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August 16, 2019

Evil Questions About Equalizers

Posted by John Baez

I have a few questions about equalizers. I have my own reasons for wanting to know the answers, but I’ll admit right away that these questions are evil in the technical sense. So, investigating them requires a certain morbid curiosity… and have a feeling that some of you will be better at this than I am.

Here are the categories:

RexRex = [categories with finite colimits, functors preserving finite colimits]

SMCSMC = [symmetric monoidal categories, strong symmetric monoidal functors]

Both are brutally truncated stumps of very nice 2-categories!

My questions:

1) Does RexRex have equalizers?

2) Does SMCSMC have equalizers?

3) We can get a functor F:RexSMCF \colon Rex \to SMC by arbitrarily choosing an initial object for each category in RexRex to serve as the unit of the tensor product and arbitrarily choosing a binary coproduct for each pair of objects to serve as their tensor product. Do this. Does the resulting FF preserve equalizers?

You see, I have some equalizers that exist in RexRex, and I know FF preserves them, but I know this by direct computation. If 1) were true I’d know these equalizers exist on general grounds, and if 3) were true I’d know they’re preserved on general grounds. This might help me avoid a bit of extra fiddliness as I walk on the dark side of mathematics and engage in some evil maneuvers… especially if there’s some reference I can just cite.

2) is just my morbid curiosity acting up. I’m pretty sure the category of symmetric monoidal categories and strict symmetric monoidal functors has equalizers, but strong?

Posted at August 16, 2019 7:31 AM UTC

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Re: Evil Questions About Equalizers

Dear John,

(1) Consider the indiscrete category AA on two objects 00 and 11. Since it is equivalent to the terminal category it has finite colimits, and any functor from AA to AA preserves them. In particular, the functor s:AAs:A \to A interchanging 00 and 11 preserves them. Any functor F:XAF:X \to A equalising ss and the identity would have to factor through their equaliser in CatCat, which is the empty category. But then XX would have to be empty itself. However no category with finite colimits can be empty – therefore the equaliser of idid and ss does not exist in RexRex.

(2) The same example and argument, viewing the finitely cocomplete category as a symmetric monoidal one, shows that SMCSMC does not have equalisers.

(3) Well, your functor FF restricts to act on strict morphisms (if you view the objects of RexRex as coming equipped with choices of colimits). Restricted to strict morphisms, in the source and target categories, it does preserve equalisers, and is indeed induced by a map of 22-monads. I haven’t thought about whether it preserves any equalisers of non-strict maps that happen to exist though – do you have some examples?

Note: if you view RexRex and SMCSMC as 22-categories they do have many nice 2-dimensional limits (pie limits, flexible limits…) but not equalisers.

Best, John.

Posted by: John Bourke on August 16, 2019 3:57 PM | Permalink | Reply to this

Re: Evil Questions About Equalizers

Thanks for deftly solving 1) and 2), John! I was trying a similar trick but it was more complicated and I got bogged down. You solved it with minimal fuss!

Can you tackle 3) with equal finesse? Here I’m looking to find an equalizer that exists in RexRex that is not preserved by Φ:RexSMC\Phi \colon Rex \to SMC.

You see, I have a bunch of equalizers that are preserved, and I think I know how to show it, but it would look a bit silly to prove this if in fact all equalizers (that exist) in RexRex are preserved. Sort of like proving “the sum of two even perfect numbers is even”.

(Actually I should check again to make sure the equalizers I think are preserved, really are.)

Posted by: John Baez on August 19, 2019 8:31 AM | Permalink | Reply to this

Re: Evil Questions About Equalizers

It seems to me that what can be proved is the following, which I’m guessing will cover your case.

Prop: Any equaliser in RexRex which is preserved by the forgetful functor to CatCat is also preserved by your functor Φ:RexSMC\Phi:Rex \to SMC.

Since your Φ\Phi commutes with the forgetful functors to Cat\Cat, to prove the above it suffices to show (*) that U:SMCCatU:SMC \to \Cat reflects equalisers: i.e., given a fork in SMCSMC sent by UU to an equaliser in CatCat then the original fork in SMCSMC is an equaliser.

Now (*) does appear to me to be true (I haven’t checked completely) if still slightly fiddly to prove carefully. I thought about more generally using 2-monads, their algebras and pseudomaps.

Hope that helps.

Posted by: John Bourke on August 20, 2019 10:35 AM | Permalink | Reply to this

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