Ok, so apparently there is an old joke (which I DO get) that says that in Hungary a mathematician is a device for turning coffee into theorems.

I found a post by Qiaochu Yuan that has the following definiton: A comathematician is a device for turning cotheorems into ffee.

Apparently this is a very funny joke. Could someone explain it to me and tell me where I could learn about the subject in question? Thank you very much in advance.


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    $\begingroup$ That's an old joke. Perhaps you should read about limits and colimits, it might help understand it. $\endgroup$
    – Asaf Karagila
    Commented Aug 16, 2014 at 19:27
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    $\begingroup$ The general joke is that given a map $f:X\to Y$, you can make a 'co' map $\text{co}f:\text{co}X\to\text{co}Y$. In this case, $\text{mathematician}:\text{coffee}\to\text{theorems}$ yields $\text{comathematician}:\text{cotheorems}\to\text{cocoffee}$. But of course coffee is just co-ffee so co-co-ffee is ffee. $\endgroup$
    – Ian Coley
    Commented Aug 16, 2014 at 19:30
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    $\begingroup$ @IanColey Should be $\operatorname{co} Y\to \operatorname{co} X$. $\endgroup$
    – user147263
    Commented Aug 16, 2014 at 19:42
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    $\begingroup$ Every nut is a coconut. $\endgroup$ Commented Aug 17, 2014 at 5:59
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    $\begingroup$ @900sit-upsaday, I'm not sure I agree. Given a product-preserving functor $\mathbf{C} \rightarrow \mathbf{D}$, we get a coproduct preserving functor $\mathbf{C}^\mathrm{op} \rightarrow \mathbf{D}^\mathrm{op}$. There is no switching. $\endgroup$ Commented Aug 17, 2014 at 6:00

1 Answer 1


A category consists of objects and arrows between them (which can be associatively composed).

We can express various constructions and theorems with the help of categories, and when we switch the direction of all arrows, we arrive to the dual constructions and dual theorems, which are usually named by the prefix 'co', and which stay valid, as the proof/construction must go through with the reversed arrows the same way.

For example, the coproduct, i.e. the dual of the cartesian product of sets (in the category of sets and functions) is the disjoint union operation.

Of course, taking the dual is involutive, that is, 'co(co X)=X' whatever X is. That's why the dual of coffee is 'ffee', so the 'definition' of the comathematician is just the dual statement of the old joke.

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    $\begingroup$ And considering that the dual statement does indeed lead to further humor, this once again proves useful the habit of considering the duals of things! $\endgroup$
    – user14972
    Commented Aug 17, 2014 at 17:57
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    $\begingroup$ Humor is selfdual. $\endgroup$ Commented Jan 13, 2015 at 2:24

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