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Stream: learning: questions

Topic: Can Hom-functors be reversed?

Julius Hamilton (Sep 08 2024 at 20:40):

For $C$, let there be a functor $Hom(A, -): C \to Set$, such that $Hom(A,-)(X \in C)$, written $Hom(A,X)$, is Hom-set $Hom(A,X) \in C$, and for $(f : X \to Y) \in C$, $F(f)$ is the function mapping each $g \in Hom(A,X)$ to $(f \circ g) \in Hom(A,Y)$.

Can a Hom-functor be “reversed”? If there was an inverse, it should send $Hom(A,X)$ to $X$, and $F(f): Hom(A,X) \to Hom(A,Y)$ to $f : X \to Y$. I don’t see initially why this shouldn’t work.

Is there an interesting use case of taking a Hom-functor a second time, on the target category of the first one?

Is there an interesting use case of treating a given category as if it were the image of a Hom-functor (or, isomorphic to that), and applying this “inverse Hom-functor”?

A representable functor is one that is naturally isomorphic to a Hom-functor. From an introductory point of view, what is useful about a functor being “representable” as a Hom-functor?

(Kind comments only.)

fosco (Sep 09 2024 at 09:19):

Can a Hom-functor be “reversed”?

It seems you're trying to ask the following question: let $C$ be a category and assume there is an object $a$ such that $\hom(a,-) : C \to Set$ is an equivalence of categories (so applying the inverse you indeed "get $x$ back out of $\hom(a,x)$"). What do I know about $a$? The answer is: this is a fairly restrictive assumption on $a$ (and on $C$, of course, which ends up being essentially the category of sets). For example, a functor that is an equivalence of categories preserves all colimits, as well as all limits: then, $\hom(a,-)$ must preserve all colimits, making $a$ into a [[tiny object]]. Now, $a$ can't be a random object: call $G_a$ the adjoint inverse of $\hom(a,-)$; since every set is a coproduct of singletons, and G is itself an equivalence of categories, for every set $X$ one has $G_aX \cong \sum_{x\in X} G_a1$, and now... [hint: relate $a$ and $G_a1$]

fosco (Sep 09 2024 at 09:22):

One can relax this (evidently too restrictive) situation; it is fairly common for $\hom(a,-)$ to admit a left adjoint, without it being an equivalence... the left adjoint of $\hom(a,-)$ is the functor sending a set $X$ to an $X$-indexed sum of copies of $a$:

$-\odot a : Set \to C : X\mapsto X\odot a := \sum_{x\in X} a$

[so that I just spoiled the previous exercise ;-) ] and it's usually called a [[copower]].

fosco (Sep 09 2024 at 09:28):

Is there an interesting use case of taking a Hom-functor a second time, on the target category of the first one?

it depends: you're considering the composition $C \xrightarrow{\hom(a,-)} Set \xrightarrow{\hom(S,-)} Set$, which is the functor

$x\mapsto \hom_{Set}(S,\hom_C(a,x)),$

which (as per my previous comment, whenever iterated coproducts exist) is isomorphic to the functor $x\mapsto \hom_C(S\odot a,x)$. Now, it depends what you want to do. I would say this is an interesting functor, but you have to see it as a particular instance of a general construction (a very simple example of a [[weighted colimit]]).

fosco (Sep 09 2024 at 09:30):

From an introductory point of view, what is useful about a functor being “representable” as a Hom-functor?

A conversation that might enlighten you on this matter is happening right now in the "classifying object" thread. Very simply put: if a functor $F$ is representable, it acts like $\hom(a,-)$ for some (uniquely determined) $a\in C$. But then, the set $Fa$ contains a special ("universal") element which specifies the action of $F$ entirely.

fosco (Sep 09 2024 at 09:36):

Out of any $F : C \to Set$ one can build the category $El(F)$ having

• objects the pairs $(a,x\in Fa)$;
• morphisms $(a,x)\to (a',y)$ those $f : a\to a'$ in $C$ such that the function $Ff : Fa\to Fa'$ sends $x$ to $y$

the category structure is exactly what you expect. Since the class of objects off $El(F)$ is the dependent sum $\sum_{a : C} Fa$, of all elements of the various $Fa$ when $a$ runs over $Obj(C)$, we call $El(F)$ the category of elements of $F$.

Now: the following two conditions are equivalent.

1. $F$ is representable, i.e. there is a natural transformation, with all components bijective, $\alpha : F \cong \hom(a,-)$ for some $a\in C$;
2. $El(F)$ admits an initial object $(a,x_0\in Fa)$.

It is particularly instructive to prove that 2 $\Rightarrow$ 1. Try!

Julius Hamilton (Sep 15 2024 at 01:02):

That is an extremely good answer, thank you hugely.