I’ve come to believe, over the past couple of years, that anyone trying to study ω \omega -categories (a.k.a. ( ∞ , ∞ ) (\infty,\infty) -categories) without knowing about coinductive definitions is going to be struggling against nature due to not having the proper tools. But although coinductive definitions are a basic notion in mathematics, for some reason they don’t seem to be taught, even to graduate students. Write something like

A 1-morphism f : x → y f\colon x\to y in an ( n + 1 ) (n+1) -category is an equivalence if there exists a 1-morphism g : y → x g\colon y\to x and equivalences 1 x → g f 1_x \to g f and f g → 1 y f g\to 1_y in the relevant hom- n n -categories; and every 1-morphism in a 0-category is an equivalence

and any mathematician (who has some inkling of what n n -categories are) will be happy. If you ask why this definition isn’t circular, since it defines the notion of “equivalence” in terms of “equivalence”, the mathematician will say “it’s an inductive definition” and expect you to stop complaining. But if you write something like

A 1-morphism f : x → y f\colon x\to y in an ω \omega -category is an equivalence if there exists a 1-morphism g : y → x g\colon y\to x and equivalences 1 x → g f 1_x \to g f and f g → 1 y f g\to 1_y in the relevant hom- ω \omega -categories

the same mathematician will object loudly, saying that this definition is circular. (In fact, not very long ago, that mathematician was me.) But actually, this latter is a perfectly valid coinductive definition.

One way of saying what an inductive definition means is that it defines the smallest class of things that is closed under some “constructor” operations. That is, the above definition of equivalences in n n -categories for finite n n says, when unraveled a bit, that the class of “equivalences in n n -categories” is the smallest class ℰ \mathcal{E} such that

Every morphism in a 0-category is in ℰ \mathcal{E} , and If f : x → y f\colon x\to y has the property that there exists g : y → x g\colon y\to x , and also morphisms 1 x → g f 1_x \to g f and f g → 1 y f g\to 1_y that are in ℰ \mathcal{E} , then f f is in ℰ \mathcal{E} .

As usual, “smallest” means “minimum”, i.e. it is contained in every other such class ℰ \mathcal{E} . How do we know that there is a smallest such class? Because of the nature of the closure conditions (each of them takes some input consisting of things in ℰ \mathcal{E} and produces something else in ℰ \mathcal{E} ), the collection of classes ℰ \mathcal{E} satisfying them is closed under intersection; thus we can take the intersection of all such ℰ \mathcal{E} to obtain the smallest one.

Inductive definitions are best-adapted to proving stuff about things which satisfy the definition. Namely, if we want to prove that all equivalences in n n -categories have property P P , all we need to do is prove that morphisms in 0-categories have property P P , and (2) given f : x → y f\colon x\to y and g : y → x g\colon y\to x , and 1 x → g f 1_x \to g f and f g → 1 y f g\to 1_y with property P P , also f f has property P P . Then the class of things satisfying P P will be one of the classes ℰ \mathcal{E} , and hence contain the class of equivalences. This is called a proof by induction.

Dually, a coinductive definition defines the largest class of things that is closed under some “destructor” operations. Thus, the above definition of equivalences in ω \omega -categories says that the class of “equivalences in ω \omega -categories” is the largest class ℰ \mathcal{E} such that

If f : x → y f\colon x\to y is in ℰ \mathcal{E} , then there exists a g : y → x g\colon y\to x , and also morphisms 1 x → g f 1_x \to g f and f g → 1 y f g\to 1_y that are in ℰ \mathcal{E} .

As usual, “largest” means “maximum”, i.e. containing every other such class ℰ \mathcal{E} . How do we know that there is a largest such class? Because of the nature of the closure conditions (each of them takes one input in ℰ \mathcal{E} and produces some number of other things in ℰ \mathcal{E} ), the collection of classes ℰ \mathcal{E} satisfying them is closed under unions; thus we can take the union of all such ℰ \mathcal{E} to obtain the largest one.

Coinductive definitions are best-adapted to proving that things do satisfy the definition. Namely, if we want to prove that some morphism f f in an ω \omega -category is an equivalence, all we need to do is prove that f f belongs to some class ℰ \mathcal{E} of morphisms with the above property. Then ℰ \mathcal{E} will be contained in the class of equivalences, so that f f is an equivalence. This is called a proof by coinduction.

The theory of ω \omega -categories is full of concepts that are naturally defined coinductively. For instance:

A functor f : C → D f\colon C\to D between ω \omega -categories is an equivalence if (1) for each y ∈ D y\in D , there exists an x ∈ C x\in C and an equivalence f ( x ) → y f(x) \to y , and (2) for each x 1 , x 2 ∈ C x_1,x_2\in C , the functor f : C ( x 1 , x 2 ) → D ( f ( x 1 ) , f ( x 2 ) ) f\colon C(x_1,x_2) \to D(f(x_1),f(x_2)) is an equivalence between ω \omega -categories.

The schematic definition of n-fibration makes perfect sense as a definition of ω \omega -fibration, if interpreted coinductively.

In fact, ω \omega -categories themselves are naturally defined coinductively!

An ω \omega -category is a category enriched over ω \omega -categories.

This requires a more general kind of coinductive definition, though, since now we are defining a structure coinductively, rather than a property of elements of some existing structure.

Here’s a way of rephrasing the inductive definition of equivalences in an n n -category. Consider the poset of “classes of morphisms in n n -categories” for finite n n , and given such a class ℰ \mathcal{E} , let F ( ℰ ) F(\mathcal{E}) be the class of all morphisms which are either (1) morphisms in 0-categories, or (2) are morphisms f : x → y f\colon x\to y such that there exists g : y → x g\colon y\to x , and also morphisms 1 x → g f 1_x \to g f and f g → 1 y f g\to 1_y that are in ℰ \mathcal{E} . Then F F is an endofunctor of this poset, and the inductive definition says that the equivalences are the initial algebra for this endofunctor, i.e. the smallest ℰ \mathcal{E} such that F ( ℰ ) ⊆ ℰ F(\mathcal{E})\subseteq \mathcal{E} .

Similarly, we can consider the poset of “classes of morphisms in ω \omega -categories” and define G ( ℰ ) G(\mathcal{E}) to be the class of all morphisms f : x → y f\colon x\to y such that there exists a g : y → x g\colon y\to x , and also morphisms 1 x → g f 1_x \to g f and f g → 1 y f g\to 1_y that are in ℰ \mathcal{E} . Then the coinductive definition says that ℰ \mathcal{E} is the terminal coalgebra for the endofunctor G G , i.e. the largest ℰ \mathcal{E} such that ℰ ⊆ G ( ℰ ) \mathcal{E}\subseteq G(\mathcal{E}) .

The generalization to structure, rather than properties, is now immediate: in general, an inductively defined gadget is an initial algebra for some endofunctor, and a coinductively defined gadget is a terminal coalgebra for some endofunctor. Of course, we need some conditions on the endofunctor to ensure that initial or terminal coalgebras exist; usually one asks them to be polynomial.

For instance, the natural numbers are the initial algebra for the endofunctor X ↦ X + 1 X\mapsto X+1 of Set Set . This automatically gives us the principle of definition by iteration or recursion.