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Hom-set adjunction


context ${\bf C},{\bf D}$ … small category
context $F$ in ${\bf D}\longrightarrow{\bf C}$
context $G$ in ${\bf C}\longrightarrow{\bf D}$
definiendum $\Phi$ in $\mathrm{it}$
postulate $\Phi$ in $\mathrm{Hom}_{\bf C}(F-,=)\cong\mathrm{Hom}_{\bf D}(-,G=)$


Here $\mathrm{Hom}_{\bf C}(F-,=),\mathrm{Hom}_{\bf D}(-,G=)$ in ${\bf Set}^{{\bf D}\times {\bf C}}$.

Observe that if $\mathrm{Hom}_{\bf C}(F-,=)\cong\mathrm{Hom}_{\bf C}(-,G=)$, then $\mathrm{Hom}_{\bf C}(F-,B)$ in ${\bf Set}^{\bf D}$ is represented by $GB$ and $\mathrm{Hom}_{\bf C}(A,G-)$ in in ${\bf Set}^{\bf C}$ is represented by $FA$.


An example in the category of sets

Let both ${\bf C}$ and ${\bf D}$ be the category ${\bf Set}$, which has products and exponential objects. Fix some objects (sets) $A$ and $Y$. Many examples can be thought of as variation of the pretty obvious relation

$\mathrm{Hom}_{\bf Set}(*\times A,Y)\cong\mathrm{Hom}_{\bf Set}(A,Y):= Y^A\cong\mathrm{Hom}_{\bf Set}(*,Y^A)$

where $*$ is a one-element set, but that's an unnecessary restriction:

Consider any set $X$. Indeed, we have

$\mathrm{Hom}_{\bf Set}(X\times A,Y)\cong\mathrm{Hom}_{\bf Set}(X,Y^A)$

and this is a hom-set adjunction

$\mathrm{Hom}_{\bf Set}(FX,Y)\cong\mathrm{Hom}_{\bf Set}(X,GY)$

if we define the Action of $F$ on object via $FX:=X\times A$ (Cartesian product) and let the action of $G$ on object be given by $GY:=Y^A$ (function space from $A$ to $Y$).


More generally, view the left adjoint $F$ as A-“thickening” of ist argument ($X$) and view $G$ as the A-indexing's of aspects of it's argument $Y$.

If ${\bf C}\neq{\bf D}$, then viewing $G$ as indexing may be harder.


Similarly, for propositions

$\left((X\land A)\implies Y\right)\leftrightarrow\left(X\implies(A\implies Y)\right)$

Here the A-“thickening” side says you have more arugments to prove $Y$ to begin with, while the “$A$-indexing's” side means you only demonstrate A-conditional truth of $Y$.

Example from Algebra

For example in the category of groups

$\mathrm{Hom}(X\otimes A,Y)\cong\mathrm{Hom}(X,\mathrm{Hom}(A,Y))$

Galois connection

$\langle A,\le\rangle$, $\langle B,\le'\rangle$ … posets, and $F:A\to B,G:B\to A$ … monotone functions, then Galois connection =

$(F(a)\le b)\leftrightarrow(a\le'G(b))$

Counit-unit adjunction

If we look at the morphisms from the corresponding Counit-unit adjunction,

$\eta_Y:{\mathrm{Hom}}(Y, GFY)$


$\epsilon_Y:{\mathrm{Hom}}(FGY, Y)$

at least for sets the way in which those must be defined should be clear from how they map

$Y$ to $(A\times Y)^A$


$(A\times Y^A)$ to $Y$.

The first can only be a direct embedding

$\eta_Y(y):=\lambda a.\, \langle a,y\rangle$

and the second is an evaluation

$\epsilon_Y(\langle a,f\rangle) := f(a)$




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