Higher category theory is a bit of a strange subject. Unlike many other areas of mathematics, which focus on the implications of a set of definitions, higher category theory finds its most daunting task to be the very first definition. What is a higher category?

The idea has taken shape from diverse and apparently disconnected applications, not yet fully forming into a single unified abstraction. A thousand and one attempts to define them have emerged over the decades, all proven to be ‘equivalent’, depending on how relaxed your idea of ‘equivalence’ is. It’s a tough one. What do we want from higher categories? What should we be able to do with them? Where do they come from? Are we thinking about them in the right way? So many questions, so few answers.

While I don’t think this blog post is going to change the subject forever, I thought I’d give my own two cents on the matter for you to consider as you wish. Disclaimer: no guarantee that these ideas are original. I’ve not bothered to check the literature that deeply.

A Span(ner) in the Works

Before anything else, we should ask where higher categories come from. Like all abstractions, they help us understand things we care about, unless they don’t, in which case we don’t care about them either.

I find that many explanations of why higher categories matter amount to throwing a few sporadic examples at you without really highlighting what the correlation’s supposed to be. What’s a higher category? Yeah, here’s a dry formal definition. What does it mean? Here’s a bunch of random examples. How are they all related? Well, they’re all higher categories, duh. Examples are data points, and giving data without a deeper discussion is poor science. I’ll try to do a bit better than that. (Just for you, my darling reader.)

Moreover, like many other areas of math, higher category theory has a habit of saying a construction is ‘well-behaved’ or ‘natural’ without explaining what in the blazes that means. Such descriptions are trying to appeal to some underlying intuition you’re assumed to already have. While that’s fine for experts that are ‘in the know’, it’s not great for newcomers. I find myself doing it too in all honesty, but I’ll try to avoid it for this post. (Again, just for you. Mwah.)

A good starting point is spans. A span $A \leftarrow B \rightarrow C$ is basically an augmented relation between sets $A$ and $C$. Each element of $B$, by the projection maps, relates an element of $A$ to one in $C$. The fact that many elements can map to the same places means elements relate in variable ways. We have for every set $A$ a trivial span $A \leftarrow A \rightarrow A$ defined by identities - this is just the trivial identity relation. We can also reverse a span by… well… flipping the diagram around. Kinda stupid, but hey, formalities. Finally, we can ‘compose’ spans by taking a pullback: a diagram $$ A \leftarrow B \rightarrow C \leftarrow D \rightarrow E $$of two connected spans yields a new span $$ A \leftarrow B \times_C D \rightarrow E $$that represents the ‘transitive’ relation. By that I mean, if $a$ relates to $b$ which relates to $c$, then $a$ relates to $c$. Simple.

We’d like an algebraic structure of these spans, so we can say things about these operations. Relations are important, which makes spans sound important. And maybe other important things are arranged similarly to spans! We’ve chosen a particular interpretation of spans, so maybe we’re missing a connection to another unexpected topic. We’d like to know these things. We are mathematicians, after all.

The most immediate structure that comes to mind is a category. We’ll take sets to be our objects and spans to be our morphisms. Identities are the trivial relation and composition is given by pullbacks. All done!

…or are we? Let’s see what happens when we compose the identity $1_A$ of a set $A$ with a generic span $f : A \leftarrow B \rightarrow C : g$. We get $$ A \leftarrow A \times_A B \rightarrow C $$Hmm… that’s not the same span as we had before. What exactly is $A \times_A B$? If we’re using the standard concrete definition of a set pullback, it’s the set $$ A \times_A B := \{(a, b) : f(b) = a\} $$ Now you might shrug and say “Eh, it’s pretty much the same as $B$. There’s a clear bijection between the two.” You might even call the bijection canonical, that is to say ‘formulaic’. Composing with the identity will always be equal to the identity in an obvious way. You might then choose to replace all morphisms with isomorphism classes of spans, where by ‘isomorphism’ we mean a bijection between the middle sets commuting with the other maps. If you did that, you’d have your category and be good to go.

But is this really the best way to understand spans? If we do this, we’re warping spans to fit the arbitrary axioms of a category, which seems a bit random. We’re not trying to make spans form a category specifically, we’re trying to find out what axioms they fit without changing them. And besides, saying $A \times_B B$ is canonically isomorphic to $B$ is a far cry from them being equal. Sometimes, we really do care what literal elements a set has! Category theory can make us forget that - ‘isomorphic’ and ‘equal’ are distinct kinds of equivalence, and each is important in its own way.

So let’s try again. We can’t expect identities to exist as we expect, nor can we expect associativity to hold. Instead, they hold up to isomorphisms between spans. Again, a morphism between spans $A \leftarrow B \rightarrow C$ and $A \leftarrow B' \rightarrow C$ will be a function $B \rightarrow B'$ commuting with the span maps. These isomorphisms will be canonical, that is, there will only be one isomorphism induced by any amount of associativity or units. This translates to a finite list of axioms that we call a bicategory. In a bicategory, we have objects, morphisms, and 2-morphisms between the morphisms. Composition operations satisfy the laws of a category up to canonical 2-isomorphisms and 2-morphisms satisfy the laws of a category.

But maybe even this isn’t enough. Two reasons come to mind. The first is that not even the definition of a pullback is unique; it’s an adjoint to a ‘diagonal’ functor, which means it’s unique up to canonical natural isomorphism. For instance, we could define $A \times_C B$ to be $$ \{(a, b) : a \in A, b \in B, f(a) = g(b)\} $$ or to be $$ \{[a, b] : a \in A, b \in B, f(a) = g(b)\} $$These are pretty much the same; I just used different notation for a pair in the second one. This is what I mean by canonically isomorphic. They’re basically the same.

But they’re not equal. They’re different sets. So we should recognize that. In the end, if we want to compose two spans, we have a groupoid of choices with precisely one isomorphism between any two options. (A category whose every hom-set is a singleton is what category theorists mean by ‘basically one thing’ - everything’s just a relabelling of everything else. Fewer isomorphisms would make things unequal. More would imply symmetries, and then we’d have ‘one thing plus a group action’.)

The second reason we might not be pleased with this bicategory is that perhaps ‘functions between spans’ aren’t the 2-morphisms we want. Spans seem pretty darn good for relating sets. Why shouldn’t we use them to relate spans? We’re certainly not obliged to of course, but maybe it’s important. What if other things are arranged this way? What if we want relations between relations?

It’s here that we come to a major mantra of higher category theory (to me):

Higher categories make morphisms into first-class citizens.

If you’ve ever learned a functional programming language like Haskell or had the pleasure of reading SICP, you’ll know what I mean. If not, consider how rich the means to compare objects are in a category. They can be isomorphic in a million different ways, which all interact with one another to give beautiful symmetries. But what about morphisms? They’re equal or they’re not. Hom-sets are just that. Sets. What if the morphisms are complex objects of their own right? Maybe we want to treat them with the same respect we treat objects.

If we want to keep up the category theory lifestyle, then by ‘compare morphisms’ we mean ‘consider morphisms between morphisms’. Should these higher morphisms allow their source and target morphisms to have different sources/targets? Probably not. It’s usually only meaningful to compare morphisms if they’re of the same ‘type’. Comparing morphisms of different types would usually happen via ‘whiskering’, which you might’ve seen in the context of natural transformations. (You’re more than welcome to ignore this constraint, but you’ll struggle to find examples! I’ll discuss that situation sometime in the future.)

In our case, the point of a ‘span between spans’ is to compare how two spans relate the same elements. So, we should require that the source and target spans of an entity share their sources and targets. This gives us globular higher morphisms. More precisely, given two spans $A \leftarrow B \rightarrow C$ and $A \leftarrow B' \rightarrow C$, a span between spans will be a span $B \leftarrow D \rightarrow B'$ commuting with the four other span maps appropriately.

So… what’s this structure? We could try and make it a bicategory like before, but now composition of 2-morphisms isn’t associative on the nose. So we need 3-morphisms between the 2-morphisms. Should these just be maps, or should they be ‘spans between spans between spans’? Yet again, perhaps spans are a more appropriate form of comparison. To treat 2-morphisms as first-class citizens, we ought to go with that. But then the 3-morphisms don’t compose associatively. So we need 4-morphisms, which for the same reasons as before, are spans between spans between spans between spans! And so on forever and ever.

This is an example of an $\infty$-category. The tower of morphisms has no peak. It begs the question: if every algebraic law only holds up to higher morphisms, how do we actually write down any algebraic laws? Before that, how do we deal with the fact that composition is only well-defined up to higher morphisms? We’ll come back to these points later. For now, take pleasure in knowing that in this structure, every relation between things is as well-treated as its source and target.

Bordisms Between Bordisms

Let’s try another example of a higher category, this time based around (smooth) manifolds. What would be a good way to relate manifolds? If you guessed ‘cobordisms’, I’d guess you’re either a geometric topologist or a physicist studying Topological Quantum Field Theories (TQFTs). We’re gonna stick with the former, since that’s closer to my own background.

Now, a category of manifolds and cobordisms suffers precisely all the issues that we highlighted in a category of sets spans - composition, given by gluing cobordisms smoothly along a shared boundary, leaves us with a range of options given by a choice of ‘collars’ around the source/target manifold. The space of such options is contractible, which is a bit fancier than any two compositions being canonically isomorphic. In the end, it’s just the infinite-dimensional analogue. (We’ll explore this fully in a future post.)

Furthermore, composition with the identity (a cylinder) doesn’t do nothing - it’s just canonically diffeomorphic to doing nothing. So, we have the same choices as before - do we take diffeomorphism classes and force a category, or do we not and observe a higher category? The latter sounds like better math.

What higher morphisms do we have between morphisms? This is actually a wonderful example of the mantra ‘morphisms are first-class citizens’. After all, cobordisms are manifolds! With boundary, anyway. In fact, one could argue that in a category of (n-1)-dimensional manifolds and n-dimensional cobordisms between them, we’re really interested in studying n-manifolds; an n-manifold in this context is an endomorphism of the empty (n-1)-manifold. The advantage such an n-manifold has over the objects of this category is that it can be ‘chopped up’ into simpler n-cobordisms, namely a series of morphisms it is a composition of. This suggests categories of cobordisms are a good structure for presenting manifolds algebraically in a generator-and-relation format. So, if cobordisms are just manifolds again, what higher morphisms should we use? Higher-dimensional cobordisms! (There are some technicalities about the ‘corners’ such a cobordism would have. Don’t lose sleep over it.)

As we slap on higher and higher dimensional cobordisms, we find that the manifolds they represent can be chopped up in more and more directions, corresponding to each direction of composition. We find the means to chop up an n-manifold in n directions, splitting it into simple copies of $\mathbb{R}^n$ that wind up corresponding to certain handle attachments and cancellations. In this situation, higher morphisms truly are first-class citizens; they are no less objects than their sources and targets.

Localization

As a final example, let’s turn to homotopy theory. In many ways, higher category theory is the pullback of category theory and homotopy theory, so it’s no surprise that homotopy would feature in this list.

It’s almost insultingly obvious how homotopy theory would generate a higher category. Take topological spaces as objects, continuous maps as 1-morphisms, homotopies as 2-morphisms, homotopies between homotopies as 3-morphisms, and so on forever and ever. Composition and identities are what you think they are. In lots of ways, this is the canonical example of an $(\infty, 1)$-category - that is to say, an $\infty$-category where every $k$-morphism for $k > 1$ is invertible (up to higher morphisms, of course). Yet again, composition isn’t so obviously defined, with a contractible space of options given by how quickly you pass through each homotopy in a composition of homotopies.

So… why exactly do we care about ‘higher homotopies’? I kind of just threw that at you, as it was once thrown at me. It’s clear enough why we care about objects, 1-morphisms and 2-morphisms - homotopy theory is important. But what’s with a homotopy of homotopies? Why is that so crucial to the theory? The ‘morphisms are first-class citizens’ argument kind of fails here, since homotopies aren’t obviously first-class citizens. We really do just care about $1$-morphisms and how they’re related.

One attempt at an answer comes from the choice of ways to compose homotopies. We should recognize that all these choices exist without any one being more ‘natural’ than the others (if you disagree, think about why composition fails to be associative!) and, while distinct, are basically the same. This demands higher homotopies that vary the way homotopies are parameterized. Homotopies need to be first-class citizens. But this isn’t the full story, as a higher homotopy can do more than just vary its source’s parameterization. What are they really there for?

Let’s step away from this higher category and turn to $\textbf{Set}$ for a moment. Consider a terminal object $\ast$ and the set $\textbf{Hom}_\textbf{Set}(\ast, X)$ for any set $X$. This is naturally isomorphic to $X$. Similarly, consider the $k$-linear category $\textbf{Vect}_k$ of $k$-vector spaces. The vector space $\textbf{Hom}_{\textbf{Vect}_k}(k, V)$ for a vector space $V$ from the terminal object $k$ is again naturally isomorphic to $V$ as a vector space. The same phenomenon happens for many topological spaces, where $\textbf{Hom}_\textbf{Top}(\ast, X)$ will be a space naturally homeomorphic to $X$. In general, the collection of morphisms from a terminal object to an object $X$ will implicitly contain all the structure of $X$. This is a microcosm of a more general principle, where morphisms will implicitly be related to one another by relations in their source and target, making many symmetric monoidal categories naturally enriched over themselves.

OK, so why does it matter? What new insight does such enrichment provide us? Well, for starters, it usually means we can extend operations on a single object in the category to the entire category itself. $k$-linear categories behave pretty similarly to vector spaces, so can be classified and studied using derivative methodologies. Go look up $2$-vector spaces if you’re interested - maybe I’ll write a post about them later. We essentially respect the fact that the entire theory carries a similar structure to its objects. For $k$-linear categories this is clearly important - there are lots of important $k$-linear categories and we wanna know how they interrelate! That stuff leads to insights in each one.

Homotopy theory, at a glance, is no different. In our higher category, we’re basically setting each hom-set to be a CW complex, ie. a weak homotopy type. Such an $\infty$-category (or $(\infty, 1)$-category), much like $k$-linear categories look like vector spaces, ends up looking a lot like a homotopy type - equivalent higher categories are entirely analogous to weakly equivalent topological spaces. Remember what I called a ‘single object’ in a category? A groupoid where every hom-set’s a singleton. Yeah, in an $\infty$-category, that becomes a contractible CW complex whose $n$-cells are $n$-morphisms. Equivalences between (higher) categories don’t need to be formally invertible, they just need to be bijective on abstract ‘things’, ie. contractible subcomplexes. That’s where the similarity to weak equivalence kicks in.

This application of higher categories, ie. situations where the theory is homotopical in nature, emerges from a correspondence between model categories and $(\infty, 1)$-categories. Model categories are basically categories with an abstract formal notion of homotopy. To really make them look like homotopy types, we can convert them formulaically into $(\infty, 1)$-categories by a process called localization. You basically add in homotopies and higher homotopies to the hom-sets, making them hom-spaces. Our example of a higher category for homotopy theory is really a localization of a model category of topological spaces.

How is this use case for higher categories related to the previous ones? Well, any higher category will look a lot like a homotopy type, no matter what. By promoting morphisms to first-class citizenship, we add in a higher dimension of homotopy data, and vice versa. Many of the ideas of higher category theory have near-perfect analogues in algebraic topology, so the connection there is fruitful to actively foster and maintain.

Conclusion

We’ve danced through a few different kinds of higher category in this post. I promised I’d try to review the underlying commonalities of these examples, so let’s do that.

To me, the important mantra is that in a higher category, morphisms are first-class citizens. Their interrelations and theory are just as complex as objects. Two morphisms may be ‘basically the same’, but not literally the same. In this situation, we need to know what an isomorphism between morphisms is. Often we find the same situation with the higher morphisms, so leading us on and on until we stop at some $n > 0$ or go all the way to $\infty$.

For instance, there are lots of situations where composition of morphisms isn’t well-defined, just up to canonical 2-isomorphism. That is to say, there’s ‘basically’ one composition, but not literally one. We saw this in every example presented, and finding more isn’t difficult. Similarly, even if we were to choose a composition operation, it wouldn’t be associative or have identities on the nose, only up to higher canonical isomorphisms. Again, it’s ‘basically’ associative, just not literally. When morphisms can be isomorphic and not equal, these things start to matter. Forcing equality by taking equivalence classes is akin to turning a groupoid into a set by taking equivalence classes. You’re imposing structure rather than observing it.

We saw in higher categories of bordisms that sometimes, the $n$-morphisms are just as interesting as the objects. Sometimes, they’re even more fundamental! This is our mantra taken a step further.

Finally, all this talk of promoting morphisms brings us right next to homotopy theory and algebraic topology; such higher morphisms stick together in a way very similar to cells in a CW complex. Our mantra makes the link with homotopy a central source of theorems and geometric insight.

In the end, higher categories have a particular purpose that manifests in diverse situations, just as any good abstraction should. They are a unifying principle for much of mathematics. It’s a shame they’re so difficult to work with right now, but the higher category theory community is unfaltering in its pursuit of better insight; in our lifetimes, I expect we’ll wield these bizarre structures with the same fluency that we use categories today.