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Stream: deprecated: id my structure

Topic: adjunction whose counit is a natural isomorphism

Bruno Gavranović (Aug 19 2022 at 13:54):

Is there name for an adjunction whose counit is a natural isomorphism?
That is, given $F : \mathcal{C} \rightleftarrows \mathcal{D} : G$, where $F \dashv G$, the following holds that for every $D : \mathcal{D}$

$$L(R(D)) \cong D$$

?

Bryce Clarke (Aug 19 2022 at 13:56):

Bruno Gavranovic said:

Is there name for an adjunction whose counit is a natural isomorphism?
That is, given $F : \mathcal{C} \rightleftarrows \mathcal{D} : G$, where $F \dashv G$, the following holds that for every $D : \mathcal{D}$

$$L(R(D)) \cong D$$

?

You're thinking of a Reflective subcategory

Bruno Gavranović (Aug 19 2022 at 13:59):

Ah, yes!

Bruno Gavranović (Aug 19 2022 at 14:00):

Thanks. I never really read in depth about reflective subcategories - but I suppose I now know what they are
Screenshot_20220819_145955.png

Jonathan Weinberger (Aug 19 2022 at 21:39):

These are also called RARI adjunctions (short for "right adjoint right inverse"), cf. Definition 1.2 in Maria Manuel Clementino and Fernando Lucatelli Nunes: Lax comma 2-categories and admissible 2-functors

John Baez (Aug 19 2022 at 23:49):

A "rari" is a bit more strict than an adjunction whose counit is a natural isomorphism: namely, the counit is an identity morphism. At least that's the definition of rari used by John Gray, who I believe introduced them. But I haven't read Clementino and Nunes' paper.

Jonathan Weinberger (Aug 20 2022 at 00:02):

Ah, that's right.

Mike Shulman (Aug 20 2022 at 04:24):

I do think some people have also appropriated the word for the case when the "inverse" is only up to isomorphism, although I don't have a reference to hand at the moment.

Oscar Cunningham (Aug 20 2022 at 13:21):

What's an example of an adjunction whose counit is an isomorphism but which isn't rari?

John Baez (Aug 20 2022 at 14:55):

Take any adjunction that's a rari and change the right adjoint R to some other functor R' that's naturally isomorphic but not equal to R.

Oscar Cunningham (Aug 20 2022 at 14:57):

Are all these things equivalent to raris?

John Baez (Aug 20 2022 at 14:58):

The ones I just described are equivalent to raris (if you change the unit and counit in the 'obvious way'). I don't know if all reflective subcategories are equivalent to raris.

John Baez (Aug 20 2022 at 15:03):

Or, let $L: \mathsf{Gp} \to \mathsf{AbGp}$ be abelianization and let $R : \mathsf{AbGp} \to \mathsf{Gp}$ send any abelian group to its underlying group. Is this a rari? The counit $LRA \to A$ maps the abelianization of the underlying group of an abelian group $A$ to $A$. Is this the identity? For this you have to decide if $LRA$ is equal to $A$. Is $A$ modulo the trivial group equal to $A$? Or merely naturally isomorphic? For this you need to look at your precise construction of quotient group.

Mike Shulman (Aug 22 2022 at 16:33):

If by "equivalent" you allow yourself to change the categories as well as the functors and transformations, then yes: if the counit of an adjunction $C\rightleftarrows D$ is an isomorphism, then $D$ is equivalent to its full image which is a reflective subcategory of $C$, and any reflective subcategory can be made a strict rari by choosing the reflector to be the identity on objects of the subcategory.

John Baez (Aug 22 2022 at 16:36):

Great!