# Is there a simple rule determining which concepts are “ordinary” vs “dual” in category theory?

In category theory, pullbacks are limits of cospans, whereas pushouts are colimits of spans.

Meanwhile, a functor has a right adjoint iff the left Kan extension of 1 exists and is absolute, and vice versa.

Naively, I'd expect the "co" labels to be chosen in such a way that all our basic limits (such as pullbacks) can be generated by diagrams that lack the "co" label, and vice versa; and I'd expect the "left" labels on things like Kan extensions and adjoint functors to be chosen such that "left" extensions give rise to "left" adjoints, and so on. But, clearly, this is not the case.

What am I to take from this? For example, which of the following states of affairs is closest to the truth?

1. The "co" label, and the "right" and "left" labels, were chosen myopically. A modern student, looking at all of category theory from a bird's-eye view, could in principle assign the labels such that basically all of the simple/natural/basic definitions can be stated in a way that treats labels like "co" or "left" in a uniform and regular fashion.
2. There is in fact a deep rule governing which things we currently call "co", and when we observe that pullbacks are limits of cospans and copullbacks are colimits of spans, this is actually indicative of an interesting parity-flip that's out there in the territory, rather than an artifact of the inconsistent labels on our map.
3. There's not a deep rule governing which things we call "co", but it's also the case that there just isn't a way to assign the labels such that the use of "co" (or "left") is uniform and regular. For example, if we were to swap the "left" and "right" labels on Kan extensions, then the definition of adjoint functors in terms of Kan extensions would feel more "forwards", but the definition of adjoint functors in terms of Kan lifts would become "backwards". While a bunch of concepts have duals, the duals don't actually cluster nicely, and we just have to pick some labels and bite some bullets.

(My current guess is (1), due to the observation that limits have a sort of "arrows coming in" flavor and colimits have a sort of "arrows going out" flavor, and that if we take this intuition at face-value it suggests that "cospan" is misnamed (and that our notation would be a bit more consistent if we renamed "cospan" to "wedge" and "span" to "cowedge"), but I am quite uncertain.)

• It’s all arbitrary. I wouldn’t think this much about it. It’s only language. – Randall Jan 5 '19 at 16:27
• @Randall I'm not so sure it is only language. Our brains mostly think in language, so if there is a pattern one can learn and follow, learning is faster. – Arthur Jan 5 '19 at 16:31

The answer is: "4) All of the above," but mostly things are "consistent" and apparent "inconsistencies" are probably not accidents, but sometimes inconsistencies are just poor terminology and other times they are to avoid hobgoblins. To directly answer which is "closest to the truth", it is 2, but the "rule" is not formal, or, to the extent that you do formalize it, doesn't cover every possible use we might want for the terminology. The "rule" is also not trivial and can be a matter of perspective.

First, whether some concept is the dual of some other concept is often not the most relevant or important aspect of it. This is why we have terms like "initial" and "terminal" or "pushout" and "pullback" and not "coterminal" and "terminal" or "copullback" and "pullback".

Typically, the "co-" prefix is used for left adjoint-y things while the prefixless version is used for right adjoint-y things. Colimits are left adjoints as are coequalizers, but not everything is determined by universal properties (and some things are determined by multiple universal properties, e.g. [binary] direct sums). Spans, for example, are commonly used while cospans are not talked about that much. (Incidentally, spans are cones and cospans are cocones, but this description will almost certainly cause more confusion than not. Indeed, a pullback is a limit span.) Nevertheless, because humans neither have the inclination nor ability to ensure global consistency, there are certainly times where it would be more "consistent" to rename which term is the dual. The most notable example is "tensors" and "cotensors", which, for the above reason among others, are also called "copowers" and "powers" respectively. (One of the other reasons is that the word "tensor" is already enormously overloaded.)

The "left"/"right" terminology, though, is even more consistently used.1 It is pretty hard to be confused between the domain and codomain.2 If you see an inversion here, it is probably reflective of something deeper. That right adjoints are left Kan extensions is not an artifact of poor choices in names. First, left Kan extensions can themselves be described as left adjoints3 (modulo some size considerations), or rather the functor which performs left Kan extension along some given functor is a left adjoint. Second, colimits are left Kan extensions. Third, left Kan extensions are weighted colimits (which can usually also be expressed as coends of pairs). So this is a case where there would be no way to "correct" this terminology without breaking "consistency" elsewhere, but, again, nothing is wrong in this case. As you allude to, we don't have this "flipping" in the Kan lift case. This no doubt has to do with the fact that pre-composition is the action of the contravariant hom-functor while post-composition is the action of the covariant hom-functor.

1 The specific words "left" and "right" come from our notation(s) for hom-sets and thus are a result of convention. One could imagine terminology like "source" and "target" adjoint that would not be sensitive to the order we write things. On the other hand, I tend to use the terminology "pre-/post-composition" rather than "right/left composition" as the order of composition is not nearly as unambiguous.

2 Which brings up another aspect that not all uses of "co-" have to do with categorical duality, or at least weren't chosen with any reference to categorical duality (e.g. because category theory didn't exist at the time).

3 This gives the notion of a global Kan extension, but that usually isn't what we want. Nevertheless, the "leftiness" still persists for local and pointwise left Kan extensions.

A nice minimal example of the unavoidability of this conflict, which is in essence where the right Kan/left adjoint thing comes from, is that an initial object is both a colimit of the empty diagram and a limit of the identity functor.

I basically agree with Derek Elkins' answer. Here is a comment on spans. I've never heard anyone describe a pushout diagram as a span. I have seen spans, with that name, used in the following two ways:

1. To describe localizations of categories.
2. To describe categories of spans. Spans are morphisms in a 2-category, and they are composed using pullbacks.

An unfortunate feature of category theory is that some of its basic concepts are so general that they can be used to model a wide variety of different things, and so it's common to have multiple names for the same thing depending on what use it's being put to. A simple example is calling a functor $$F : C \to D$$ a "diagram of shape $$C$$ inside $$D$$" to indicate that you're about to take its limit or colimit. Similarly here, if I'm going to take a pushout I'd call the diagram involved a pushout diagram, not a span.