Return to Answer

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 a sieve $Sec$ on $c$, and we demand that there is some cover $h_j:c_j \to c$ such that each $h_j \in Sec$.

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:

There exists a cover of $c$ such that each $\sigma_j$ is in its associated sieve.

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 $d$, is a cover of $c$.

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

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 a sieve $Sec$ on $c$, and we demand that there is some cover $h_j:c_j \to c$ such that each $h_j \in Sec$.

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 $c$, is a cover of $d$.

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

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 $d$, is a cover of $c$.

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