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# Do Categorical Quotients andPreserve Covering Maps?

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SETTINGS:

Let $C$ be a category with fiber products and $B$ be a closed subcategory of $C$ (i.e. $B$ contains any isomorphism of $C$, and any base change of any morphism in $B$ is also a morphism in $B$).

Let $T$ be the topology on $C$ associated to $B$ (i.e. $Cov T$ consists of universal effective epimorphic families {$f_{i}:U_{i}\to U$} in $C$ such that each $f_{i}$ is a morphism in $B$).

Moreover, assume that the topology $T$ satisfies the following two conditions:

Condition I

Let $f,g,h$ be arbitrary morphisms in $C$, and assume that $h=gf$ and $h$ is in $B$.

1. If $g \in B$, then $f \in B$.

2. If ${f} \in CovT$, then $g \in B$.

Condition II

For any family of maps {$f_{i}:U_{i}\to X$} in $C$ for which there exists "disjoint union" $\coprod_{i} U_{i}$, then the induced map $\coprod_{i}U_{i}\to X$ is in $B$ if and only if $f_{i} \in B$ for all $i$.

Under the situation above,

Let $R$ be a categorical equivalence relation on an object $U \in C$ such that the two canonical projections $\pi_{i}: R\to U$ ($i=1,2$) are both covering maps of $U$ in the topology $T$.

And assume that this categorical equivalence has $T$-quotient $X$. This means that there exists a categorical quotient $p:U\to X$ of $\pi_{i}:R\to U$ such that the induced morphism of associated sheaves (on $T$) $p_{\ast}:h_{U} \to h_{X}$ is a categorical quotient of $\pi_{i \ast}:h_{R}\to h_{U}$ in the category of sheaves of sets on $T$.

(Then, one can prove that $R \cong U\times_{X}U$.)

Now, this is my QUESTION:

Is the map $p:U\to X$ a covering map in $T$ ?

In other words, is $p$ a universal effective epimorphism and satisfying $p\in B$ ?

I could prove a "converse" statement i.e. if $p:U\to X$ is a covering map of the topology $T$, then $\pi_{i}: U\times_{X}U \to X$ U$($i=1,2$) is a categorical equivalence relation such that each$\pi_{i} \in CovT$and has$p$as a$T$-quotient. But I could not prove the statement in my question. Please give me any advice. 1 # Categorical Quotients and Covering Maps Before asking a question, please let me write down settings. SETTINGS: Let$C$be a category with fiber products and$B$be a closed subcategory of$C$(i.e.$B$contains any isomorphism of$C$, and any base change of any morphism in$B$is also a morphism in$B$). Let$T$be the topology on$C$associated to$B$(i.e.$Cov T$consists of universal effective epimorphic families {$f_{i}:U_{i}\to U$} in$C$such that each$f_{i}$is a morphism in$B$). Moreover, assume that the topology$T$satisfies the following two conditions: Condition I Let$f,g,h$be arbitrary morphisms in$C$, and assume that$h=gf$and$h$is in$B$. 1. If$g \in B$, then$f \in B$. 2. If${f} \in CovT$, then$g \in B$. Condition II For any family of maps {$f_{i}:U_{i}\to X$} in$C$for which there exists "disjoint union"$\coprod_{i} U_{i}$, then the induced map$\coprod_{i}U_{i}\to X$is in$B$if and only if$f_{i} \in B$for all$i$. Under the situation above, Let$R$be a categorical equivalence relation on an object$U \in C$such that the two canonical projections$\pi_{i}: R\to U$($i=1,2$) are both covering maps of$U$in the topology$T$. And assume that this categorical equivalence has$T$-quotient$X$. This means that there exists a categorical quotient$p:U\to X$of$\pi_{i}:R\to U$such that the induced morphism of associated sheaves (on$T$)$p_{\ast}:h_{U} \to h_{X}$is a categorical quotient of$\pi_{i \ast}:h_{R}\to h_{U}$in the category of sheaves of sets on$T$. (Then, one can prove that$R \cong U\times_{X}U$.) Now, this is my QUESTION: Is the map$p:U\to X$a covering map in$T$? In other words, is$p$a universal effective epimorphism and satisfying$p\in B$? I could prove a "converse" statement i.e. if$p:U\to X$is a covering map of the topology$T$, then$\pi_{i}: U\times_{X}U \to X$($i=1,2$) is a categorical equivalence relation such that each$\pi_{i} \in CovT$and has$p$as a$T\$-quotient.

But I could not prove the statement in my question.