Category Theory

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May 30, 2025

Fibrations and Cofibrations

Previously: Subfunctor Classifier.

We are used to thinking of a mapping as either being invertible or not. It’s a yes or no question. A mapping between sets is invertible if it’s both injective and surjective. It means that it never merges two elements into one, and it covers the whole target set.

But if you are willing to look closer, the failures of invertibility are a whole fascinating area of research. Things get a lot more interesting if you consider mapping between topological spaces, which have to skillfully navigate around various holes and tears in the target space, and shrink or glue together parts of the source space. I will be glossing over some of the topological considerations, concentrating on the big-picture intuitions (both culinary and cinematographic).

Fibrations

In what follows, we’ll be considering a function $p \colon E \to B$ . We’ll call $p$ a projection from the total set $E$ to the base set $B$ .

Let’s start by considering the first reason for a failure of invertibility: multiple elements being mapped into one. Even though we can’t invert such a mapping, we can use it to fibrate the source $E$ .

To each element $y \in B$ we’ll assign a fiber, the set of elements of $E$ that are mapped to $y$ . By abuse of notation, we call such a fiber $p^{-1} y$ :

$p^{-1} y = \{ x |\; p x = y \}$

For some $y$ ‘s, this set may be empty $\emptyset$ ; for others, it may contain lots elements.

Notice that $p$ is an isomorphism if and only if every fiber is a singleton set. This property gives rise to a very useful definition of equivalence in homotopy type theory, where we ask for every fiber to be contractible.

A set-theoretic union of all fibers reconstructs the total set $E$ .

Things get more interesting when we move to topological spaces and continuous functions. To begin with, we can define a path in $B$ as a continuous mapping from the unit interval $I = [0, 1]$ to $B$ .

$\gamma \colon I \to B$

We can then ask if it’s possible to lift this path to $E$ , that is to construct $\tilde{\gamma} \colon I \to E$ that lies above $\gamma$ . To do this, we pick the starting point $e \in E$ that lies above the starting point of $\gamma$ , that is $p \,e = \gamma \,0$ .

This setup can be summarized in the commuting square:

The lifting $\tilde{\gamma}$ is then the diagonal arrow that makes both triangles commute:

It’s helpful to imagine a path as a trajectory of a point moving through a topological space. The parameter $t \in I$ is then interpreted as time.

In homotopy theory we generalize this idea to the movement, or deformation, of arbitrary shapes, not just points.

If we describe such a shape as the image of some topological space $X$ , its deformation is a mapping $h \colon X \times I \to B$ . Such potentially “fat” paths are called homotopies. In particular, if we replace $X$ by a single point, we recover our “thin” paths.

A homotopy lifting property is expressed as the existence of the diagonal function $\tilde h$ in this commuting square, such that the resulting triangles commute:

In other words, given a homotopy $h$ of the shape $X$ in the base, and an embedding $e$ of this shape in $E$ above $h \, 0$ , we can construct a homotopy in $E$ whose projection is $h$ .

Cofibrations

In category theory, every construction has its dual, which is obtained by reversing all the arrows. In topology, there is an analogous relation called the Eckmann-Hilton duality. Besides reversing the arrows, it also dualizes products to exponentials (using the currying adjunction), and injections to surjections.

When dualizing the homotopy lifting diagram, we replace the trivial injection of $X \times \{0\} \cong X$ into $X \times I$ by a surjection $\varepsilon_0$ of the exponential $X^I$ onto $X$ . Here, $X^I$ is the set of functions $I \to X$ , or the path space of $X$ . The surjection $\varepsilon_0$ maps every path $\gamma \colon I \to X$ to its starting point by evaluating it at zero. (The evaluation map $\varepsilon_t \colon \gamma \mapsto \gamma \,t$ is continuous in the compact-open topology of $X^I$ .)

The homotopy lifting property is thus dualized to the homotopy extension property:

Corresponding to the fibration that deals with the failure of the injectivity of $p$ , the homotopy extension property deals with the failure of the surjectivity of $i$ .

If the mapping $i$ has the extension property for all topological spaces $X$ , it is called a cofibration.

Intuitively, a cofibration is an embedding of $B$ in $E$ such that any “spaghettification” of $B$ , that is embedding it in the path space $X^I$ , can be extended to a spaghettification of $E$ . $X$ plays the role of an ambient space where these operations can be visualized. Later we’ll see how to construct a minimal such ambient space called a mapping cylinder.

Let’s deconstruct the meaning of the outer commuting square.

$i$ embeds $B$ into $E$ .
$e$ further embeds $E$ into $X$ , and we end up with the embedding of $B$ inside $X$ given by $e \circ i$ .
$h$ embeds $B$ into the path space of $X$ (the dotted paths below).

The commuting condition for this square means that the starting points of all these paths, the result of applying $\varepsilon_0$ to each of them, coincide with the embedding of $B$ into $X$ given by $e \circ i$ . It’s like extruding a bunch of spaghetti, each strand corresponding to a point in $B$ .

With this setup, we postulate the existence of the diagonal mapping $\tilde h \colon E \to X^I$ that makes the two triangles commute. In other words, $E$ is mapped to a family of paths in $X$ which, when restricted to the image of $B$ coincide with the original mapping $h$ .

The spaghetti extruded through the $E$ shape contain the spaghetti extruded through the $B$ shape.

Another intuitive description of this situation uses the idea of homotopy as animation. The strands of spaghetti become trajectories of particles.

We start by setting up the initial scene. We embed $E$ in the big space $X$ using $e \colon E \to X$ .

We have the embedding $i \colon B \to E$ , which induces the embedding of $B$ into $X$ :

$b = e \circ i$

Then we animate the embedding of $B$ using the homotopy

$h \colon B \times I \to X$

The initial frame of this animation is given by $b$ :

$h(x, 0) = b \, x$

We say that $i$ is a cofibration if every such animation can be extended to the bigger animation:

$\tilde h \colon E \times I \to X$

whose starting frame is given by $e$ :

$\tilde h (x, 0) = e \, x$ .

The commuting condition (for the lower triangle) means that the two animations coincide for all $x \in B$ and $t \in I$ :

$h (x, t) = \tilde h (i \,x, t)$

Just like a fiber is an epitome of non-injectiveness, one can define a cofiber as an epitome of non-surjectiveness. It’s essentially the part of $E$ that is not covered by $i$ .

As a topological space it’s the result of shrinking the image of $B$ inside $E$ to a point (the resulting topology is called the quotient topology).

Notice that, unlike with fibers, there is just one cofiber for a given cofibration (up to homotopy equivalence).

Lifting Property

A category theorist looking at the two diagrams that define, respectively, homotopy lifting and extension, will immediately ignore all the superfluous details. She will turn the second diagram upside down and merge it with the first diagram to get:

This is how we read the new diagram: If for any morphisms $f$ and $g$ that make the outer square commute, there exist a diagonal morphism $h$ that makes the two triangles commute, then we say that $i$ has the left lifting property with respect to $p$ . Equivalently, $p$ has the right lifting property with respect to $i$ .

Or we can say that the two morphisms are orthogonal to each other.

The motivation for this nomenclature is interesting. In the category of sets, the archetypal non-surjective function is the “absurd” $0 \to 1$ (or, in set notation, $\emptyset \to \{*\}$ ). It turns out that all surjective functions are its right liftings. In other words, all surjections are right-orthogonal to the simplest non-surjective function.

Indeed, the function at the bottom $y \colon 1 \to Y$ picks an element $y \in Y$ . Similarly, the diagonal function picks an element $x \in X$ . The commuting triangle tells us that for every $y$ there exists an $x$ such that $y = p \, x$ .

Similarly, we can show that all injective functions are orthogonal to the archetypal non-injective function $2 \to 1$ (or $\{x_1, x_2\} \to \{*\}$ ).

Indeed, assume that $i$ maps two different elements $a_1, a_2 \in A$ to a single element $b \in B$ . We could then pick $f$ such that $f \, a_1 = x_1$ and $f \, a_2 = x_2$ . The diagonal $h$ can map $b$ to either $x1$ or $x2$ but not both, so we couldn’t make the upper triangle commute.

Incidentally, injections are also right-orthogonal to the archetypal non-injection.

Next: (Weak) Homotopy Equivalences.

April 21, 2025

Subfunctor Classifier

Posted by Bartosz Milewski under Category Theory, Topology, Topos | Tags: Category Theory, math, Topology, Topos |
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Previously: Subobject Classifier.

In category theory, objects are devoid of internal structure. We’ve seen however that in certain categories we can define relationships between objects that mimic the set-theoretic idea of one set being the subset of another. We do this using the subobject classifier.

We would like to define a subobject classifier in the category of presheaves, so we could easily characterize subfunctors of a given presheaf. This will help us work with sieves, which are subfunctors of the hom-functor $C(-, a)$ ; and coverages, which are special kinds of sieves.

Recall that a presheaf $S$ is a subfunctor of another presheaf $P \colon C^{op} \to Set$ if it satisfies two conditions.

For every object $a$ , we have a set inclusion: $S a \subseteq P a$ ,
For every morphism $f \colon c \to a$ , the function $S f \colon S a \to S c$ is a restriction of the function $P f \colon P a \to P c$ . In other words, $P f$ and $S f$ must agree on the subset $S a$ .

As category theory goes, this is a very low-level definition. We need something more abstract: We need to construct a subobject classifier in the category of presheaves. Recall that a subobject classifier is defined by the following pullback diagram:

This time, however, the objects are presheaves and the arrows are natural transformations.

To begin with we have to define a terminal presheaf, $1 \colon C^{op} \to Set$ that satisfies the condition that, for any presheaf $P$ , there is a unique natural transformation $! \colon P \to 1$ . This will work if every component $!_a \colon P a \to 1 a$ of this natural transformation is unique, which is true if we choose $1 a$ to be the terminal singleton set $\{ * \}$ . Thus the terminal presheaf maps all objects to the terminal set, and all morphisms to the identity on $\{ * \}$ .

Next, let’s instantiate the subobject classifier diagram at a particular object $a$ .

Here, the component $true_a$ picks a special “True” element in the set $\Omega_a$ . If the presheaf $S$ is a subfunctor of $P$ , the set $S a$ is a subset of $P a$ . The function $\chi_a$ must therefore map the whole subset $S a$ to “True”. This is consistent with our definition of the subobject classifier for sets.

The second condition in the definition of a subfunctor is more interesting. It involves the mapping of morphisms.

The restriction condition

We have to consider all morphisms converging on our object of interest $a$ . For instance, lets take $f \colon c \to a$ . The presheaf $P$ lifts it to a function $P f \colon P a \to P c$ . If $S$ is a subfunctor of $P$ , $S f$ is a restriction of $P f$ .

Specifically the restriction condition tells us that, if we pick an element $x \in S a$ , then both $P f$ and $S f$ will map it to the same element of $S c$ . In fact, when defining a subobject, we only care if the embedding of $S c$ in $P c$ is injective (monomorphic). It’s okay if it permutes the elements of $S c$ . So it’s enough that, for all $x \in S a$ , the following condition is satisfied:

$(P f) x \in S c$

Now consider an arbitrary $x \in P a$ (not necessarily an element of $S a$ ). We can gather all arrows $f$ converging on $a$ for which the subset-mapping condition is satisfied:

$(P f) x \in S c$

If $S$ is a subfunctor of $P$ , these arrows form a sieve on $a$ , as any composition $f \circ g$ also satisfies the subset-mapping condition:

Moreover, if $x$ is in fact an element of $S a$ , this sieve is the maximal sieve. A maximal sieve on $a$ is a collection of all arrows converging on $a$ .

We can now define a function $\chi_a$ that assigns to each $x \in P a$ the sieve of arrows that satisfy the subset-mapping condition.

$\chi_a x = \{f \colon c \to a \, | \, (P f) x \in S c\}$

The function $\chi_a$ has the property that, if $x$ is an element of $S a$ , the result is the maximal sieve on $a$ .

It makes sense then to define $\Omega_a$ as a set of sieves on $a$ , and “True” as the maximal sieve on $a$ . (Thus $\Omega_a$ is a set whose elements are sets.)

The mapping $\Omega \colon a \to \Omega_a$ can be made into a presheaf by defining its action on morphisms. The lifting of $f \colon c \to a$ takes a sieve $s_a \in \Omega_a$ to a sieve $s'_{c} \in \Omega c$ , defined as a set of arrows $h \colon c' \to c$ , such that $f \circ h \in s_a$ .

Notice that the resulting sieve $s_c'$ is maximal if and only if $f \in \Omega_a$ . (Hint: If a sieve is maximal, then it contains identity.)

It can be shown that the the functions $\chi_a$ combine to form a natural transformation $\chi \colon P \to \Omega$ .

What remains to be shown is that this $\chi$ is a unique such natural transformation that completes the pullback:

To show that, let’s assume that there is another natural transformation $\theta \colon P \to \Omega$ making this diagram into a pullback. Let’s redraw the subfunctor condition for arrows, replacing $\chi$ with $\theta$ :

Let’s pick an $x \in P a$ and call $y = (P f) x$ . We’ll follow a set of equivalences.

The pullback condition:

tells us that $y \in S c$ is equivalent to $\theta_c y = true_c$ . In other words:

$\theta_c ((P f) x) = true_c$

Using naturality of $\theta$ :

we can rewrite it as:

$(\Omega f) (\theta_a x) = true_c$ .

Both sides of this equation are sieves. By definition, the lifting of $f$ , $\Omega f$ , acting on $\theta_a x$ is a sieve defined by the following set of arrows:

$(\Omega f) (\theta_a x) = \{ h \colon c' \to c \, | \, f \circ h \in \theta_a x \}$

Since $true_c$ is a maximal sieve, it must be that $f \in \theta_a x$ .

We have shown that the condition $(P f) x \in S c$ is equivalent to $f \in \theta_a x$ . But the first condition is exactly the one we used to define $\chi_a x$ . Therefore $\chi$ is the only function that makes the subobject classifier diagram into a pullback.

Subfunctor classifier

The subobject classifier in the category of presheaves is thus a presheaf $\Omega$ that maps objects to sieves, together with the natural transformation $true \colon 1 \to \Omega$ that picks maximal sieves.

Every natural transformation $\chi \colon P \to \Omega$ defines a subfunctor of the presheaf $P$ . The components of this natural transformation serve as characteristic functions for the sets $P a$ . A given element $x$ is in the subset $S a$ iff $\chi_a$ maps it to the maximal sieve on $a$ .

But there’s not one but many different ways of failing the subset test. They are given by non-maximal sieves. We may think of them as satisfying the Anna Karenina principle, “All happy families are alike; each unhappy family is unhappy in its own way.”

Why sieves? Because once an element of a set $P a$ is mapped by $P f$ to an element of a subset $S c$ , it will continue to be mapped into consecutive subsets $S c'$ , etc. The network of “happy” morphisms keeps growing outward. By contrast, the “unhappy” elements of $x \in P a$ have at least one morphism $f \colon c \to a$ , whose lifting maps it outside the subset $S c$ . That’s the morphism that’s absent from the non-maximal sieve $\chi_a$ . Finally, naturality of $\chi$ ensures that subset conditions propagate coherently from object to object.

Next: Fibrations and Cofibrations.

March 20, 2025

Subobject Classifier

Posted by Bartosz Milewski under Category Theory, Topology, Topos | Tags: Category Theory, math, Topology |
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Proviously Sieves and Sheaves.

We have seen how topology can be defined by working with sets of continuous functions over coverages. Categorically speaking, a coverage is a special case of a sieve, which is defined as a subfunctor of the hom-functor $C(-, a)$ .

We’d like to characterize the relationship between a functor and its subfunctor by looking at them as objects in the category of presheaves. For that we need to introduce the idea of a subobject.

We’ll start by defining subobjects in the category of sets in a way that avoids talking about elements. Here we have two options.

The first one uses a characteristic function. It’s a predicate that answers the question: Is some element $x$ a member of a given subset or not? Notice that any Boolean-valued function uniquely defines a subset of its domain, so we don’t really need to talk about elements, just a function.

But we still have to define a Boolean set. Let’s call this set $\Omega$ , and designate one of its element as “True.” Selecting “True” can be done by defining a function $true \colon 1 \to \Omega$ , where $1$ is the terminal object (here, a singleton set). In principle we should insist that $\Omega$ contains two elements, “True” and “False,” but that would make it more difficult to generalize.

The second way to define a subset $S \subseteq P$ is to provide an injective function $m \colon S \rightarrowtail P$ that embeds $S$ in $P$ . Injectivity guarantees that no two elements are mapped to the same element. The image of $m$ then defines the subset of $P$ . In a general category, injective functions are replaced by monics (monomorphisms).

Notice that there can be many injections that define the same subset. It’s okay for them to permute the image of $m$ as long as it covers exactly the same subset of $P$ . (These injections form an equivalence class.)

The fact that the two definitions coincide can be summarized by one commuting diagram. In the category of sets, given a characteristic function $\chi$ , the subset $S$ and the monic $m$ are uniquely (up to isomorphism) defined as a pullback of this diagram.

We can now turn the tables and use this diagram to define the object $\Omega$ called the subobject classifier, together with the monic $true \colon 1 \rightarrowtail \Omega$ . We do it by means of a universal construction. We postulate that: For every monic $S \rightarrowtail P$ between two arbitrary objects there exist a unique arrow $\chi \colon P \to \Omega$ such that the above diagram constitutes a pullback.

This is a slightly unusual definition. Normally we think of a pullback as defining the northwest part of the diagram given its southeast part. Here, we are solving a sudoku puzzle, trying to fill the southeast part to uniquely complete a pullback diagram.

Let’s see how this works for sets. To construct a pullback (a.k.a., a fibered product $P \times_{\Omega} 1$ ) we first create a set of pairs $(x, *)$ where $x \in P$ and $* \in 1$ (the only element of the singleton set). But not all $x$ ‘s are acceptable, because we have a pullback condition, which says that $\chi x = \text{True}$ , where $\text{True}$ is the element of $\Omega$ pointed to by $true$ . This tells us that $S$ is isomorphic to the subset of $P$ for which $\chi$ is $\text{True}$ .

The question is: What happens to the other elements of $P$ ? They cannot be mapped to $\text{True}$ , so $\Omega$ must contain at least one more element (in case $m$ is not an isomorphism). Can it contain more?

This is where the universal construction comes into play. Any monic $m$ (here, an injective function) must uniquely determine a $\chi$ that completes the pullback. In particular, we can pick $S$ to be a singleton set and $P$ to be a two-element set. We see that if $\Omega$ contained only $\text{True}$ and nothing else, no $\chi$ would complete the pullback. And if $\Omega$ contained more than two elements, there would be not one but at least two such $\chi$ ‘s. So, by the Goldilock principle, $\Omega$ must have exactly two elements.

We’ll see later that this is not necessarily true in a more general category.

Next: Subfunctor Classifier.

January 4, 2025

Legalizing Comonad Composition

Posted by Bartosz Milewski under Category Theory, Haskell, Programming | Tags: Category Theory, Comonad, Game of Life, Haskell |
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The yearly Advent of Code is always a source of interesting coding challenges. You can often solve them the easy way, or spend days trying to solve them “the right way.” I personally prefer the latter. This year I decided to do some yak shaving with a puzzle that involved looking for patterns in a grid. The pattern was the string XMAS, and it could start at any location and go in any direction whatsoever.

My immediate impulse was to elevate the grid to a comonad. The idea is that a comonad describes a data structure in which every location is a center of some neighborhood, and it lets you apply an algorithm to all neighborhoods in one fell swoop. Common examples of comonads are infinite streams and infinite grids.

Why would anyone use an infinite grid to solve a problem on a finite grid? Imagine you’re walking through a neighborhood. At every step you may hit the boundary of a grid. So a function that retrieves the current state is allowed to fail. You may implement it as returning a Maybe value. So why not pre-fill the infinite grid with Maybe values, padding it with Nothing outside of bounds. This might sound crazy, but in a lazy language it makes perfect sense to trade code for data.

I won’t bore you with the details, they are available at my GitHub repository. Instead, I will discuss a similar program, one that I worked out some time ago, but wasn’t satisfied with the solution: the famous Conway’s Game of Life. This one actually uses an infinite grid, and I did implement it previously using a comonad. But this time I was more ambitious: I wanted to generate this two-dimensional comonad by composing a pair of one-dimensional ones.

The idea is simple. Each row of the grid is an infinite bidirectional stream. Since it has a specific “current position,” we’ll call it a cursor. Such a cursor can be easily made into a comonad. You can extract the current value; and you can duplicate a cursor by creating a cursor of cursors, each shifted by the appropriate offset (increasing in one direction, decreasing in the other).

A two-dimensional grid can then be implemented as a cursor of cursors–the inner one extending horizontally, and the outer one vertically.

It should be a piece of cake to define a comonad instance for it: extract should be a composition of (extract . extract) and duplicate a composition of (duplicate . fmap duplicate), right? It typechecks, so it must be right. But, just in case, like every good Haskell programmer, I decided to check the comonad laws. There are three of them:

extract . duplicate = id
fmap extract . duplicate = id
duplicate . duplicate = fmap duplicate . duplicate

And they failed! I must have done something illegal, but what?

In cases like this, it’s best to turn to basics–which means category theory. Compared to Haskell, category theory is much less verbose. A comonad is a functor $W$ equipped with two natural transformations:

$\varepsilon \colon W \to \text{Id}$

$\delta \colon W \to W \circ W$

In Haskell, we write the components of these transformations as:

extract :: w a -> a
duplicate :: w a -> w (w a)

The comonad laws are illustrated by the following commuting diagrams. Here are the two counit laws:

and one associativity law:

These are the same laws we’ve seen above, but the categorical notation makes them look more symmetric.

So the problem is: Given a comonad $W$ , is the composition $W \circ W$ also a comonad? Can we implement the two natural transformations for it?

$\varepsilon_c \colon W \circ W \to \text{Id}$

$\delta_c \colon W \circ W \to W \circ W \circ W \circ W$

The straightforward implementation would be:

$W \circ W \xrightarrow{\varepsilon \circ W} W \xrightarrow{\varepsilon} \text{Id}$

corresponding to (extract . extract), and:

$W \circ W \xrightarrow{W \circ \delta} W \circ W \circ W \xrightarrow{\delta \circ W \circ W} W \circ W \circ W \circ W$

corresponding to (duplicate . fmap duplicate).

To see why this doesn’t work, let’s ask a more general question: When is a composition of two comonads, say $W_2 \circ W_1$ , again a comonad? We can easily define a counit:

$W_2 \circ W_1 \xrightarrow{\varepsilon_2 \circ W_1} W \xrightarrow{\varepsilon_1} \text{Id}$

The comultiplication, though, is tricky:

$W_2 \circ W_1 \xrightarrow{W_2 \circ \delta_1} W_2 \circ W_1 \circ W_1 \xrightarrow{\delta_2 \circ W} W_2 \circ W_2 \circ W_1 \circ W_1$

Do you see the problem? The result is $W_2^2 \circ W_1^2$ but it should be $(W_2 \circ W_1)^2$ . To make it a comonad, we have to be able to push $W_2$ through $W_1$ in the middle. We need $W_2$ to distribute over $W_1$ through a natural transformation:

$\lambda \colon W_2 \circ W_1 \to W_1 \circ W_2$

But isn’t that only relevant when we compose two different comonads–surely any functor distributes over itself! And there’s the rub: Not every comonad distributes over itself. Because a distributive comonad must preserve the comonad laws. In particular, to restore the the counit law we need this diagram to commute:

and for the comultiplication law, we require:

Even if the two comonad are the same, the counit condition is still non-trivial:

The two whiskerings of $\varepsilon$ are in general not equal. All we can get from the original comonad laws is that they are only equal when applied to the result of comultiplication:

$(\varepsilon \circ W) \cdot \delta = (W \circ \varepsilon) \cdot \delta$ .

Equipped with the distributive mapping $\lambda$ we can complete our definition of comultiplication for a composition of two comonads:

$W_2 \circ W_1 \xrightarrow{W_2 \circ \delta_1} W_2 \circ W_1^2 \xrightarrow{\delta_2 \circ W} W_2^2 \circ W_1^2 \xrightarrow{W_2 \circ \lambda \circ W_1} (W_2 \circ W_1)^2$

Going back to our Haskell code, we need to impose the distributivity condition on our comonad. There is a type class for it defined in Data.Distributive:

class Functor w => Distributive w where
  distribute :: Functor f => f (w a) -> w (f a)

Thus the general formula for composing two comonads is:

instance (Comonad w2, Comonad w1, Distributive w1) => 
  Comonad (Compose w2 w1) where
    extract = extract . extract . getCompose
    duplicate = fmap Compose . Compose .
                fmap distribute . duplicate . fmap duplicate .
                getCompose

In particular, it works for composing a comonad with itself, as long as the comonad distributes over itself.

Equipped with these new tools, let’s go back to implementing a two-dimensional infinite grid. We start with an infinite stream:

data Stream a = (:>) { headS :: a
                     , tailS :: Stream a}
  deriving Functor

infixr 5 :>

What does it mean for a stream to be distributive? It means that we can transpose a “matrix” whose rows are streams. The functor f is used to organize these rows. It could, for instance, be a list functor, in which case you’d have a list of (infinite) streams.

  [   1 :>   2 :>   3 .. 
  ,  10 :>  20 :>  30 ..
  , 100 :> 200 :> 300 .. 
  ]

Transposing a list of streams means creating a stream of lists. The first row is a list of heads of all the streams, the second row is a list of second elements of all the streams, and so on.

  [1, 10, 100] :>
  [2, 20, 200] :>
  [3, 30, 300] :>
  ..

Because streams are infinite, we end up with an infinite stream of lists. For a general functor, we use a recursive formula:

instance Distributive Stream where
    distribute :: Functor f => f (Stream a) -> Stream (f a)
    distribute stms = (headS  stms) :> distribute (tailS  stms)

(Notice that, if we wanted to transpose a list of lists, this procedure would fail. Interestingly, the list monad is not distributive. We really need either fixed size or infinity in the picture.)

We can build a cursor from two streams, one going backward to infinity, and one going forward to infinity. The head of the forward stream will serve as our “current position.”

data Cursor a = Cur { bwStm :: Stream a
                    , fwStm :: Stream a }
  deriving Functor

Because streams are distributive, so are cursors. We just flip them about the diagonal:

instance Distributive Cursor where
    distribute :: Functor f => f (Cursor a) -> Cursor (f a)
    distribute fCur = Cur (distribute (bwStm  fCur)) 
                          (distribute (fwStm  fCur))

A cursor is also a comonad:

instance Comonad Cursor where
  extract (Cur _ (a :> _)) = a
  duplicate bi = Cur (iterateS moveBwd (moveBwd bi)) 
                     (iterateS moveFwd bi)

duplicate creates a cursor of cursors that are progressively shifted backward and forward. The forward shift is implemented as:

moveFwd :: Cursor a -> Cursor a
moveFwd (Cur bw (a :> as)) = Cur (a :> bw) as

and similarly for the backward shift.

Finally, the grid is defined as a cursor of cursors:

type Grid a = Compose Cursor Cursor a

And because Cursor is a distributive comonad, Grid is automatically a lawful comonad. We can now use the comonadic extend to advance the state of the whole grid:

generations :: Grid Cell -> [Grid Cell]
generations = iterate $ extend nextGen

using a local function:

nextGen :: Grid Cell -> Cell
nextGen grid
  | cnt == 3 = Full
  | cnt == 2 = extract grid
  | otherwise = Empty
  where
      cnt = countNeighbors grid

You can find the full implementation of the Game of Life and the solution of the Advent of Code puzzle, both using comonad composition, on my GitHub.

November 16, 2024

Sieves and Sheaves

Posted by Bartosz Milewski under Category Theory, Topology | Tags: Category Theory, math, mathematics, Topology |
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Previously: Covering Sieves.

We’ve seen an intuitive description of presheaves as virtual objects. We can use the same trick to visualize natural transformations.

A natural transformation can be drawn as a virtual arrow $\alpha$ between two virtual objects corresponding to two presheaves $S$ and $P$ . Indeed, for every $s_a \in S a$ , seen as an arrow $a \to S$ , we get an arrow $a \to P$ simply by composition $\alpha \circ s_a$ . Notice that we are thus defining the composition with $\alpha$ , because we are outside of the original category. A component $\alpha_a$ of a natural transformation is a mapping between two arrows.

Untitled Artwork

This composition must be associative and, indeed, associativity is guaranteed by the naturality condition. For any arrow $f \colon a \to b$ , consider a zigzag path from $a$ to $P$ given by $\alpha \circ s_b \circ f$ . The two ways of associating this composition give us $\alpha_a \circ S f = P f \circ \alpha_b$ .

Untitled Artwork

Let’s now recap our previous definitions: A cover of $u$ is a bunch of arrows converging on $u$ satisfying certain conditions. These conditions are defined in terms of a coverage. For every object $u$ we define a whole family of covers, and then combine them into one big collection that we call the coverage.

A sheaf is a presheaf that is compatible with a coverage. It means that for every cover $\{u_i\}$ , if we pick a compatible family of $x_i \in P u_i$ that agrees on all overlaps, then this uniquely determines the element (virtual arrow) $x \in P u$ .

Untitled Artwork

A covering sieve of $u$ is a presheaf that extends a cover $\{u_i\}$ . It assigns a singleton set to each $u_i$ and all its open subsets (that is objects that have arrows pointing to $u_i$ ); and an empty set otherwise. In particular, the sieve includes all the overlaps, like $u_i \cap u_j$ , even if they are not present in the original cover.

The key observation here is that a sieve can serve as a blueprint for, or a skeleton of, a compatible family $\{ x_i \}$ . Indeed, $S_u$ maps all objects either to singletons or to empty sets. In terms of virtual arrows, there is at most one arrow going to $S_u$ from any object. This is why a natural transformation from $S_u$ to any presheaf $P$ produces a family of arrows $x_i \in P u_i$ . It picks a single arrow from each of the hom-sets $u_i \to P$ .

Untitled Artwork

The sieve includes all intersections, and all diagrams involving those intersections necessarily commute. They commute because the category we’re working with is thin, and so is the category extended by adding the virtual object $S_u$ . Thus a family generated by a natural transformation $\alpha \in Nat (S_u, P)$ is automatically a compatible family. Therefore, if $P$ is a sheaf, it determines a unique element $x \in P u$ .

This lets us define a sheaf in terms of sieves, rather than coverages.

A presheaf $P$ is a sheaf if and only if, for every covering sieve $S_u$ of every $u$ , there is a one-to-one correspondence between the set of natural transformations $Nat (S_u, P)$ and the set $P u$ .

In terms of virtual arrows, this means that there is a one-to-one correspondence between arrows $\alpha \colon S_u \to P$ and $x \colon u \to P$ .

Untitled Artwork
Next: Subobject Classifier

October 31, 2024

Covering Sieves

Posted by Bartosz Milewski under Category Theory, Topology | Tags: Category Theory, math, mathematics, Topology |
[2] Comments

Previously: Sheaves as Virtual Objects.

In order to define a sheaf, we have to start with coverage. A coverage defines, for every object $u$ , a family of covers that satisfy the sub-coverage conditions. Granted, we can express coverage using objects and arrows, but it would be much nicer if we could use the language of functors and natural transformations.

Let’s start with the idea that, categorically, a cover of $u$ is a bunch of arrows converging on $u$ . Each arrow $p_i \colon u_i \to u$ is a member of the hom-set $\mathcal C (u_i, u)$ . Now consider the fact that $\mathcal C (-, u)$ is a presheaf, $\mathcal C^{op} \to \mathbf{Set}$ , and ask the question: Is a cover a “subfunctor” of $\mathcal C (-, u)$ ?

A subfunctor of a presheaf $P$ is defined as a functor $S$ such that, for each object $v$ , $S v$ is a subset of $P v$ and, for each arrow $f \colon v \to w$ , the function $S f \colon S w \to S v$ is a restriction of $P f$ .

Untitled Artwork

In general, a cover does not correspond to a subfunctor of the hom-functor. Let’s see why, and how we can fix it.

Let’s try to define $S$ , such that $S u_i$ is non-empty for any object $u_i$ that’s in the cover of $u$ , and empty otherwise. As a presheaf, we could represent it as a virtual object with arrows coming from all $\{ u_i \}$ ‘s.

Untitled Artwork

Now consider an object $v$ that is not in the cover, but it has an arrow $f \colon v \to u_k$ connecting it to some element $u_k$ of the cover. Functoriality requires the (virtual) composition $s_k \circ f$ to exist. Untitled Artwork

Thus $v$ must be included in the cover–if we want $S$ to be a functor.

In particular, if we are looking at a category of open sets with inclusions, this condition means that all (open) sub-sets of the covering sets must also be included in the cover. Such a “downward closed” family of sets is called a sieve.

Imagine sets in the cover as holes in a sieve. Smaller sets that can “pass through” these holes must also be parts of the sieve.

If you start with a cover, you can always extend it to a covering sieve by adding more arrows. It’s as if you started with a few black holes, and everything that could fall into them, would fall.

We have previously defined sheaves in terms of coverings. In the next installment we’ll see that they can equally well be defined using covering sieves.

Next Sieves and Sheaves.

October 24, 2024

Sheaves as Virtual Objects

Posted by Bartosz Milewski under Category Theory, Topology | Tags: Category Theory, math, mathematics, Topology |
[3] Comments

Previously: Coverages and Sites

The definition of a sheaf is rather complex and involves several layers of abstraction. To help us navigate this maze we can use some useful intuitions. One such intuition is to view objects in our category as some kind of sets (in particular, open sets, when we talk about topology), and arrows as set inclusions. An arrow from $v$ to $u$ means that $v$ is a subset of $u$ .

A cover of $u$ is a family of arrows $\{ p_i \colon u_i \to u \}$ . A coverage assigns a collection of covers to every object, satisfying the sub-coverage conditions described in the previous post. A category with coverage is called a site.

The next layer of abstraction deals with presheaves, which are set-valued contravariant functors. Interestingly, there is a way to interpret a presheaf as an extension of the original category. I learned this trick from Paolo Perrone.

We may represent a presheaf $P$ using virtual hom-sets. First we add one virtual object, let’s call it $\bullet$ , to our category. The set $P u$ is then interpreted as the set of arrows from $u$ to $\bullet$ .

Untitled Artwork

Moreover, we can represent the action of $P$ on arrows as simple composition. Take an arrow $f \colon v \to u$ . The presheaf lifts it to a function between sets: $P f \colon P u \to P v$ (contravariance means that the arrow is reversed). For any $h \in P u$ we can define the composition $h \circ f$ to be $(P f) h$ .

Untitled Artwork

Incidentally, if the functor $P$ is representable, it means that we can replace the virtual object $\bullet$ with an actual object in our category.

Notice that, even though the category of open sets with inclusions is a poset (hom-sets are either singletons or empty, and all diagrams automatically commute), the added virtual hom-sets usually contain lots of arrows. In topology these hom-sets are supposed to represent sets of continuous functions over open sets.

We can interpret the virtual object $\bullet$ as representing an imaginary open set that “includes” all the objects $u$ for which $P u$ is non-empty, but we have to imagine that it’s possible to include an object in more than one way, to account for multiple arrows. In fact, in what follows we won’t be assuming that the underlying category is a poset, so virtual hom-sets are nothing special.

To express the idea of intersections of open sets, we use commuting diagrams. For every pair of objects $u_i$ and $u_j$ that are in the cover of $u$ , an object $v$ is in their intersection if the following diagram commutes:

Untitled Artwork

Note that in a poset all diagrams commute, but here we’re generalizing this condition to an arbitrary category. We could say that $v$ is in the intersection of $u_i$ and $u_j$ seen as covers of $u$ .

Equipped with this new insight, we can now express the sheaf condition. We assume that there is a coverage defined in our category. We are adding one more virtual object $\bullet$ for the presheaf $P$ , with bunches of virtual arrows pointing to it.

For every cover $\{ p_i \colon u_i \to u \}$ we try to select a family of virtual arrows, $s_i \colon u_i \to \bullet$ . It’s as if the objects $u_i$ , besides covering $u$ , also covered the virtual object $\bullet$ .

We call the family $\{s_i \}$ a matching family, if this new covering respects the existing intersections. If $v$ is in the intersection of $u_i$ and $u_j$ (as covers of $u$ , see the previous diagram), then we want the following diagram to also commute:
Untitled Artwork
In other words, the $\{u_i\}$ ‘s intersect as covers of $\bullet$ .

A presheaf $P$ is a sheaf if, for every covering family $p_i$ and every matching family $s_i$ there exists a unique $s \colon u \to \bullet$ that factorizes those $s_i$ ‘s:
Untitled Artwork
Translating it back to the language of topology: There is a unique global function $s$ defined over $u$ whose restrictions are $s_i$ ‘s.

The advantage of this approach is that it’s easy to imagine the sheafification of an arbitrary presheaf by freely adding virtual arrows (the $s$ ‘s and their compositions with $p_i$ ‘s in the above diagram) to all intersection diagrams.

Next: Covering Sieves

August 18, 2024

Sheaves and Topology

Posted by Bartosz Milewski under Category Theory, Topology | Tags: Category Theory, Topology |
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Previously: Presheaves and Topology.

In all branches of science we sooner or later encounter the global vs. local duality. Topology is no different.

In topology we have the global definition of continuity: counter-images of all open sets are open. But we perceive a discontinuity as a local jump. How are the two pictures related, and can we express this topologically, that is without talking about sizes and distances?

All we have at our disposal are open sets, so exactly what properties of open sets are the most relevant? They do form a (thin) category with inclusions as arrows, but so does any set of subsets. As it turns out open sets can be stitched together to create coverings. Such coverings let us zoom in on finer and finer details, thus creating the bridge between the global and the local picture.

Open sets are plump–they can easily fill the bulk of space. They are also skinless, so they can’t touch each other without some overlap. That makes them perfect for constructing covers.

Covering, unlike tiling, requires overlapping. To create a leak-free roof, you need your tiles to overlap. The idea is that, if we were defining functions over a tiling, it would be possible for them to make sudden jumps at tile boundaries. Open coverings overlap, so such functions have to flow continuously.

An open cover of a set $u$ is a family of open sets $\{u_i\}$ such that $u$ is their union:

$u = \bigcup_{i \in I} u_i$

Here $I$ is a set used for indexing the family.

If we have a continuous function $f$ defined over $u$ , then all its restrictions $f|_{u_i}$ are also continuous (this follows from the condition that an intersection of open sets is open). Thus going from global to local is easy.

The converse is more interesting. Suppose that we have a family of functions $f_i$ , one per each open set $u_i$ , and we want to reconstruct the global function $f$ defined over the set $u$ covered by $u_i$ ‘s. This is only possible if the individual functions agree on overlaps.

Take two functions: $f_i$ defined over $u_i$ , and $f_j$ defined over $u_j$ . If the two sets overlap, each of the functions can be restricted to the overlap $u_i \cap u_j$ . We want these restrictions to be equal:

$f_i|_{u_i \cap u_j} = f_j|_{u_i \cap u_j}$

If all individual continuous functions agree on the overlaps then they uniquely determine a global continuous function $f$ defined over the whole set $u$ . You can stitch or collate functions that are defined locally.

In the language of category theory we talk about functions in bulk. We define a functor–a presheaf $P$ –that maps all open sets to sets of continuous functions. In this language, to an open cover $\{u_i\}$ corresponds a family of functions $\{f_i\}$ that are members of the individual sets $P u_i$ . Every such selection forms a giant $I$ -indexed tuple, that is an element of the cartesian product:

$\{f_i | i \in I\} \in \prod_{i} P u_i$

Similarly, we can gather functions that are defined over the intersections of sets into a product:

$\prod_{i j} P (u_i \cap u_j)$

(Notice that every empty intersection corresponds to a single trivial function that we call absurd in Haskell.)

Set inclusions generate function restrictions. In particular, for every intersection $u_i \cap u_j$ we have a pair of restrictions:

$f_i \mapsto f_i|_{u_i \cap u_j}$

$f_j \mapsto f_j|_{u_i \cap u_j}$

These restrictions can be seen as functions between sets:

$P u_i \to P (u_i \cap u_j)$

$P u_j \to P (u_i \cap u_j)$

If all such restrictions are pairwise equal, we call $\{f_i\}$ a matching family, and for every such matching family there is a unique element $f \in P u$ such that $f_i = f|_{u_i}$ , for all $i$ .

These pairs of restrictions define two mappings between our big products:

$p, q : \prod_i P u_i \rightrightarrows \prod_{i j} P (u_i \cap u_j)$

Think of each function as acting on a tuple $\{f_k\}$ and producing a matrix indexed by elements of $I$ :

$(p\; \{f_k\})_{i j} = f_i|_{u_i \cap u_j}$

$(q\; \{f_k\})_{i j} = f_j|_{u_i \cap u_j}$

Our matching condition can be expressed in the language of category theory by saying that the following diagram is an equalizer of $p$ and $q$ (the two parallel arrows):

$P u \xrightarrow{e} \prod_i P u_i \rightrightarrows \prod_{i j} P (u_i \cap u_j)$

Here $e$ is defined as mapping a function $f \in P u$ to a tuple of its restrictions $\{ f|{u_i}\}$ . These restrictions are then required to match when further restricted by $p$ and $q$ to all possible intersections.

A presheaf $P$ is called a sheaf if, for every open covering $\{u_i\}$ , a matching family $\{f_i\}$ uniquely determines the element of $P u$ of the equalizer above. This element corresponds to the function $f$ that is the result of stitching of individual functions.

Notice that, even though we tried to use the categorical language as much as possible, we still had to rely on the language of sets to define coverings. To abstract away from set theory and traditional topology, we need to talk about sites.

Next: Coverages and Sites .

August 7, 2024

Presheaves and Topology

Posted by Bartosz Milewski under Category Theory, Topology | Tags: Category Theory, Topology |
[7] Comments

Previously: Topology as a Dietary Choice.

Category theory lets us change the focus from individual objects to relationships between them. Since topology is defined using open sets, we’d start by concentrating on relations between sets.

One such obvious relation is inclusion. It imposes a categorical structure on the subsets of a given set $X$ . We draw arrows between two sets whenever one is a subset of the other. These arrows satisfy the axioms of a category: there is an identity arrow for every object (every set is its own subset) and arrows compose (inclusion is transitive). Not every pair of objects is connected by an arrow–some sets are disjoint, others overlap only partially. We may include the whole space as the terminal object (with arrows coming from every subset) and the empty set $\emptyset$ as the initial object (with arrows going to every set). As categories go, this is a thin category, because there is at most one arrow between any two objects.

Every topological space gives thus rise to a thin category that abstracts the structure of its open sets. But the real reason for defining a topology is to be able to talk about continuous functions. These are functions between topological spaces such that the inverse image of every open set is open. Here, again, category theory tells us not to think about the details of how these functions are defined, but rather what we can do with them. And not just one function at a time, but the whole bunch at once.

So let’s talk about sets of functions. We have our topological space $X$ , and to each open subset $u$ we will assign a set of continuous function on it. These could be functions to real or complex numbers, or whatever–we don’t care. All we care about is that they form a set.

Since open sets in $X$ form a (thin) category, we are talking about assigning to each object (open set) $u$ its own set (of continuous functions) $P u$ . Notice however that these sets of functions are not independent of each other. If one open set is a subset of another, it inherits all the functions defined over the larger set. These are the same functions, the only difference being that their arguments are restricted to a smaller subset. For instance, given two sets $v \subseteq u$ and a function $f \colon u \to \mathbb R$ , there is a function $f|_{v} \colon v \to \mathbb R$ such that $f|_{v} = f$ on $v$ .

Let’s restate these statements in the categorical language. We already have a category $X$ of open sets with inclusion. The sets of functions on these open sets are objects in the category $\mathbf{Set}$ . We have defined a mapping $P$ between these two categories that assigns sets of functions to open sets.

Notice that we are dealing with two different categories whose objects are sets. One has inclusions as arrows, the other has functions as arrows. (To confuse matters even more, the objects in the second category represent sets of functions.)

To define a functor between categories, we also need a mapping of arrows to accompany the mapping of objects. An arrow $v \to u$ means that $v \subseteq u$ . Corresponding to it, we have a function $P u \to P v$ that assigns to each $f \in P u$ its restriction $f|_{v} \in P v$ .

Together, these mappings define a functor $P \colon X^{op} \to \mathbf{Set}$ . The “op” notation means that the directions of arrows are reversed: the functor is “contravariant.”

A functor must preserve the structure of a category, that is identity and composition. In our case this follows from the fact that an identity $u \subseteq u$ maps to a trivial do-nothing restriction, and that restrictions compose: $(f|_v)|_w = f|_w$ for $w \subseteq v \subseteq u$ .

There is a special name for contravariant functors from any category $\mathcal C$ to $\mathbf{Set}$ . They are called presheaves, exactly because they were first introduced in the context of topology as precursors of “shaves.” Consequently, the simpler functors $\mathcal C \to \mathbf{Set}$ had to be confusingly called co-presheaves.

Presheaves on $\mathcal C$ form their own category, often denoted by $\hat{\mathcal C}$ , with natural transformations as arrows.

Next: Sheaves and Topology.

March 24, 2024

Neural Networks, Pre-lenses, and Triple Tambara Modules, Part II

Posted by Bartosz Milewski under Category Theory, Lens, Neural Networks, Programming | Tags: AI, Category Theory, Lens, Neural Networks, Optics, Profunctors, Tambara Modules |
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I will now provide the categorical foundation of the Haskell implementation from the previous post. A PDF version that contains both parts is also available.

The Para Construction

There’s been a lot of interest in categorical foundations of deep learning. The basic idea is that of a parametric category, in which morphisms are parameterized by objects from a monoidal category $\mathcal P$ :

Screenshot 2024-03-24 at 15.00.20
Here, $p$ is an object of $\mathcal P$ .

When two such morphisms are composed, the result is parameterized by the tensor product of the parameters.

Screenshot 2024-03-24 at 15.00.34

An identity morphism is parameterized by the monoidal unit $I$ .

If the monoidal category $\mathcal P$ is not strict, the parametric composition and identity laws are not strict either. They are satisfied up to associators and unitors of $\mathcal P$ . A category with lax composition and identity laws is called a bicategory. The 2-cells in a parametric bicategory are called reparameterizations.

Of particular interest are parameterized bicategories that are built on top of actegories. An actegory $\mathcal C$ is a category in which we define an action of a monoidal category $\mathcal P$ :

$\bullet \colon \mathcal P \times \mathcal C \to C$

satisfying some obvious coherency conditions (unit and composition):

$I \bullet c \cong c$

$p \bullet (q \bullet c) \cong (p \otimes q) \bullet c$

There are two basic constructions of a parametric category on top of an actegory called $\mathbf{Para}$ and $\mathbf{coPara}$ . The first constructs parametric morphisms from $a$ to $b$ as $f_p = p \bullet a \to b$ , and the second as $g_p = a \to p \bullet b$ .

Parametric Optics

The $\mathbf{Para}$ construction can be extended to optics, where we’re dealing with pairs of objects from the underlying category (or categories, in the case of mixed optics). The parameterized optic is defined as the following coend:

$O \langle a, da \rangle \langle p, dp \rangle \langle s, ds \rangle = \int^{m} \mathcal C (p \bullet s, m \bullet a) \times \mathcal C (m \bullet da, dp \bullet ds)$

where the residues $m$ are objects of some monoidal category $\mathcal M$ , and the parameters $\langle p, dp \rangle$ come from another monoidal category $\mathcal P$ .

In Haskell, this is exactly the existential lens:

data ExLens a da p dp s ds = 
  forall m . ExLens ((p, s)  -> (m, a))  
                    ((m, da) -> (dp, ds))

There is, however, a more general bicategory of pre-optics, which underlies existential optics. In it, both the parameters and the residues are treated symmetrically.

The PreLens Bicategory

Pre-optics break the feedback loop in which the residues from the forward pass are fed back to the backward pass. We get the following formula:

$\begin{aligned}O & \langle a, da \rangle \langle m, dm \rangle \langle p, dp \rangle \langle s, ds \rangle = \\ &\mathcal C (p \bullet s, m \bullet a) \times \mathcal C (dm \bullet da, dp \bullet ds) \end{aligned}$

We interpret this as a hom-set from a pair of objects $\langle s, ds \rangle$ in $\mathcal C^{op} \times C$ to the pair of objects $\langle a, da \rangle$ also in $\mathcal C^{op} \times C$ , parameterized by a pair $\langle m, dm \rangle$ in $\mathcal M \times \mathcal M^{op}$ and a pair $\langle p, dp \rangle$ from $\mathcal P^{op} \times \mathcal P$ .

To simplify notation, I’ll use the bold $\mathbf C$ for the category $\mathcal C^{op} \times \mathcal C$ , and bold letters for pairs of objects and (twisted) pairs of morphisms. For instance, $\bold f \colon \bold a \to \bold b$ is a member of the hom-set $\mathbf C (\bold a, \bold b)$ represented by a pair $\langle f \colon a' \to a, g \colon b \to b' \rangle$ .

Similarly, I’ll use the notation $\bold m \bullet \bold a$ to denote the monoidal action of $\mathcal M^{op} \times \mathcal M$ on $\mathcal C^{op} \times \mathcal C$ :

$\langle m, dm \rangle \bullet \langle a, da \rangle = \langle m \bullet a, dm \bullet da \rangle$

and the analogous action of $\mathcal P^{op} \times \mathcal P$ .

In this notation, the pre-optic can be simply written as:

$O\; \bold a\, \bold m\, \bold p\, \bold s = \bold C (\bold m \bullet \bold a, \bold p \bullet \bold b)$

and an individual morphism as a triple:

$(\bold m, \bold p, \bold f \colon \bold m \bullet \bold a \to \bold p \bullet \bold b)$

Pre-optics form hom-sets in the $\mathbf{PreLens}$ bicategory. The composition is a mapping:

$\mathbf C (\bold m \bullet \bold b, \bold p \bullet \bold c) \times \mathbf C (\bold n \bullet \bold a, \bold q \bullet \bold b) \to \mathbf C (\bold (\bold m \otimes \bold n) \bullet \bold a, (\bold q \otimes \bold p) \bullet \bold c)$

Indeed, since both monoidal actions are functorial, we can lift the first morphism by $(\bold q \bullet -)$ and the second by $(\bold m \bullet -)$ :

$\mathbf C (\bold m \bullet \bold b, \bold p \bullet \bold c) \times \mathbf C (\bold n \bullet \bold a, \bold q \bullet \bold b) \xrightarrow{(\bold q \bullet) \times (\bold m \bullet)}$

$\mathbf C (\bold q \bullet \bold m \bullet \bold b, \bold q \bullet \bold p \bullet \bold c) \times \mathbf C (\bold m \bullet \bold n \bullet \bold a,\bold m \bullet \bold q \bullet \bold b)$

We can compose these hom-sets in $\mathbf C$ , as long as the two monoidal actions commute, that is, if we have:

$\bold q \bullet \bold m \bullet \bold b \to \bold m \bullet \bold q \bullet \bold b$

for all $\bold q$ , $\bold m$ , and $\bold b$ .
The identity morphism is a triple:

$(\bold 1, \bold 1, \bold{id} )$

parameterized by the unit objects in the monoidal categories $\mathbf M$ and $\mathbf P$ . Associativity and identity laws are satisfied modulo the associators and the unitors.

If the underlying category $\mathcal C$ is monoidal, the $\mathbf{PreOp}$ bicategory is also monoidal, with the obvious point-wise parallel composition of pre-optics.

Triple Tambara Modules

A triple Tambara module is a functor:

$T \colon \mathbf M^{op} \times \mathbf P \times \mathbf C \to \mathbf{Set}$

equipped with two families of natural transformations:

$\alpha \colon T \, \bold m \, \bold p \, \bold a \to T \, (\bold n \otimes \bold m) \, \bold p \, (\bold n \bullet a)$

$\beta \colon T \, \bold m \, \bold p \, (\bold r \bullet \bold a) \to T \, \bold m \, (\bold p \otimes \bold r) \, \bold a$

and some coherence conditions. For instance, the two paths from $T \, \bold m \, \bold p\, (\bold r \bullet \bold a)$ to $T \, (\bold n \otimes \bold m)\, (\bold p \otimes \bold r) \, (\bold n \bullet \bold a)$ must give the same result.

One can also define natural transformations between such functors that preserve the two structures, and define a bicategory of triple Tambara modules $\mathbf{TriTamb}$ .

As a special case, if we chose the category $\mathcal P$ to be the trivial one-object monoidal category, we get a version of (double-) Tambara modules. If we then take the coend, $P \langle a, b \rangle = \int^m T \langle m, m\rangle \langle a, b \rangle$ , we get regular Tambara modules.

Pre-optics themselves are an example of a triple Tambara representation. Indeed, for any fixed $\bold a$ , we can define a mapping $\alpha$ from the triple:

$(\bold m, \bold p, \bold f \colon \bold m \bullet \bold a \to \bold p \bullet \bold b)$

to the triple:

$(\bold n \otimes \bold m, \bold p, \bold f' \colon (\bold n \otimes \bold m) \bullet \bold a \to \bold p \bullet (\bold n \bullet \bold b))$

by lifting of $\bold f$ by $(\bold n \bullet -)$ and rearranging the actions using their commutativity.
Similarly for $\beta$ , we map:

$(\bold m, \bold p, \bold f \colon \bold m \bullet \bold a \to \bold p \bullet (\bold r \bullet \bold b))$

to:

$(\bold m , (\bold p \otimes \bold r), \bold f' \colon \bold m \bullet \bold a \to (\bold p \otimes \bold r) \bullet \bold b)$

Tambara Representation

The main result is that morphisms in $\mathbf {PreOp}$ can be expressed using triple Tambara modules. An optic:

$(\bold m, \bold p, \bold f \colon \bold m \bullet \bold a \to \bold p \bullet \bold b)$

is equivalent to a triple end:

$\int_{\bold r \colon \mathbf P} \int_{\bold n \colon \mathbf M} \int_{T \colon \mathbf{TriTamb}} \mathbf{Set} \big(T \, \bold n \, \bold r \, \bold a, T \, (\bold m \otimes \bold n) \, (\bold r \otimes \bold p) \, \bold b \big)$

Indeed, since pre-optics are themselves triple Tambara modules, we can apply the polymorphic mapping of Tambara modules to the identity optic $(\bold 1, \bold 1, \bold{id} )$ and get an arbitrary pre-optic.

Conversely, given an optic:

$(\bold m, \bold p, \bold f \colon \bold m \bullet \bold a \to \bold p \bullet \bold b)$

we can construct the polymorphic mapping of triple Tambara modules:

$\begin{aligned} & T \, \bold n \, \bold r \, \bold a \xrightarrow{\alpha} T \, (\bold m \otimes \bold n) \, \bold r \, (\bold m \bullet \bold a) \xrightarrow{T \, \bold f} T \, (\bold m \otimes \bold n) \, \bold r \, (\bold p \bullet \bold b) \xrightarrow{\beta} \\ & T \, (\bold m \otimes \bold n) \, (\bold r \otimes \bold p) \, \bold b \end{aligned}$

Bibliography

Brendan Fong, Michael Johnson, Lenses and Learners,
Brendan Fong, David Spivak, Rémy Tuyéras, Backprop as Functor: A compositional perspective on supervised learning, 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) 2019, pp. 1-13, 2019.
G.S.H. Cruttwell, Bruno Gavranović, Neil Ghani, Paul Wilson, Fabio Zanasi, Categorical Foundations of Gradient-Based Learning
Bruno Gavranović, Compositional Deep Learning
Bruno Gavranović, Fundamental Components of Deep Learning, PhD Thesis. 2024

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Neural Networks, Pre-lenses, and Triple Tambara Modules, Part II

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