I am trying to understand the semidirect product of groupoids, as defined in this answer by Theo Johnson-Freyd. Part of my difficulty is that although the definition of a 2-group makes sense, I am quite unfamiliar with that concept. So, as a test, I 'computed' the crossed product of two (arbitrary) sets $X,Y$ seen as discrete groupoids. My understanding is that the 2-group of automorphism in this case is the trivial monoidal category $(\mathbf{1}, \otimes, \ast)$, with $\mathbf{1}$ the trivial 1-object category, $\otimes:\mathbf{1}\times\mathbf{1}\rightarrow\mathbf{1}$, and unit the unique object $\ast$.

The object set of $X \rtimes Y$ is then $X \times Y$, and there is an arrow $(x_1,y_1) \rightarrow (x_2,y_2)$ iff $x_1=y_1$ and $x_2=y_2$, a long winded way of saying that in this particular case, $X \rtimes Y$ is (exactly) $X\times Y$ as a discrete groupoid. Question 1: is this correct?

There is another case that I have not been able to work as successfully: given an equivalence relation $E_1$ on (set) $X$ seen as a groupoid, and equivalence relation $E_2$ on $Y$, what are $E_1 \rtimes E_2$ and $E_2 \rtimes E_1$ ?

[Edit: I meant semidirect product, not crossed]

I am trying to understand the semidirect product of groupoids, as defined in this answer by Theo Johnson-Freyd. Part of my difficulty is that although the definition of a 2-group makes sense, I am quite unfamiliar with that concept. So, as a test, I 'computed' the crossed product of two (arbitrary) sets $X,Y$ seen as discrete groupoids. My understanding is that the 2-group of automorphism in this case is the trivial monoidal category $(\mathbf{1}, \otimes, \ast)$, with $\mathbf{1}$ the trivial 1-object category, $\otimes:\mathbf{1}\times\mathbf{1}\rightarrow\mathbf{1}$, and unit the unique object $\ast$.

The object set of $X \rtimes Y$ is then $X \times Y$, and there is an arrow $(x_1,y_1) \rightarrow (x_2,y_2)$ iff $x_1=y_1$ and $x_2=y_2$, a long winded way of saying that in this particular case, $X \rtimes Y$ is (exactly) $X\times Y$ as a discrete groupoid. Question 1: is this correct?

There is another case that I have not been able to work as successfully: given an equivalence relation $E_1$ on (set) $X$ seen as a groupoid, and equivalence relation $E_2$ on $Y$, what are $E_1 \rtimes E_2$ and $E_2 \rtimes E_1$ ?

[Edit: I meant semidirect product, not crossed]

I am trying to understand the crossed product of groupoids, as defined in this answer by Theo Johnson-Freyd. Part of my difficulty is that although the definition of a 2-group makes sense, I am quite unfamiliar with that concept. So, as a test, I 'computed' the crossed product of two (arbitrary) sets $X,Y$ seen as discrete groupoids. My understanding is that the 2-group of automorphism in this case is the trivial monoidal category $(\mathbf{1}, \otimes, \ast)$, with $\mathbf{1}$ the trivial 1-object category, $\otimes:\mathbf{1}\times\mathbf{1}\rightarrow\mathbf{1}$, and unit the unique object $\ast$.

The object set of $X \rtimes Y$ is then $X \times Y$, and there is an arrow $(x_1,y_1) \rightarrow (x_2,y_2)$ iff $x_1=y_1$ and $x_2=y_2$, a long winded way of saying that in this particular case, $X \rtimes Y$ is (exactly) $X\times Y$ as a discrete groupoid. Question 1: is this correct?

There is another case that I have not been able to work as successfully: given an equivalence relation $E_1$ on (set) $X$ seen as a groupoid, and equivalence relation $E_2$ on $Y$, what are $E_1 \rtimes E_2$ and $E_2 \rtimes E_1$ ?

I am trying to understand the semidirect product of groupoids, as defined in this answer by Theo Johnson-Freyd. Part of my difficulty is that although the definition of a 2-group makes sense, I am quite unfamiliar with that concept. So, as a test, I 'computed' the crossed product of two (arbitrary) sets $X,Y$ seen as discrete groupoids. My understanding is that the 2-group of automorphism in this case is the trivial monoidal category $(\mathbf{1}, \otimes, \ast)$, with $\mathbf{1}$ the trivial 1-object category, $\otimes:\mathbf{1}\times\mathbf{1}\rightarrow\mathbf{1}$, and unit the unique object $\ast$.

The object set of $X \rtimes Y$ is then $X \times Y$, and there is an arrow $(x_1,y_1) \rightarrow (x_2,y_2)$ iff $x_1=y_1$ and $x_2=y_2$, a long winded way of saying that in this particular case, $X \rtimes Y$ is (exactly) $X\times Y$ as a discrete groupoid. Question 1: is this correct?

There is another case that I have not been able to work as successfully: given an equivalence relation $E_1$ on (set) $X$ seen as a groupoid, and equivalence relation $E_2$ on $Y$, what are $E_1 \rtimes E_2$ and $E_2 \rtimes E_1$ ?

[Edit: I meant semidirect product, not crossed]