Is it true that Nature promotes products?

I hope this question is not unreasonable.

We all know how to take products of numbers, this generalises to a huge amount of different types of products in mathematics. In a certain sense this notion is "on the surface". On the other hand there is a different operation, called "co-product" that seem to be a less intuitive notion (for example, things like co-algebras seem to be not so intuitive).

Is there a real asymmetry between these two notions (or is this explained by the human perception?)?

This question might turn out to be silly but I have not seen it anywhere and it seem to be a natural question...

Motivation. This is not quite a motivation, but what led me to the question is the reflection on the (well known) fact that there is no product operation for homologies but there is one for cohomologies.

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If you're interested in AG, you may be interested to know that descent data is in fact controlled by a coalgebra and its comodules. I'm hardly an authority on the subject, but this seems to be purely a historical accident and not necessarily the "natural" way of things. –  Jon Beardsley May 14 '13 at 23:14
That is to say, the fact that it's not (at least around me) first talked about in that way seems to be historical. –  Jon Beardsley May 14 '13 at 23:15
Oh, it is a mathematical question after all. I read the title, Nature, produce, ... Well, most(?) of us are used to the SET category. And SET is not too symmetric, is not self-dual. –  Włodzimierz Holsztyński May 14 '13 at 23:42
I was about to suggest publishing.mathforge.org just on seeing the title of this question... –  darij grinberg May 15 '13 at 1:43

You need to distinguish between "coproduct" and "comultiplication." The categorical coproduct is just a generalization of addition and is intuitive in many contexts.

Comultiplications are more interesting. Actually it turns out that many familiar mathematical objects are canonically equipped with comultiplications. In fact, in any category $C$ with finite products every object $c$ has a canonical comultiplication given by the diagonal map $\Delta : c \to c \times c$. On sets this is the diagonal map $x \mapsto (x, x)$. In other words, in such a category $C$ every object is canonically a coalgebra. Usually this structure is invisible, but coalgebras are preserved by monoidal functors, so for example the free vector space on a set is canonically a coalgebra in vector spaces. Usually this structure is also invisible, but it appears for example in the Hopf algebra structure on group rings...

In particular, every topological space is canonically a coalgebra. Now, homology (over a field) is a covariant monoidal functor, so it sends coalgebras to coalgebras. But cohomology is a contravariant monoidal functor, so...

Another source of comultiplications is combinatorics. Where multiplications are intuitively like "building things up," comultiplications are intuitively like "breaking things down." And this allows you to define many natural coalgebras by breaking down combinatorial structures into parts. For example, $R[x]$ has a natural $R$-linear comultiplication given by

$$x^n \mapsto \sum_k x^k \otimes x^{n-k}$$

and this reflects the process of breaking apart a string of length $n$ into two strings of length $k$ and $n-k$. The dual of this coalgebra is the algebra of ordinary generating functions over $R$. Similarly, there is another comultiplication given by

$$x^n \mapsto \sum_k {n \choose k} x^k \otimes x^{n-k}$$

and this reflects the process of breaking apart a set of size $n$ into two subsets of size $k$ and $n-k$. The dual of this coalgebra is the algebra of exponential generating functions over $R$.

In the same spirit, Rota noticed that the incidence algebra of a poset is naturally regarded as the dual of an incidence coalgebra, namely the free vector space on intervals $[a, b]$ in the poset with comultiplication given by

$$[a, b] \mapsto \sum_{a \le c \le b} [a, c] \otimes [c, b]$$

and this reflects the process of breaking apart an interval into two subintervals.

Yet another source of comultiplications comes from objects in a category $C$ which represent a (covariant) functor to the category of monoids. For example, the fundamental group functor $\text{hTop}_{\ast} \to \text{Grp}$ is represented by the object $S^1$ in the (pointed) homotopy category, and this is because $S^1$ naturally has a cogroup structure. Similarly, the multiplicative group functor $\text{CRing} \to \text{Grp}$ is represented by the object $\mathbb{Z}[x, x^{-1}]$, which also naturally has a cogroup structure (in fact a commutative Hopf algebra is precisely a cogroup object in $\text{CRing}$, or equivalently is precisely the ring of functions on an affine group scheme).

Comultiplications only seem unfamiliar because they were historically not pointed out, but actually they are all over the place. See also What is a coalgebra intuitively?

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Thank you, this is a sharp answer! –  aglearner May 14 '13 at 23:18