Why does mathematics seem to have a polarity bias?

Question

Why does mathematics seem to have a polarity bias, i.e., why are products more common than coproducts, algebras more common than coalgebras, limits more common than colimits, monads more common than comonads, topoi more common than cotopoi, etc. despite each pair being in a formal duality?

Answer 1

It is difficult to answer this question, because the answer is probably different for each of the examples the OP mentions. I think for each instance of "co" there are several possibilities:

• A. One thing was discovered because it's very natural, then people realized you can dualize and create the co-thing, which has fewer applications but is still interesting. Examples include topos vs cotopos, homotopy vs cohomotopy, monad vs comonad, group vs cogroup, etc. This was also pointed out in the comments.

• B. The co-thing is strictly more complicated than the thing, and to understand it requires that you first understand the thing. Mathematical concepts that are easier and less technical often are more common in the literature than more complicated concepts. Examples include homology vs cohomology. Also, in many cases, a coalgebra starts with algebra structure, then adds more.

• C. The original thing is based on our intuition from early math, where learning is probably driven by what's useful to a wide segment of society, and hence feels more intuitive than the co-thing. This was also mentioned in the comments. This is probably the case with algebras vs coalgebras, or products vs coproducts if you mean in the algebraic sense.

• D. The thing is actually not "more common" than the co-thing. They are both extremely natural and you could learn them in either order. This is how I think about limit vs colimit, or product vs coproduct in the categorical sense.

Let me write a bit more about (A). I don't think a cogroup is any harder to understand than a group, but we learn about groups first because they are ubiquitous. Same for representable functors vs corepresentable functors. Of course, this choice in the education system becomes a self-fulfilling prophecy, and every year more work is done on the thing than on the co-thing because people understand the thing better or feel like it's more natural. Once you've learned about the "thing" then thinking about the co-thing is harder because it requires reversing all arrows, hence there are fewer papers about the co-thing. But, if the order of discovery (or, the applications) had pushed us to invent the co-thing first, then the "thing" would be harder for our brains. Neither is intrinsically more natural than the other. This might be a good place to point out that in cases where two notions are both very natural, sometimes the "co" is on the wrong one, for historical reasons, like with covariant functors, which actually do NOT involve reversing the direction of the arrows. This isn't an example of a thing vs co-thing, because there are no "variant functors". Instead, functors that reverse arrows are called contravariant. But, if these notions were being invented today, what we know as covariant functors would be the thing, and contravariant functors would be the co-thing.

Let me write a bit more about (B). Sometimes the co-thing implies the existence of the thing. For example, in algebra, it's common to fix a monoid $A$ and an $A$-coring $B$, then look at $B$-comodules in $A$-modules. In this case, a comodule is already a module. We consider this setting in Section 6.6 of this recent paper of mine with Donald Yau, which also cites other places where this kind of thing is studied in algebra. Something similar happens with comodules over a Hopf algebroid, which we recall in 6.9 and 7.4 of that paper.

Let me write a bit more about (C). It might be a special case of A, but I put it separately because it could also be driven by applications more than order of discovery or which thing is more natural (if you can even quantify that). So, while you could think of the natural numbers as either a monoid or a comonoid, one way involves addition, which is a fundamental application of math, so we bias in that direction. I didn't start thinking of it as a comonoid until logic programming in college. I spent many years writing papers about algebra structure, and recently wrote one about co-algebra structure, and while doing the literature review I was struck by how common coalgebras are after all. In many cases they are actually even more common than algebras, e.g., in a cartesian category, every object is a comonoid with respect to the diagonal map. Another example is that when you study closed symmetric monoidal categories, you could start with either the tensoring or the cotensoring (meaning, the internal hom). There are categories that are closed but not monoidal, and categories that are monoidal but not closed. I've always felt the monoidal structure was more natural because it feels like things I've learned before, but I know plenty of category theorists and homotopy theorists who would argue that the co-thing (the closed structure) is actually the more natural one to consider. One place this comes up is how there are two ways to check the pushout product axiom or the SM7 axiom in model categories, and a non-trivial proportion of researchers use the co-thing approach via hom objects.

Lastly, let me write more about (D). In my work, I see colimits more than limits, and I work with cofibrations more than with fibrations. I think both limits and colimits encode extremely common human ways of thinking. I think about the colimit as what I can build from all the pieces in front of me, the global behavior trying to emerge from the local behavior, or "where things are going." I think of limits as where things came from, since the limit of the diagram maps to each object in it. I previously wrote an answer, based on David Spivak's book Category theory for the sciences, that got into the human ways of thinking that concepts in category theory are encoding.

Note that my way of thinking about limits and colimits is the opposite of the comment about creating vs destroying. I think that comment is more correct for algebra vs coalgebra, e.g., creating new numbers by adding up old numbers, vs breaking a number down into its constituent pieces. I'd argue that both goals are really fundamental to the thinking process of humans, and neither is "more common" or "more natural" than the other.

Answer 2

Coproducts of sets are introduced earlier in mathematical education than products of sets, under the name "union" or "disjoint union". Also, addition is of course introduced earlier than multiplication.

The reason for the naming choice in that case seems to be that products in a wide variety of concrete categories (e.g. algebras for a variety in the sense of universal algebra, various sorts of manifolds) correspond to products of the underlying sets with the "obvious"s structure, while coproducts correspond to coproducts of the underlying sets less often (only in the manifolds case, not the algebras), so it makes sense to name categorical products by analogy with a more familiar construction and coproducts by duality with products than vice versa.

Answer 3

I'm not sure that what you are saying is even true, e.g. "limits/products are more common than colimits/coproducts", what does "common" even really mean?

In a literal sense the statement is false because of opposite categories: e.g., a cone is a limit iff the corresponding cocone in the opposite category is a colimit. This might seem silly to point out but remember that mathematicians have a choice of orientation of any category, so when given the choice they will pick the orientation that is most familiar to them. I think this entirely explains why e.g., cotopoi is not a common notion: any time someone would work with a category whose opposite is a topos, they would just work with the opposite category to re-use their intuition about toposes. For example, the Schanuel topos is often defined as sheaves on the opposite caetgory of finite sets and injections (https://ncatlab.org/nlab/show/Schanuel+topos) rather than being defined as the cotopos of co-sheaves on the category of finite sets and injections.

Then if we acknowledge that mathematicians have this degree of freedom, then the question is the simpler "why are common categories generally asymmetric in what universal properties they have" and this is something where there are fundamental mathematical obstructions. For instance, any category that is both cartesian closed and whose opposite is cartesian closed is a preorder, as originally noted by Joyal.

Answer 4

I don't have any deep insights to offer, but I'd like to suggest that there are two separate questions being mixed together here.

The first question is why there often appears to be an asymmetry between "primal" and "dual" in mathematical structures that arise in practice.

The second question is a linguistic one, having to do with which structure we deem to be the "primal" one and which one we deem to be the "dual" one.

That structures we care about are often not perfectly symmetric between primal and dual should not be too surprising, since category theory captures only some features of whatever mathematical thing we're studying.

The linguistic question has many possible explanations, as others have pointed out, and is perhaps less interesting since it doesn't necessarily imply anything of mathematical substance.

There is potentially a third question that could be asked, which is whether seemingly unrelated mathematical structures $S_1$ and $S_2$ are nevertheless asymmetric in the same way. But I'm not sure that this question actually makes sense. If $S_1$ and $S_2$ are truly unrelated, then in what sense can we claim that we've chosen the primal and dual labels "in the same way"? Conversely, if $S_1$ and $S_2$ are related, then that presumably explains why their labels are correlated (by analogy, physical magnets all have a north pole and a south pole, and we can label them all consistently because north poles all repel each other, even though which one we call north and which one we call south is arbitrary).

Answer 5

$\newcommand\Set{\mathrm{Set}}$Rather than what we use more or less than what, I've wondered at why while category theory is a completely symmetrical theory, many dual things feel very different from each other in the actual math we use category theory for. Limits in concrete categories are generally just limits of underlying sets whereas colimits are often much more complicated. Relatedly, forgetful functors are usually quite simple and are right adjoints, and free functors are often quite complicated and are left adjoints.

I think that a whole bunch of asymmetry stems from the asymmetry of the category of sets. Most notably (imo) is that it is cartesian closed, and not cocartesian coclosed. So much of math is built off of Set so it's not surprising that there ends up being so much asymmetry, given the asymmetry of Set.

But why the asymmetry of $\Set$? I always found it curious that even in classical set theory, $\Set$ is a cartesian closed category which through categorical semantics is associated with intuitionistic propositional logics.

(Please forgive me as I'm rusty on the technical details. I believe much of these arguments can be made completely formal through the duality of values and continuations and through some of the machinary around linear logic. The duality of abstraction has pretty clear exposition on some of this and references a lot of the literature that I don't have access to.)

I believe that this asymmetry of $\Set$ is because we like to work forwards rather than backwards [1]. We have values $1 \to A$ and want to achieve a result so we work forward, applying functions to get to that result. Dually we could start with a request $A \to P(1)$ or a continuation (think a predicate as in $\mathrm{CABA} = \Set^\text{op}$, or a linear form) and work backwards, precomposing or "coapplying" functions, until we end up at some starting point. With the former, we get a cartesian closed structure, whereas with the latter we get the less familiar cocartesian coclosed structure. Importantly, you can't have both at once as that collapses your category into a poset by a simple argument given in the above paper.

$\Set is based on taking the forward value-oriented side of the duality, so it fundamentally breaks the symmetry at the start, before we define our concrete categories, our groups and rings and topological spaces and presheaves on top of them. I believe linear logic and related machinery can provide a more symmetrical starting point for whatever that is worth.

[1] And I think because we like to think in terms of "things" rather than whatever is the "right" way to think of continuations or covalues. Requests, results, a hole waiting for a block of the right shape? Predicates $A \to P(1)$ (like elements of a complete atomic boolean algebra) or something like linear forms, the dual of values/vectors in vector spaces?