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It is standard in category theory to study things 'from the top down' -- to study structured sets we use categories, to study structured categories we use bicategories, to study structured bicategories we use Gray-categories, and to study all of them at once we can use $\infty$-categories.

Unless I'm mistaken, some of Grothendieck's initial work on fibered categories was implicitly using bicategorical ideas (indexed categories) without the bicategorical vernacular, and the development of $2$-category theory allowed us to more succinctly express the ideas in play.

This has obviously been fruitful, but it seems counterintuitive to study simpler things using more complicated things. I can write down the definition of a set in a few seconds (primitive), the definition of a category in about a minute, the definition of a bicategory takes maybe a few minutes if rushed, and a tricategory with all of its coherence pasting diagrams explicitly written out would take 27 pages (if I remember correctly, I'm moving and the GPS book with the definition is still in the old house). Ideally, I would like to be able to study the more complicated things using conceptually simpler ones.

Has there been any work done attempting to seriously study higher categorical notions using lower categorical ones?

For example, we can consider the $1$-category of pseudonatural transformations and modifications (in full generality or between two fixed pseudofunctors) and study it using the machinery of $1$-category theory, perhaps gaining insight into pseudofunctor $2$-categories. We could also study the category of bicategories and pseudofunctors. These examples both require prior knowledge of the definitions involved in the higher stages of the categorical hierarchy though, which is not ideal.

There are also some negative results in this direction; for example, we can't form a category of tricategories and trihomomorphisms since composition of trihomomorphisms isn't associative on the nose; the best we can hope for is a bicategory (see page $4$ of the linked Garner/Gurski paper). This obstacle seems to imply that an approach based on this method would be limited to using levels just below the one we want to study.

Another type of example would be attempting to study the ingredients of higher categories at lower levels. For example, the definition of a bicategory uses categories, functors and natural transformations. While I'm not aware of a way to compile all three of these into one structure without using bicategories, we can collect the top two levels into a category ${\bf Fun}$ of functors and natural transformations -- functor categories between fixed categories $\mathcal{C}$ and $\mathcal{D}$ can be obtained as the full subcategory of ${\bf Fun}$ on functors whose domain is $\mathcal{C}$ and whose codomain is $\mathcal{D}$.

While the composition of functors and Godement product of natural transformations together don't form any kind of named structure I'm aware of on ${\bf Fun}$, if we restrict out attention to endofunctor categories $\mathcal{C}^\mathcal{C}$ for a fixed category $\mathcal{C}$ then composition and Godement products constitute a tensor product, turning $\mathcal{C}^\mathcal{C}$ into a strict monoidal category; whatever structure composition constitutes on ${\bf Fun}$, it's one one that restricts to a strict tensor product on certain 'well-behaved' full subcategories.

This can be seen as a structure 'endowed' on the endofunctor category by virtue of the fact that $\mathfrak{Cat}$ is a strict $2$-category, but we could also in theory make the observations about $\mathcal{C}^\mathcal{C}$ and ${\bf Fun}$ first and study them, gaining insight into horizontal composition of $2$-cells at the $2$-categorical level using $1$-categorical machinery. This is just a rough example of what I'm looking for, but it's the best I've come up with after a week or so thinking about it. Any assistance is appreciated.

Linked paper: arXiv:0711.1761v2 [math.CT]


EDIT: For another example in the second vein, we have the following lemma about sets and functions that essentially yields the intuituion for the Yoneda lemma, and can be used to prove it.

Lemma For sets $X,Y$, let $Hom(X,Y)$ denote the set of all functions from $X$ to $Y$. For any two functions $f:A\to B$ and $g:C\to D$, the following diagram commutes
where $g\circ$ and $\circ f$ denote postcomposition with $g$ and precomposition with $f$, respectively.

We can thusly study a central piece of $1$-category theory using sets and functions ($0$-category theory).

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    $\begingroup$ You might be interested in Riehl and Verity's work which studies $(\infty,1)$ (and even $(\infty,n)$) categories primarily by studying the homotopy strict 2-category of all such. $\endgroup$ – Noah Snyder Jan 13 at 19:34
  • $\begingroup$ @NoahSnyder Thanks for the reference, I’ll give it a look. $\endgroup$ – Alec Rhea Jan 13 at 22:47
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Well, what is an $\infty$-category? Most people who study $\infty$-categories would, if pressed for an actual definition, say something like "a quasicategory" or "a complete Segal space". But any such definition consists of a collection of sets and functions between them satisfying certain axioms. So in that sense, we always study the more complicated $\infty$-categories using the simpler sets. Even homotopy type theory, which treats $\infty$-groupoids or $\infty$-categories as "synthetic" objects that aren't built out of sets, uses set-like structures at the meta-level, with definitional equalities between terms.

Quillen model categories, and related tools like fibration categories or cofibrant replacement comonads, are another way of studying $\infty$-categories using 1-categories. The Riehl-Verity followers who say "$\infty$-category" to mean "object of some $\infty$-cosmos" are using this approach, since an $\infty$-cosmos is a certain kind of fibration category. Homotopy categories and homotopy 2-categories have already been mentioned as another way to study higher-categorical things with lower-categorical ones; "derivators" are an enhanced sort of homotopy category that's especially good for this purpose. (I wrote a blog post about some of these approaches.)

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I'm not sure I could write down the definition of a set in a few seconds since to do this explicitly means writing out a set of axioms, such as ZFC, and I'm not even sure - actually I am sure - that those axioms are not on the tip of my fingers, so to speak. Of course the notion of a set is intuitively easy to grasp, as well as a category or indeed higher categories and it's this that takes a few seconds.

Have you come across Batanins description of infinity categories? They are described as algebras over certain globular operads. These operads in turn are monoids in a certain monoidal category called Coll.

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  • $\begingroup$ Extensionality, regularity, comprehension, infinity, powerset, pairing, union, replacement, choice if you like. (I didn't see your answer till about a minute ago ;). More seriously though, my point is that even if you have the definition sitting right next to you, the time it takes to copy it down (let alone digest it) increases superexponentially past the bicategorical level, to the point where just writing down the algebraic definition of a tetracategory is intimidating (see Todd Trimbles notes for an example, it takes 51 pages: math.ucr.edu/home/baez/trimble/tetracategories.html) $\endgroup$ – Alec Rhea Dec 14 '20 at 18:53
  • $\begingroup$ I am (vaguely) aware that there are various models for $\infty$-categories, but I prefer algebraic definitions and it's still in the air what an algebraic definition of a fully weak infinity category should be. $\endgroup$ – Alec Rhea Dec 14 '20 at 18:56
  • $\begingroup$ @Alec Rhea: In that case can I just say ZFC :-)? I know exactly what you mean. I heard about the intimidating definition of Trimbles tetracategories probably from the same source years ago. Luckily, Trimble says that his approach is 'primative' and hence recognises the need for more sophistication to bring down the diagram count. There are other approaches - Emily Riehl & Dominic Verity are attempting a model independebt approach whilst the homotopy type approach sell themselves as infinity categories for undergraduates. I'll wait & see ;-). $\endgroup$ – Mozibur Ullah Dec 14 '20 at 19:02
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    $\begingroup$ @Alec Rhea: I think the globular approach is considered an algebraic approach - at least according to Nlab - but there's still some work to go. $\endgroup$ – Mozibur Ullah Dec 14 '20 at 19:06
  • $\begingroup$ That's interesting, I'll take a closer look at the globular approach. (I'd also be interested in looking at the Riehl/Verity work) One method of reducing the diagram work taken by Mike Shulman is to rely on the internal logic of higher categories to express diagrams using set-theory like notation in the internal language, with some nice results. $\endgroup$ – Alec Rhea Dec 14 '20 at 20:05

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