The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes.
$\color{Green}{Background:}$
$\textbf{(1)}$ $\textbf{Motivation and Definition 1:}$
We have seen that a one-object category is a monoid, let us now see ia sense in which any category is a $\textit{generalized}$ monoid. Given $\textbf{K},$ let
$$\textbf{M}_{\textbf{k}}=\coprod_{A,B\in {Obj}\textbf{K}}\textbf{K}(A,B)$$
be the collection of all $\textbf{K}-$morphisms. Composition definees a $\textit{partial}$ function
$$\textbf{M}_{\textbf{k}}\times\textbf{M}_{\textbf{k}}\to\textbf{M}_{\textbf{k}}:(f,g)\mapsto g\circ f.$$
Let us see how we may recapture the identities of $\textbf{K}$ from this function: We say that $u$ is an identity if $g\circ u=g$ whenever $g\circ u$ is defined, and if $u\circ f=f$ whenever $u\circ f$ is defined. As before, each $f$ has exactly one identity $u$ such that $u\circ f$ is defined, call it $C(f)$ for the identity of the $\textit{codomain}$ of $f$; and exactly one $v$ such that $f\circ v$ is defined, call it $D(f)$ for the $\textit{domain}$ of $f.$ Thus our partial function $\textbf{M}_{\textbf{k}}\times \textbf{M}_{\textbf{k}} \to \textbf{M}_{\textbf{k}}$ satisfies the conditions:
$1.$ There are total functions $$C,D: \textbf{M}_{\textbf{k}}\to \text{identities of } \textbf{M}_{\textbf{k}}$$
for which
$$D(D(f))=D(f)=C(D(f));$$ $$\quad D(C(f))=C(f)=C(C(f)); \quad \text{and}$$ $$f\circ D(f)=f=C(f)\circ f.$$
$2.$ $g\circ f$ is defined iff $C(f)=D(g).$ If $g\circ f$ is defined then $D(g\circ f)=D(f)$ and $C(g\circ f)=C(g).$ Moreover, if either $(h\circ g)\circ f$ or $h\circ (g\circ f)$ is defined, then both are defined and $(h\circ g)\circ f=h\circ (g\circ f).$ [In fact, given a partial function satisfying $1$ and $2$ we can reconstitute a category, with objects being the identities.]
$\textbf{(2)}$ $\textbf{Definition 2:}$
Generalized monoids cry out for generalized homomorphisms $H:\textbf{M}_{\textbf{K}}\to \textbf{M}_{\textbf{L}}$ and the obvious demand is that
$$f \text{ an identity } \Longrightarrow Hf \text{ an identity}\quad (1)$$
$$g\cdot f \text{ definedy } \Longrightarrow H(g\cdot f)=Hg\cdot Hf.\quad (2)$$
For each object $A$ of $\textbf{K}$ denotes by $HA$ the object on which $H(\text{id}_A)$ is the identity, i.e.such that $H(\text{id}_A)=\text{id}_{HA}.$ Since $f\cdot\text{id}_A$ is defined for $f:A\to B,$ we have from $(2)$ that $$Hf=H(f\cdot \text{id}_A)=Hf\cdot H(\text{id}_A)=Hf\cdot\text{id} _\color{purple}{AH}.$$
Hence $Hf,$ being composable with $\text{id}_\color{purple}{AH},$ must have domain $\color{purple}{AH}.$ Similarly, it must have codomain $\color{purple}{BH}$ since $\text{id}_B\cdot f$ is defined. Thus $(2)$ says, in short, $H(A\xrightarrow{f}B)=HA\xrightarrow{Hf}HB.$
$\textbf{(3)}$ $\textbf{Definition 3:}$ A functor $H$ from a category $\textbf{K}$ to a category $\textbf{L}$ is a function which maps $\text{Obj}\textbf{(K)}\to \text{Obj}\textbf{(L)}:A\mapsto HA,$ and which for each pair $A,B$ of objects $\textbf{K}$ naos $\textbf{K}(A,B)\to \textbf{L}(HA, HB):f\mapsto Hf,$ while satisfying the two conditions:
$$H(\text{id}_A)=\text{id}_{HA}\quad\text{ for every }A\in\text{Obj}\textbf{(K)}$$ $$H(g\cdot f)=Hg\cdot Hf \quad\text{ whenever }g\cdot f\text{ is defined in }\textbf{K}.$$
We say that $H$ is an $\textbf{isomorphism}$ if $A\mapsto HA$ and each $\textbf{K}(A,B)\to \textbf{L}(HA, HB)$ are bijections.
$\color{Red}{Questions:}$
- (possible misprint?) For where I colored coded $\color{purple}{AH},$ $\color{purple}{BH}$ Should it not be $HA$ and $HB$ instead?
2a.) I understand in the definition of $H$ in $\textbf{(2)}$ $\textbf{Definition 2},$ that $H$ is a homomorphic map of generalized monoids. But in $\textbf{(3)}$ $\textbf{Definition 3},$ $H$ only designated to stand for being a function?
2b) Also in the context of functors, what do $HA,$ and $HB,$ mean? I know that in $\textbf{(1)}$ $H$ kind of serve similar roles as $D(f)$, and $C(f)$, but is never specific what it means in terms of mapping/functions for functors.
- (about isomorphisms) I am unclear about the statement: "We say that $H$ is an $\textbf{isomorphism}$ if $A\mapsto HA$ and each $\textbf{K}(A,B)\to \textbf{L}(HA, HB)$ are bijections."
In mathematical notations: does it mean we have maps: $p:A\mapsto HA$ and $q:\textbf{K}(A,B)\to \textbf{L}(HA, HB),$ and both of them being bijective?
If so, $p$ being a bijection would mean that it is both surjective and injective. But for the definition of surjection, what are elements in $HA$? where $HA$ is never really explicitly defined, because it should start with something like: For any elements $w\in {HA},$ there exists a....
For the case of $q:\textbf{K}(A,B)\to \textbf{L}(HA, HB)$, since $\textbf{K}(A,B)$ denotes the collection of morphisms, say $\{f_i\}_{i\in I}\in \textbf{K}(A,B)$ between objects $A$ to $B$ in the category $\textbf{K}$; and $\textbf{L}(HA, HB),$ denotes the collection of morphisms, say $\{Hf_i\}_{i\in I}\in \textbf{L}(HA, HB)$ between objects $HA$ to $HB$ in the category of $\textbf{L}.$ Then $q$ being injective would mean: If $q(\textbf{L}_1(HA, HB))=q(\textbf{L}_2(HA, HB)),$ then $\textbf{K}_1(A,B)=\textbf{K}_2(A,B).$ But does $q(\textbf{L}_1(HA, HB))=q(\textbf{L}_2(HA, HB)),$ mean: $\{Hf_i\}_{i\in I}$ is in both $\textbf{L}_1(HA, HB),$ and $\textbf{L}_2(HA, HB),$ similarly, $\textbf{K}_1(A,B)=\textbf{K}_2(A,B)$ would mean that $\{f_i\}_{i\in I}$ is in both $\textbf{K}_1(A,B)$ and $\textbf{K}_2(A,B)?$ Similarly for $q$ being surjective would mean: For any $Hf\in \textbf{L}(HA, HB),$ there exists a $f\in \textbf{K}(A,B)?$
Thank you in advance.
