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Stream: learning: questions

Topic: composing exponential objects

Richie Yeung (Mar 10 2021 at 15:27):

In a category with exponentials you can apply the evaluation map on $B^A \otimes A$ to get $B$. Can you "compose" two exponentials? How would that look like?
Is there an arrow $C^B \otimes B^A \to C^A$ for all objects $A, B, C$?

Fawzi Hreiki (Mar 10 2021 at 15:34):

Yes. This is the internal composition morphism.

Joshua Meyers (Mar 10 2021 at 15:36):

$(C^B\otimes B^A)\otimes A \to C^B\otimes (B^A\otimes A ) \to C^B\otimes B \to C$ induces a map $C^B\otimes B^A\to C^A$ by lambda-abstraction.

Richie Yeung (Mar 10 2021 at 17:18):

thank you for this. @Fawzi Hreiki so since a closed monoidal categoryis defined to have this pair of adjoint functors $(-) \otimes X \dashv (-)^X$, the evaluation map and the composition map come for free?

The evaluation map $Y^X \otimes X \to Y$ comes from the ---- adjoint of $Id_{Y^X}: Y^X \to Y^X$
and the composition map $Z^Y \otimes Y^X \to Z^X$ comes from the ---- adjoint of the two evaluation maps above?

Richie Yeung (Mar 10 2021 at 17:19):

I find it hard to remember / work out when something is a left / right adjoint. Any tips?

Morgan Rogers (he/him) (Mar 10 2021 at 17:24):

In what context? You'll notice that when you write out the correspondence defining an adjunction, the left adjoint functor appears on the left of the arrow (assuming it's pointing from left to right). That is, an adjunction $L \dashv R$ give a correspondence between morphisms $LX \to Y$ and morphisms $X \to RY$.

Morgan Rogers (he/him) (Mar 10 2021 at 17:24):

Joshua's explanation for the map you're looking for was pretty clear, I think, as long as you know that "lambda abstraction" means "transposing across the adjunction (in the relevant direction)".

D.G. Berry (Mar 10 2021 at 17:36):

Richie Yeung said:

I find it hard to remember / work out when something is a left / right adjoint. Any tips?

The evaluation and abstraction maps come from the (co)unit of the adjunction. From these it is clear which functor is the left/right adjoint. One just then needs to remember the order of composition of the functors for the (co)unit which is obvious when you think about the hom-set adjunction but this requires knowing which functor is which; this is hardly surprising that one is obvious from the other given that they are equivalent.

John Baez (Mar 10 2021 at 17:50):

Richie Yeung said:

I find it hard to remember / work out when something is a left / right adjoint. Any tips?

There are dozens of tricks, and it's good to have a lot of them.

Between categories of sets with extra structure (groups, rings, topological spaces, etc.) left adjoints tend to be "free" and right adjoints tend to be "forgetful". A good mnemonic is that politically, the leftists want more freedom while the rightists want us to forget. (It doesn't need to be politically accurate, it's just a mnemonic).

Matteo Capucci (he/him) (Mar 10 2021 at 17:56):

Morgan Rogers (he/him) said:

In what context? You'll notice that when you write out the correspondence defining an adjunction, the left adjoint functor appears on the left of the arrow (assuming it's pointing from left to right). That is, an adjunction $L \dashv R$ give a correspondence between morphisms $LX \to Y$ and morphisms $X \to RY$.

I just noticed that $\dashv$ looks like an arrow $\to$, whose direction is the correct one (from the left adjoint to the right adjoint)

Matteo Capucci (he/him) (Mar 10 2021 at 17:57):

I used to be very confused about adjunctions and $\dashv$ until one day I started reading it without problem, like, overnight. It was very weird. I'm finding reasons a fortiori to find it intuitive.

Morgan Rogers (he/him) (Mar 10 2021 at 17:59):

Weird to think that some early category theorists could have chosen any other suitably asymmetric symbol and we would all be using that instead.

John Baez (Mar 10 2021 at 18:04):

I used to hate ⊣, and I still sort of do. From logic I had a nice intuition about ⊢, and then all of a sudden I fell into some crowd of people who would use ⊣ to denote adjunctions, and I could never figure out why, or which side was supposed to be the left adjoint. It's like waking up one day and suddenly people are talking funny.