One way we can define the product of two objects $A$ and $B$ in some category $\mathcal C$ is as a representation of the contravariant functor $(\to A) \times (\to B)$ in $[\mathcal C^{\mathrm{op}}, Set]$.
The analogous definition of coproducts is as a corepresentation of the covariant functor $(A \to) \times (B \to)$ in $[\mathcal C, Set]$.
Notably, AFAICT one can't get products by asking for a corepresentation of $(A \to) + (B \to)$, nor coproducts by asking for a representation of $(\to A) + (\to B)$, where here I'm using $+$ to denote coproducts.
I find this asymmetry curious, and my main question is the title question. That question's a bit broad and vague, though. To be a bit more specific, aspects of this question that I'm particularly curious about include:
- What are these other objects? ((co)representations of $(A \to) + (B \to)$) Are they well-studied? Alternatively, are there good reasons to consider them boring and ignore them?
- Whence the asymmetry? It's almost as if $Set$ is insisting that $\times$ is "more primary" than $+$. Of course, $Set^{\mathrm{op}}$ will sing the opposite tune, but I still find the situation surprising. Is there a good intuition for why the "correct" way to ask for coproducts in arbitrary categories is equivalent to asking for a corepresentation of Set-theoretic products, rather than (say) a direct representation of Set-theoretic coproducts?
- Given a construction in $Set$ (such as a product or a coproduct or an exponential), what determines whether that construction in going to be "representable-ish" versus "corepresentable-ish"? We could of course just try both and see which one works and classify our constructions accordingly, but is there any way to look at products and coproducts in $Set$ and notice ab initio that products are going to be the "representy" ones and coproducts the "corepresenty" ones? (Alternatively, what are the keywords I should be searching to read up on this?)
(lmn if I should expand on my notation more here. Note that I'm following the nlab's convention when distinguishing between representable functors and corepresentable functors, and that by eg $(\to A) \times (\to B)$ I mean $(X \mapsto \mathrm{Hom}_{\mathcal C}(X, A) \times \mathrm{Hom}_{\mathcal C}(X, B))$.)
Best Answer
Suppose we had, for some objects $A$ and $B$, an object $C$ such that $(C,-)$ is naturally isomorphic to $(A,-)+(B,-)$. Then the category in which this happens cannot have a terminal object, since such an object, call it $T$, would have only one element in $(C,T)$ but two in $(A,T)+(B,T)$.
More generally, any functor of the form $(C,-)$ preserves all the products that exist in your category. So $(A,-)+(B,-)$ would have to preserve products, and this leads to lots of problems since $(A,-)$ and $(B,-)$ individually also preserve products. More generally yet, you'll have a problem with preservation of limits.
I'm not sure how far one can carry this line of reasoning, but it certainly prevents the existence of such representing objects $C$ in the categories that people usually want to work with. (At the moment, I can't think of any category where such $A,B,C$ exist, but that may be just a deficiency of my imagination.)