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I believe the definition should be as follows:

Let $C$ be a Grothendieck site. Suppose $f:c \to d$ in $C$. Let $a$ be a cover $\left(a_i:d_i \to d\right)$ of $d$. Let $Sec(f)_a$ denote the set of maps $\sigma_i:d_i \to c$ is a map such that $f \circ \sigma_i=a_i$. Say that $f$ is a submersion if the set $\underset{a} \cup Sec(f)_a$, that is all maps $\sigma_i:d_i \to d$, ranging over all covers of $d$, is a cover of $c$.

It is easy to check that this gives the same definition you gave for manifolds.

EDIT: I made a mistake. What I actually should say is:

The collection $\sigma_i:d_i \to c$ defines

There exists a sieve $Sec$ on $c$, and we demand that there is some cover of $h_j:c_j \to c$ such that each $h_j \\sigma_j$ is in Sec$its associated sieve. 3 fixed stuff I believe the definition should be as follows: Let$C$be a Grothendieck site. Suppose$f:c \to d$in$C$. Let$a$be a cover$\left(a_i:d_i \to d\right)$of$d$. Let$Sec(f)_a$denote the set of maps$\sigma_i:d_i \to c$is a map such that$f \circ \sigma_i=a_i$. Say that$f$is a submersion of if the set$\underset{a} \cup Sec(f)_a$, that is all maps$\sigma_i:d_i \to d$, ranging over all covers of$d$, is a cover of$c$. It is easy to check that this gives the same definition you gave for manifolds. EDIT: I made a mistake. What I actually should say is: The collection$\sigma_i:d_i \to c$defines a sieve$Sec$on$c$, and we demand that there is some cover$h_j:c_j \to c$such that each$h_j \in Sec$. 2 edited body I believe the definition should be as follows: Let$C$be a Grothendieck site. Suppose$f:c \to d$in$C$. Let$a$be a cover$\left(a_i:d_i \to d\right)$of$d$. Let$Sec(f)_a$denote the set of maps$\sigma_i:d_i \to c$is a map such that$f \circ \sigma_i=a_i$. Say that$f$is a submersion of the set$\underset{a} \cup Sec(f)_a$, that is all maps$\sigma_i:d_i \to d$, ranging over all covers of$c$, d$, is a cover of $d$.c\$.

It is easy to check that this gives the same definition you gave for manifolds.

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