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?
- 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.
- 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.
- 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.)