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Stream: community: general

Topic: Clever Proofs in Category Theory

Ellis D. Cooper (Jul 22 2020 at 19:47):

Do you agree that in category theory, in general, theorem statements are more clever than proofs? I would like to know what you consider to be theorems in category theory with "tricky" or "clever" proofs. (A proof involving routine checking of the conditions of definitions is not "clever." On the other hand, a delicate use of intertwined inductions might be considered "tricky." The Rabinowich Trick often used to prove Hilbert's Nullstellensatz is "clever." What about the statement and proof(s) of Giraud's Theorem? )

Peter Arndt (Jul 23 2020 at 00:13):

The proof that an elementary topos - i.e. a category which has finite limits and a subobject classifier and is cartesian closed - is finitely cocomplete is pretty clever! You can find it in MacLane/Moerdijk's "Sheaves in geometry and logic".
It doesn't go through for infinity categories: the corresponding statement is false by work of Nima Rasekh. That is a bit of an indicator that the proof involves tricks rather than just canonical constructions.

David Michael Roberts (Jul 23 2020 at 00:31):

There's the proof that relies on the contravariant power object functor being monadic, that's pretty tricky, and then there's a proof written up by @Todd Trimble based on an idea of Tierney that uses an elementary approach, but which is not entirely trivial either.

Peter Arndt (Jul 23 2020 at 00:34):

Ah, cool, I didn't know about Trimble's proof.

David Michael Roberts (Jul 23 2020 at 05:27):

@Peter Arndt In case you haven't seen it, it's detailed here: https://ncatlab.org/toddtrimble/published/An+elementary+approach+to+elementary+topos+theory

Simon Burton (Jul 23 2020 at 12:00):

Emily Riehl's book has a whole chapter devoted to big results in category theory as counter examples to the claim that CT is all shallow.

Jules Hedges (Jul 23 2020 at 13:55):

I think Eckmann-Hilton (monoid of endomorphisms of unit in any monoidal category is always commutative, plus higher dimensional versions) could be seen as a trick, or as a deep fact, depending on your perspective

Mike Shulman (Jul 28 2020 at 19:33):

On cleverness in category theory see also https://golem.ph.utexas.edu/category/2010/03/a_perspective_on_higher_catego.html

Ellis D. Cooper (Jul 29 2020 at 21:32):

@Mike Shulman A propos Tom Leinster's remark, "if cleverness is the first quality that comes to mind then it suggests to me that something is insufficiently understood," Saunders Mac Lane in "Mathematics Form and Function" deduces Hamilton's equations in the usual way from Lagrange's equations via a Legendre transformation. And he writes, "Analysis is full of ingenious changes of coordinates, clever substitutions, and astute manipulations. In some of these cases, one can find a conceptual background. When so, the ideas so revealed help us to understand what's what. We submit that this aim of understanding is a vital aspect of mathematics." Then again, in the beautiful talk by Simon Willerton, see https://johncarlosbaez.wordpress.com/2020/05/26/the-legendre-transform-a-category-theoretic-perspective/
he summarized the result: the Legendre transform "arises out of nothing more than the pairing between a vector space and its dual in the same way that the many classical dualities (e.g. in Galois theory or algebraic geometry) arise from a relation between sets." It is in thermodynamics, however, that the Legendre transform peaks its value: "Everybody knows that the variables in an equation can be changed by simply substituting an equation for one of the variables in terms of a new variable. Not everybody knows that a derivative can be introduced as a new variable, but Gibbs did." (R. Alberty, "Use of Legendre transforms in chemical thermodynamics")

Ellis D. Cooper (Aug 03 2020 at 11:04):

Regarding the Legendre transform, please consider the following

Theorem. If $v=v(m(v))$ and $m=m(v(m))$, then the following are equivalent:

1. (a) $l(v)=m(v)\cdot v - h(m(v))$ and (b) $m(v)=\frac{dl}{dv}(v)$;
2. (a) $h(m)=m\cdot v(m)-l(v(m))$ and (b) $v(m)=\frac{dh}{dm}(m)$.

Proof. By symmetry, it suffices to prove (1) implies (2). In (1a), substitute $v(m)$ for $v$:

$$l(v(m))=m(v(m))\cdot v(m) -h(m(v(m)))=m\cdot v(m)-h(m)$$

so $h(m)=m\cdot v(m)-l(v(m))$, which is (2a). Also,

$$\frac{dh}{dm}(m)=m\cdot\frac{dv}{dm}(m)+v(m)\cdot \frac{dm}{dm}(m)-\frac{dl}{dv}(v(m))\cdot\frac{dv}{dm}(m) .$$

In (1b) substitute $v(m)$ for $v$, so

$$m=m(v(m))=\frac{dl}{dv}(v(m))$$

hence $\frac{dh}{dm}(m)=v(m)$, which is (2b).

Does that work for you?

fosco (Aug 13 2020 at 07:30):

I would like to know what you consider to be theorems in category theory with "tricky" or "clever" proofs.

I would define the entire mathematical production of PJ Freyd "clever", but it would be a huge understatement.