# Is the thingie/cothingie distinction absolute?

Is there some inherent quality of a mathematical object that marks it as being "naturally" a thingie or a cothingie?

Suppose, for example, that two mathematical concepts, say, doodad and doohickey, originally defined far, far away from the reaches of category theory, are later discovered to be–who knew?–categorical duals of each other.

Would it then be entirely arbitrary whether to rename doohickeys as "codoodads" or rename doodads as "codoohickeys"?

The fact that, for me at least, cothingies are typically far more exotic than their duals suggests that there is no intrinsic "coquality" that distinguishes cothingies from thingies, and therefore, all other things being equal, it is the more exotic member of the pair that gets the coname. (Otherwise the extreme underrepresentation of cothingies in my prior mathematical education is hard to explain.) But occasionally I come across comments that suggest that some mathematicians at least have an intuitive sense of how to classify arbitrary mathematical entities as either thingies or cothingies. Therein the nundrum.

Thanks!

PS: Wikipedia's extremely useful List of mathematical jargon desperately needs a knowledgeable entry on the term cointuition.

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Well, sometimes a cogadget is the natural thing, and its exotic twin is a contragadget... – Henning Makholm Feb 25 '12 at 21:53
I believe the more general one is adapted, or the one with most scope for generalization. – Ravi Feb 25 '12 at 21:56
All this talk reminds me of a joke I heard: "comathematicians are devices for turning cotheorems into ffee." Off-topic. – anon Feb 25 '12 at 21:59
I believe cointuition is that early stage when a man and a woman merely think at the same time about having babies. – Will Jagy Feb 26 '12 at 0:28

An algebraic category like $\text{Grp}$ or $\text{Ring}$ often has a forgetful functor to $\text{Set}$ which has a left adjoint (the free object functor) but generally not a right adjoint. It follows that the forgetful functor preserves limits but generally not colimits. That's one reason you might consider limits more basic than colimits, at least if you like algebra, since limits look more familiar from $\text{Set}$ (e.g. the underlying set of the product is the Cartesian product in $\text{Grp}$ whereas the underlying set of the coproduct is very different from the disjoint union).

On the other hand, arguably the status of the Yoneda embedding $C \to \text{Set}^{C^{op}}$ as the free cocompletion of a category is evidence suggesting that colimits are more fundamental.

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On that last point, I must disagree: the fact that the Yoneda embedding is the free cocompletion means that colimits are malleable. If the colimits are "wrong" in $\mathcal{C}$, simply find a suitable subcategory of $[\mathcal{C}^\textrm{op}, \textbf{Set}]$ giving the right answer instead! This is one way to motivate the functor of points approach to schemes. – Zhen Lin Feb 26 '12 at 19:24
@ZhenLin: "colimits are malleable", with its tacit implication that limits are less so, is a good example of the kinds of remarks that give me the impression that there is an intrinsic "co-" nature. Of course, I realize that such interpretation entails a fair bit of unwarranted extrapolation on my part. But at the very least your comment does suggest that duality is anything but symmetrical: the cothingie may have properties that the thingie would have never led one to suspect, and vice-versa. This is something I know to be the case, but still haven't fully digested... – kjo Feb 26 '12 at 20:27
Well, everything I said can be dualised: if $\mathcal{C}$ has the wrong limits, then embed it into $[\mathcal{C}, \textbf{Set}]^\textrm{op}$ and take a suitable subcategory. The main difference is that $[\mathcal{C}, \textbf{Set}]^\textrm{op}$ is a somewhat less well-understood category... – Zhen Lin Feb 26 '12 at 21:30
@ZhenLin: "The main difference is that $[\mathcal{C}, \textbf{Set}]^\textrm{op}$ is a somewhat less well-understood category..." This "asymmetry of understanding" I find interesting in itself, and may very well be an example of what's at the root of what I'm trying to pin down. Why, for example, are operations with coequalizers/quotients/coproducts so much more unintuitive than those with equalizers/subsets/products? At least for me they are, and judging by how textbooks invariably start with the latter and give the former a short shrift $\cdots$ [cont'd] – kjo Feb 27 '12 at 13:25
@kjo: That part of your question is what the first paragraph of Qiaochu's answer addresses: in many categories of interest, the underlying set functor preserves limits but not colimits. – Zhen Lin Feb 27 '12 at 18:40

In category theory, cartesian squares, products, kernels, and limits are all maps into a given diagram, while cocartesian squares, coproducts, cokernels and colimits are all maps from a given diagram. Homology and cohomology follow this rule as well (when taking resolutions for covariant functors).

As Henning pointed out, this shouldn't be confused with co-vs-contra.

To answer the title of the post, I don't believe "co-ness" is absolute, but the above seems to be a pretty good guiding principle. Sometimes it may just be a matter of which is discovered first.

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I'll just give one example showing that the distinction is indeed not absolute.

Traditionally in differential geometry you define tangent vectors to a manifold $V$ at a point $P$ first, for example as the vector space $T_P(V)$ of equivalence classes of differentiable curves through $P$ .
And then you define the cotangent space $T^*_P(V)$ as its dual vector space.

Zariski realized that in algebraic geometry, if you consider a variety $X$ and a point $x\in X$, it was more natural to first define the cotangent space of $X$ at $x$ as $\Omega^1_x(X)=\mathfrak m_x/\mathfrak m_x^2$.
(In this formula $m_x$ denotes the maximal ideal of the local ring $\mathcal O_{X,x}$ of $X$ at $x$)
The tangent space is then defined as the dual of the cotangent space $\Omega^1_x(X)$, seen as a vector space over the field $\mathcal O_{X,x}/\mathfrak m_x$.
This approach (published by Zariski in 1947) has technical advantages, especially in the case when $x$ is a singular point of $X$ or for varieties in characteristic $p\gt 0$.

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There's good reason to call $\Omega^1_X$ the cotangent sheaf though – the maps induced by a regular map $f : X \to Y$ will go the other way, $\Omega^1_Y \to \Omega^1_X$. – Zhen Lin Feb 25 '12 at 22:45
@ZhenLin, ...if you regard the spaces as the central thing. – Mariano Suárez-Alvarez Feb 26 '12 at 5:32