Using the definition of a "congruence relation" on a category (as defined here), I was wondering if there is a notion of the congruence relation $$ \{E_{x,y}\subset \mathrm{Hom}_C(x,y)^{\times 2} \}_{(x,y) \in \mathrm{Obj}(C)^{\times 2}} $$ on a category $C$ generated by a given collection of relations $$ \{R_{x,y}\subset \mathrm{Hom}_C(x,y)^{\times 2} \}_{(x,y) \in \mathrm{Obj}(C)^{\times 2}}, $$ on each hom-set of $C$?
Indeed, if I'm not mistaken, the "pair-object-wise intersection" of a family of congruence relations is again a congruence relation. In detail: if $C$ is a category, and $\{\{E^i_{x,y}\subset \mathrm{Hom}_C(x,y)^{\times 2} \}_{(x,y) \in \mathrm{Obj}(C)^{\times 2} } \}_{i\in I}$ is an indexed collection of congruence relations on $C$, then $\{\bigcap_{i\in I} E^i_{x,y} \}_{x,y}$ is again a congruence relation on $C$.
Using this, it should follow that for any given collection of hom-set relations $R = \{R_{x,y}\}$ on a category $C$, the "pair-object-wise intersection" of all possible congruence relations on $C$ containing $R$ gives the "smallest" congruence relation containing $R$.
I was wondering if there is an explicit formula for the congruence relation generated by $R$? Below I've posted what I think is a possible construction, but I'm not fully sure it's valid.
My idea for a possible construction is as follows:
- Let $R = \{R_{x,y} \subset \mathrm{Hom}_C(x,y)^{\times 2}\}_{x,y\in\mathrm{Obj}(C)}$ be a given collection of relations on each hom-set of a given category $C$.
- Let $S_{x,y}$ be the equivalence relation on $\mathrm{Hom}_C(x,y)$ generated by $R_{x,y}$, in the set-theoretic sense.
- Let $T_{x,z}$ be the relation on $\mathrm{Hom}_C(x,z)$ defined as follows:
for $f,f' \in \mathrm{Hom}_C(x,z)$
we say $f,f'$ are $T_{x,z}$-related
iff there exist
some $k\in\mathbb{Z}_{\geq0}$
and $y_1,y_2,\ldots,y_k \in \mathrm{Obj}(C)$
and morphisms $x=:y_0 \xrightarrow{f_0} y_1 \xrightarrow{f_1} y_2 \cdots y_k \xrightarrow{f_k} y_{k+1} := z$ and $x=:y_0 \xrightarrow{f_0'} y_1 \xrightarrow{f_1'} y_2 \cdots y_k \xrightarrow{f_k'} y_{k+1} := z$ in $C$,
such that:
- $f = f_kf_{k-1} \cdots f_1f_0$ and $f' = f_k'f_{k-1}' \cdots f_1'f_0'$ ,
- and $f_i,f_i'$ are $S_{y_i,y_{i+1}}$-related for each $i\in\{0,1,\ldots,k\}$ .
- Finally, let $E_{x,y}$ be the equivalence relation on $\mathrm{Hom}_C(x,y)$ generated by $T_{x,y}$, in the set-theoretic sense.
My claim is $E = \{E_{x,y}\}$ now forms a congruence relation on $C$, and furthermore it is "contained" by every possible congruence relation "containing" $R$. (And $E$ contains $R$. Pf: Already $E$ contains $T$, which from the definition we see contains $S$ by "taking $k=0$" when comparing $f,f'$; and $S$ contains $R$.)
I'm trying to figure out a proof or counterexample for this, but I keep getting confused on the details. Is my above reasoning and construction valid? Any help would be much appreciated!