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Stream: theory: category theory

Topic: contravariance and infinite regress

sarahzrf (Aug 19 2021 at 21:26):

something is bugging me about contravariance. im not sure i have a specific question but i do have a vague feeling of something being strange or confusing or off, & wondering if anyone had any input on it:

sarahzrf (Aug 19 2021 at 21:28):

so theres a certain phenomenon which ive seen described on this zulip at least once as some structure "riding" another structure—the particular example i have in mind, and where i think i saw the term, is the case of talking about a symmetry for a monoidal structure. to say that a braiding is a symmetry is approximately to say that it is an involution, except that's not quite right, because a braiding is not an endomorphism at all so it isn't even well-typed to ask if it is an involution

sarahzrf (Aug 19 2021 at 21:29):

rather, you want to say that the braiding is inverse to the whiskering of the braiding by the functor swap : C × C → C × C

sarahzrf (Aug 19 2021 at 21:29):

so it's an "involution" riding on the actual involution swap : C × C → C × C

sarahzrf (Aug 19 2021 at 21:29):

im fine with this so far, but...

sarahzrf (Aug 19 2021 at 21:31):

consider the notion of a "contravariant involution". formally speaking, this clearly suffers from the same issue as calling a symmetry an "involution"—a contravariant "endofunctor" is not truly an endofunctor at all

sarahzrf (Aug 19 2021 at 21:31):

so when i'm confronted with a functor F : C^op → C and told that it is an "involution", i'm tempted to seek an actual involution happening at a higher level, which F is an "rides on" to be an involution

sarahzrf (Aug 19 2021 at 21:32):

the clear candidate is -^op...

sarahzrf (Aug 19 2021 at 21:32):

...but ^op isn't an involution either!

sarahzrf (Aug 19 2021 at 21:32):

-^op is not Cat → Cat, it's Cat^co → Cat!

sarahzrf (Aug 19 2021 at 21:32):

...but it too seems to clearly be "an involution"...

sarahzrf (Aug 19 2021 at 21:33):

i suppose you could say that it is riding on -^co... but that isn't an involution either!

sarahzrf (Aug 19 2021 at 21:33):

so at every level you have something "involutive", but you can never actually phrase that property as "inverse to a 'shifted' version of itself by a higher-level Actual Involution", because the higher-level thing is never an actual involution either!

sarahzrf (Aug 19 2021 at 21:34):

you can escape by considering -^op as only an endofunctor on the 1-category Cat → Cat, but that seems like a cop-out somehow?

sarahzrf (Aug 19 2021 at 21:34):

...anyway, like i said, i don't have a specific question, but this is just bothering me

sarahzrf (Aug 19 2021 at 21:35):

does anyone have, like, any coherent thoughts on this that might make me less bothered about this?

John Baez (Aug 19 2021 at 22:21):

I think I was the one who mentioned this "riding" idea - it's a phrase due to James Dolan, and connected to the microcosm principle. It happens a lot.

John Baez (Aug 19 2021 at 22:23):

For example, what's a monoid? It's a thing you can define in any monoidal category. But what's a monoidal category? Well, it's like a monoid in Cat... really it's a weak monoid in Cat, and you can define this sort of thing in any monoidal 2-category. And so on.

John Baez (Aug 19 2021 at 22:25):

Or: how do you think of "binary product" as a functor on a category $C$? Well, it's a functor from $C \times C$ to $C$, where $\times$ is the binary product in Cat!

John Baez (Aug 19 2021 at 22:25):

And so on.

John Baez (Aug 19 2021 at 22:25):

But I guess what's bugging you about "op" is that it gets a bit more twisted each time you go up a level.

John Baez (Aug 19 2021 at 22:27):

I don't have anything to make you less bothered by this, except: it's not so surprising, "op" is an inherently twisty concept.

Fawzi Hreiki (Aug 19 2021 at 22:49):

It doesn't seem like there is any legitimate circularity here though. All of these concepts can be defined without reference to the higher versions, but it just puts things in context to consider the higher versions.

Fawzi Hreiki (Aug 19 2021 at 22:53):

Either you have to pick some level to stop at and just axiomatise (which will depend on your specific uses), or maybe there is some sort of stabilising effect which happens when you pass all the way to $\infty$ (however you end up axiomatising that).

Mike Shulman (Aug 20 2021 at 00:43):

FWIW, my paper Contravariance through enrichment includes a notion of "contravariance relative to a group action". If you go all the way to $\omega$-categories, with a duality action by $(\mathbb{Z}/2)^\omega$, then at least you can phrase the notion of involution as relative to itself rather than to some "higher" version of itself.

Mike Shulman (Aug 20 2021 at 00:44):

Alternatively, you can consider $(-)^{\rm op}$ as an involution of the (2,1)-category Cat, which at least isn't evil, even if it's not maximally satisfying.