. Mat. Proc. Cam. Phil. Soc. 3, pp. 447-468, 1996. Available here.

Bartosz Milewski

Title: Traversal optics and profunctors

Description: In functional programming, optics are ways to zoom into a specific part of a given data type and mutate it.  Optics come in many flavors such as lenses and prisms and there is a well-studied categorical viewpoint, known as profunctor optics.  Of all the optic types, only the traversal has resisted a derivation from first principles into a profunctor description. This project aims to do just this.

Background reading:

  1. Bartosz Milewski. Profunctor optics, categorical View. Available here.
  2. Craig Pastro, Ross Street. Doubles for monoidal categories. arXiv:0711.1859

Mehrnoosh Sadrzadeh

Title: Formal and experimental methods to reason about dialogue and discourse using categorical models of vector spaces

Description: Distributional semantics argues that meanings of words can be represented by the frequency of their co-occurrences in context. A model extending distributional semantics from words to sentences has a categorical interpretation via Lambek’s syntactic calculus or pregroups. In this project, we intend to further extend this model to reason about dialogue and discourse utterances where people interrupt each other, there are references that need to be resolved, disfluencies, pauses, and corrections. Additionally, we would like to design experiments and run toy models to verify predictions of the developed models.

Background reading:

  1. Gerhard Jager.  A multi-modal analysis of anaphora and ellipsis. Available here.
  2.  Matthew Purver, Ronnie Cann, Ruth Kempson. Grammars as parsers:    Meeting the dialogue challenge. Available here.

David Spivak

Title: Toward a mathematical foundation for autopoiesis

Description: An autopoietic organization—anything from a living animal to a political party to a football team—is a system that is responsible for adapting and changing itself, so as to persist as events unfold. We want to develop mathematical abstractions that are suitable to found a scientific study of autopoietic organizations. To do this, we’ll begin by using behavioral mereology and graphical logic to frame a discussion of autopoiesis, most of all what it is and how it can be best conceived. We do not expect to complete this ambitious objective; we hope only to make progress toward it.

Background reading:

  1. Fong, Myers, Spivak. Behavioral mereology.  arXiv:1811.00420.
  2. Fong, Spivak. Graphical regular logic.  arXiv:1812.05765.
  3. Luhmann. Organization and Decision, CUP. (Preface)

School structure

All of the participants will be divided up into groups corresponding to the projects.  A group will consist of several students, a senior researcher, and a TA. Between January and June, we will have a reading course devoted to building the background necessary to meaningfully participate in the projects. Specifically, two weeks are devoted to each paper from the reading list. During this two week period, everybody will read the paper and contribute to a discussion in a private online chat forum.  There will be a TA serving as a domain expert and moderating this discussion. In the middle of the two week period, the group corresponding to the paper will give a presentation via video conference. At the end of the two week period, this group will compose a blog entry on this background reading that will be posted to the n-category cafe.

After all of the papers have been presented, there will be a two-week visit to Oxford University from 15 – 26 July 2019.  The first week is solely for participants of the ACT2019 School. Groups will work together on research projects, led by the senior researchers.  

The second week of this visit is the ACT2019 Conference, where the wider applied category theory community will arrive to share new ideas and results. It is not part of the school, but there is a great deal of overlap and participation is very much encouraged. The school should prepare students to be able to follow the conference presentations to a reasonable degree.

How to apply

To apply please send the following to act2019school@gmail.com

  • Your CV
  • A document with:
    • An explanation of any relevant background you have in category theory or any of the specific projects areas
    • The date you completed or expect to complete your Ph.D. and a one-sentence summary of its subject matter.
    • Order of project preference
    • To what extent can you commit to coming to Oxford (availability of funding is uncertain at this time)
  • A brief statement (~300 words) on why you are interested in the ACT2019 School. Some prompts:
    • how can this school contribute to your research goals
    • how can this school help in your career?

Also, have sent on your behalf to act2019school@gmail.com a brief letter of recommendation confirming any of the following:

  • your background
  • ACT2019 School’s relevance to your research/career
  • your research experience

Questions?

For more information, contact either

  • Daniel Cicala. cicala (at) math (dot) ucr (dot) edu
  • Jules Hedges. julian (dot) hedges (at) cs (dot) ox (dot) ac (dot) uk

Flies

It was a hot evening and, as is usual at that time of the year, there were quite a few flies buzzing around us. Understandably, I was annoyed, as they were interfering with my meditation.

“Master,” I said, “You keep telling me that everything in this world has a purpose, but I can’t figure out the purpose of these flies. All they do is break my concentration. Can we move indoors already, behind the screens, so that we can continue the lessons in peace?”

The Master looked at me the way he usually does when I say something that shows my lack of understanding — which unfortunately happens a lot.

“The flies are here to teach us about meditation,” he said.

“How so?” I said. “Are you trying to tell me that I should be able to quiet my mind even when there’s constant interference?”

“That would be the ultimate goal,” said the Master, “but for now I’d like you to observe the way these flies move. Can you do that?”

“Of course, Master,” I said and started watching the flies criss-crossing the air in front of me.

“What do you see?” asked the Master after a while.

“I see them zig-zagging constantly. They never seem to fly in a straight line for longer than a fraction of a second.”

“You are very astute, my Disciple,” said the Master. “Now, why would you say they’re doing that?”

“I think they are doing that to avoid being caught” I said. “Those flies that were, long ago, flying in straight lines were eliminated by predators, and only those that employed more elaborate movement schemes survived long enough to produce offspring. Evolution in action!” I said not without some pride at my cleverness.

“Quite so, my Disciple,” said the Master, “quite so…”

“But suppose,” he continued, “that, in some other universe, there is a colony of flies that live confined within the boundaries a hostile environment. Their life is short and full of suffering. But there is a benevolent being that can set individual flies free, to live a happy and productive life. The trouble is that she has to catch them first. And, in the beginning, it was easy, since they were all flying in straight lines. Almost all. The benevolent being was able to remove the straight-flying flies and make them happy. But there remained a few flies that, for one reason or another, kept zig-zagging. They survived long enough to produce offspring, some of which also kept zig-zagging. Soon enough, all flies in that hostile and unhappy environment developed this new survival strategy that prevented them from escaping their horrible fate. That’s evolution in action, too.”

“That’s a pretty sad story,” I said. “It shows that evolution is a cruel mistress. It doesn’t care if we are happy or not, as long as we produce offspring.”

“But what does it have to do with meditation?” I said, a little confused.

“The flies are our thoughts,” said the Master.

The Pipe

It was a pleasant evening and I was enjoying the warm breeze coming from the mountains bringing with it the smell of pine and something else.

“Why are you smoking a pipe, Master? It’s bad for your health.”

“This is not a pipe,” said the Master.

“This is most definitely a pipe. You must have bought it in a pipe shop,” I said.

“This is not a pipe,” said the Master.

I thought for a moment.

“Oh, I see. You are making reference to the famous painting by Rene Magritte, right? Ceci n’est pas une pipe! But what Magritte meant was that it wasn’t a pipe–it was a picture of a pipe. He played on our confusion between the object itself and its representation. But here you are holding an actual pipe.”

“What makes you think it’s a pipe?” asked the Master.

“Well, I can look up the definition of a pipe and you’ll see that it describes the object you are holding. Do you want me to do that?” I asked.

“Please do,” said the Master.

I pulled out my tablet and started tapping.

“What’s the wi-fi password for today, Master?” I asked.

“That which cannot be named,” said the Master.

“Oh, I know that one,” I said and continued tapping. I stopped after a few tries and looked up.

“I tried Tao and Dao, upper- and lowercase, but it didn’t work. Is it ‘the Tao,’ with ‘the’?”

“You are much too clever, my Disciple,” said the Master.

“Oh, you mean it’s literally ‘that which cannot be named’?” I started tapping again.

“Okay, here it is. According to the OED, a pipe is…” I hesitated for a moment.

“Wait, you don’t really want me to read the definition,” I said.

“No,” said the Master.

After a moment of silence, I said:
“You have corrected me, Master, because by naming the pipe I focused on just one small aspect of it. Its relationship to other pipes. By doing that I ignored its relationship to you, to me, to our conversation, to this lovely sunset, to Magritte and–now I get it–to the Tao.”

“But, Master, I’m confused,” I said after a while. “When you say that ‘The Tao that can be named is not the eternal Tao,’ you are giving it a name, aren’t you? You are calling it the Tao.”

“This is not a name,” said the Master.

“I have the feeling that if I say that this is indeed a name, I would be hitting a dead end,” I said.

We sat there for a moment while I was organizing my thoughts. Then it occurred to me.

“By calling it the Tao, you are not separating it from everything else, because the Tao is in everything. Neither are you ignoring its relationship to yourself or to me, because you are the Tao and I am the Tao. And the lovely sunset, and the pipe, and Magritte, it’s all the eternal Tao.

“This is not the Tao,” said the Master.

Fallout

“I watched a movie last night,” I said. “And it made me think.”

“Movies often make us think,” said the Master. “Good movies, like life itself, ask a lot of questions, but rarely provide answers.”

“Well, that’s the thing, Master” I said. “Maybe you know the answer to this question, or maybe you can steer me towards the answer. I’m sure this problem has been analyzed before by many people much wiser than yours truly.

“It’s a problem of moral nature. In the movie, agent Hunt faces a dilemma. His friend is in immediate mortal danger. Hunt can save him, but at the risk of endangering the lives of thousands of innocent people. He makes a choice, saves the friend but, in the process, the terrorists get hold of plutonium, which they use to make nuclear bombs. Of course, in the movie, he’s ultimately able to avert the disaster, disabling the bombs literally one second before they’re about to go off.

“Sorry if I spoiled the movie for you, Master.”

“Don’t worry, I’ve seen the movie,” said the Master.

“So what do you think, Master? Was agent Hunt acting recklessly, risking uncountable lives to save one?”

“And what’s your opinion?” asked the Master.

“I think the answer is clear. It’s just simple math: one life against thousands. I would probably feel guilty for the rest of my life for sacrificing a friend, but what right do I have to risk thousands of innocent lives?”

“You say it’s simple math,” said the Master. “I presume there is an equation that calculates the moral value of an act, based on the number of lives saved or lost.”

“It’s not an exact science, but I guess one could make some rough estimates,” I said. “I’ve read some articles that mostly deal with pulling levers to divert trolleys. So this seems like one of these problems, where your friend is tied up on one track, and thousands of people on another. A runaway trolley is going to kill your friend, and you pull the switch to divert it to the other track, possibly killing thousands of people.”

“If this is simple math, then why do you say you’d feel guilty? Shouldn’t you feel satisfied, like when you solve a difficult equation?”

“I don’t know. I think I would always speculate: What if? What if I saved my friend and, just like in the movie, were able to avert the disaster? I’d never know.”

“And what if you saved your friend’s life and the bomb exploded?” asked the Master.

“I guess I’d feel terrible for the rest of my life. And I would probably be the most despised person on Earth.”

“And what if that explosion prevented an even bigger disaster in the future?” asked the Master.

“And what if that bigger disaster prevented an even bigger disaster?” I asked. “Where does this end? Are you saying that, since we cannot predict the results of our actions on a global scale, then there is no moral imperative?”

“Would that satisfy you?” asked the Master.

“No, it wouldn’t!”

“Would you like to have a small set of simple rules to guide all moral decisions in your life?” asked the Master.

“When you put it this way, I’m not sure. I think there’s been many attempts at rule-based ethics, and they all have exhibited some pretty disastrous failure modes. It vaguely reminds me of the Goedel’s incompleteness theorem. No matter what moral axioms you choose, there will be a situation in which they fail.

“On the other hand, rejecting the axioms may lead to an even bigger tragedy, like in the case of Raskolnikov.”

“Do you see similarities between Raskolnikov and agent Hunt?” asked the Master.

“They both reject the ‘Thou shalt not kill’ commandment. They both feel intense loyalty to their friends and family. But Raskolnikov had a lot of time to think about his choices, he even published an article about it; whereas Hunt acted impulsively, following his gut feelings. One was rational, the other irrational.”

“But you said that Raskolnikov had no axioms,” said the Master. “So how could he rationally justify his actions?”

“I see your point,” I said. “He was trying to do the math. Solve the ethical equation. His hubris was not in rejecting the accepted axioms, but in believing that he can come up with a better set. So, in a way, agent Hunt had the advantage of being a moral simpleton.”

“He was the uncarved wood,” said the Master.