This begs the question: What's a "graded category"? A graded category is a category $\mathcal{C}$ enriched in $(\mathcal{V},\otimes)$ that is equipped with a decomposition $\mathcal{C}(x,y) = \bigoplus_{q\in Q} \mathcal{C}_q(x,y)$ such that $\operatorname{id}_x \in \mathcal{C}_1(x,x)$ and composition of morphisms decomposes accordingly as $\circ: \mathcal{C}_p \otimes \mathcal{C}_q \to \mathcal{C}_{pq}$.
In particular: Considering only the morphisms in $\mathcal{C}_1$ gives us another category (with the same set of objects).
A graded category is crossed if every $\mathcal{C}_q(x,x)$ contains an isomorphism.
Every group extension $1\to K\to G\to Q\to 1$ gives a $Q$-grading on the one-object category $G$ if we let $(\mathcal{V},\oplus,\otimes)$ be $(\mathsf{Set},\sqcup,\times)$, namely the decomposition into cosets of $K$. These gradings are always crossed.
A graded $k$-algebra $A=\bigoplus_{q\in Q}$ gives a $Q$-grading on the one-object category $A$ if we let $(\mathcal{V},\oplus,\otimes)$ be $(k\mathsf{-mod},\oplus,\otimes)$. These gradings are crossed iff they are crossed in the usual sense, i.e. if all $A_q$ contain at least one unit.
A crossed $Q$-graded category induces an outer action $Q\to\operatorname{Out}_{\mathcal{V}\mathsf{-cat}}(\mathcal{C}_1)$ by choosing 1-isomorphisms $u_{q,x}\in \mathcal{C}_q(x,x)$ for every $x\in Ob(\mathcal{C})$ and $q\in Q$ and mapping $q$ to the equivalence class $[\alpha_q]\in\operatorname{Out}(\mathcal{C}_1)$ of the "conjugation"-functor $\alpha_q: \mathcal{C}_1\to\mathcal{C}_1, (x\xleftarrow{f}y) \mapsto (x \xleftarrow{u_{q,x} \circ f \circ u_{q,y}^{-1}} y)$.
- For group extensions $1\to K\to G\to Q\to 1$ this is exactly the outer action $Q\to\operatorname{Out}(K)$ induced by conjugation.
- For crossed, $Q$-graded algebras this is also the outer action $Q\to\operatorname{Out}(A_1)$ induced by conjugation.
Now, what precisely does the theorem do? For every fixed group homomorphism $\omega: Q \to \pi_1(\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1))$ it establishes really two pairs bijection between the following three sets:
$Q$-graded, crossed categories extending $\mathcal{C}_1$ that induce $\omega$
$$\uparrow \downarrow$$
2-group morphisms $\mathscr{Q}\to\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1)$ up to equivalence that induce $\omega$ on the $\pi_1$-groups, where $\mathscr{Q}$ is the 2-group upgrade of $Q$.
$$\uparrow \downarrow$$
non-abelian cohomology $H^2(Q,\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1),\omega)$, i.e. 2-cocycles $(\alpha,\chi)$ up to 2-coboundaries
- $(\alpha,\chi)$ being a 2-cocycle means $\alpha: Q \to \{1\text{-morphisms}\}, \chi: Q^2 \to \{2\text{-morphisms}\}$ with $\chi_{p,q}: \alpha_{p} \alpha_{q} \Rightarrow \alpha_{pq}$, $\alpha_q$ is in the equivalence class $\omega(q)$, and the 2-coycle condition holds: $\chi_{xy,z}\circ\chi_{x,y} = \chi_{x,yz}\circ(\alpha_x(\chi_{y,z}))\circ\text{associator}$ as 2-morphisms $(\alpha_x\circ\alpha_y)\circ\alpha_z \Rightarrow \alpha_{xyz}$
- A 2-coboundary $(\alpha,\chi) \to (\alpha',\chi')$ is map $\lambda: Q\to\{2\text{-morphisms}\}$ such that $\lambda_x: \alpha_x \Rightarrow \alpha'_x$ with $\lambda_{xy}\circ\chi_{x,y} = \chi'_{x,y}\circ(\lambda_x\lambda_y)$ as 2-morphism $\alpha_x \circ \alpha_x \Rightarrow \alpha'_{xy}$.
And finally: How does one prove all of that? If I have not made any huge mistakes, the classical proofs from Schreier theory all carry over if one replaces all equations by the appropriate commutative diagrams.
There is even a regular action of the group $H^2(Q,Z(\mathcal{C}_1)^\times)$ on the set $H^2(Q,\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1),\omega)$ if it is non-empty given by $[\zeta] . [\alpha,\chi] := [\alpha, \zeta.\!\chi]$ where $\zeta.\!\chi$ means the 2-morphism $\alpha_p\circ\alpha_q \overset{\chi_{p,q}}{\Longrightarrow} \alpha_{pq} \overset{1_{\alpha_{pq}} \ast \zeta_{p,q}}{\Longrightarrow} \alpha_{pq}$
This begs the question: What's a "graded category"? A graded category is a category $\mathcal{C}$ enriched in $(\mathcal{V},\otimes)$ that is equipped with a decomposition $\mathcal{C}(x,y) = \bigoplus_{q\in Q} \mathcal{C}_q(x,y)$ such that $\operatorname{id}_x \in \mathcal{C}_1(x,x)$ and composition of morphisms decomposes accordingly as $\circ: \mathcal{C}_p \otimes \mathcal{C}_q \to \mathcal{C}_{pq}$.
In particular: Considering only the morphisms in $\mathcal{C}_1$ gives us another category (with the same set of objects).
A graded category is crossed if every $\mathcal{C}_q(x,x)$ contains an isomorphism.
Every group extension $1\to K\to G\to Q\to 1$ gives a $Q$-grading on the one-object category $G$ if we let $(\mathcal{V},\oplus,\otimes)$ be $(\mathsf{Set},\sqcup,\times)$, namely the decomposition into cosets of $K$. These gradings are always crossed.
A graded $k$-algebra $A=\bigoplus_{q\in Q}$ gives a $Q$-grading on the one-object category $A$ if we let $(\mathcal{V},\oplus,\otimes)$ be $(k\mathsf{-mod},\oplus,\otimes)$. These gradings are crossed iff they are crossed in the usual sense, i.e. if all $A_q$ contain at least one unit.
A crossed $Q$-graded category induces an outer action $Q\to\operatorname{Out}_{\mathcal{V}\mathsf{-cat}}(\mathcal{C}_1)$ by choosing 1-isomorphisms $u_{q,x}\in \mathcal{C}_q(x,x)$ for every $x\in Ob(\mathcal{C})$ and $q\in Q$ and mapping $q$ to the equivalence class $[\alpha_q]\in\operatorname{Out}(\mathcal{C}_1)$ of the "conjugation"-functor $\alpha_q: \mathcal{C}_1\to\mathcal{C}_1, (x\xleftarrow{f}y) \mapsto (x \xleftarrow{u_{q,x} \circ f \circ u_{q,y}^{-1}} y)$.
- For group extensions $1\to K\to G\to Q\to 1$ this is exactly the outer action $Q\to\operatorname{Out}(K)$ induced by conjugation.
- For crossed, $Q$-graded algebras this is also the outer action $Q\to\operatorname{Out}(A_1)$ induced by conjugation.
Now, what precisely does the theorem do? For every fixed group homomorphism $\omega: Q \to \pi_1(\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1))$ it establishes really two pairs bijection between the following three sets:
$Q$-graded, crossed categories extending $\mathcal{C}_1$ that induce $\omega$
$$\uparrow \downarrow$$
2-group morphisms $\mathscr{Q}\to\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1)$ up to equivalence that induce $\omega$ on the $\pi_1$-groups, where $\mathscr{Q}$ is the 2-group upgrade of $Q$.
$$\uparrow \downarrow$$
non-abelian cohomology $H^2(Q,\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1),\omega)$, i.e. 2-cocycles $(\alpha,\chi)$ up to 2-coboundaries
- $(\alpha,\chi)$ being a 2-cocycle means $\alpha: Q \to \{1\text{-morphisms}\}, \chi: Q^2 \to \{2\text{-morphisms}\}$ with $\chi_{p,q}: \alpha_{p} \alpha_{q} \Rightarrow \alpha_{pq}$, $\alpha_q$ is in the equivalence class $\omega(q)$, and the 2-coycle condition holds: $\chi_{xy,z}\circ\chi_{x,y} = \chi_{x,yz}\circ(\alpha_x(\chi_{y,z}))\circ\text{associator}$ as 2-morphisms $(\alpha_x\circ\alpha_y)\circ\alpha_z \Rightarrow \alpha_{xyz}$
- A 2-coboundary $(\alpha,\chi) \to (\alpha',\chi')$ is map $\lambda: Q\to\{2\text{-morphisms}\}$ such that $\lambda_x: \alpha_x \Rightarrow \alpha'_x$ with $\lambda_{xy}\circ\chi_{x,y} = \chi'_{x,y}\circ(\lambda_x\lambda_y)$ as 2-morphism $\alpha_x \circ \alpha_x \Rightarrow \alpha'_{xy}$.
And finally: How does one prove all of that? If I have not made any huge mistakes, the classical proofs from Schreier theory all carry over if one replaces all equations by the appropriate commutative diagrams.
This begs the question: What's a "graded category"? A graded category is a category $\mathcal{C}$ enriched in $(\mathcal{V},\otimes)$ that is equipped with a decomposition $\mathcal{C}(x,y) = \bigoplus_{q\in Q} \mathcal{C}_q(x,y)$ such that $\operatorname{id}_x \in \mathcal{C}_1(x,x)$ and composition of morphisms decomposes accordingly as $\circ: \mathcal{C}_p \otimes \mathcal{C}_q \to \mathcal{C}_{pq}$.
In particular: Considering only the morphisms in $\mathcal{C}_1$ gives us another category (with the same set of objects).
A graded category is crossed if every $\mathcal{C}_q(x,x)$ contains an isomorphism.
Every group extension $1\to K\to G\to Q\to 1$ gives a $Q$-grading on the one-object category $G$ if we let $(\mathcal{V},\oplus,\otimes)$ be $(\mathsf{Set},\sqcup,\times)$, namely the decomposition into cosets of $K$. These gradings are always crossed.
A graded $k$-algebra $A=\bigoplus_{q\in Q}$ gives a $Q$-grading on the one-object category $A$ if we let $(\mathcal{V},\oplus,\otimes)$ be $(k\mathsf{-mod},\oplus,\otimes)$. These gradings are crossed iff they are crossed in the usual sense, i.e. if all $A_q$ contain at least one unit.
A crossed $Q$-graded category induces an outer action $Q\to\operatorname{Out}_{\mathcal{V}\mathsf{-cat}}(\mathcal{C}_1)$ by choosing 1-isomorphisms $u_{q,x}\in \mathcal{C}_q(x,x)$ for every $x\in Ob(\mathcal{C})$ and $q\in Q$ and mapping $q$ to the equivalence class $[\alpha_q]\in\operatorname{Out}(\mathcal{C}_1)$ of the "conjugation"-functor $\alpha_q: \mathcal{C}_1\to\mathcal{C}_1, (x\xleftarrow{f}y) \mapsto (x \xleftarrow{u_{q,x} \circ f \circ u_{q,y}^{-1}} y)$.
- For group extensions $1\to K\to G\to Q\to 1$ this is exactly the outer action $Q\to\operatorname{Out}(K)$ induced by conjugation.
- For crossed, $Q$-graded algebras this is also the outer action $Q\to\operatorname{Out}(A_1)$ induced by conjugation.
Now, what precisely does the theorem do? For every fixed group homomorphism $\omega: Q \to \pi_1(\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1))$ it establishes really two pairs bijection between the following three sets:
$Q$-graded, crossed categories extending $\mathcal{C}_1$ that induce $\omega$
$$\uparrow \downarrow$$
2-group morphisms $\mathscr{Q}\to\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1)$ up to equivalence that induce $\omega$ on the $\pi_1$-groups, where $\mathscr{Q}$ is the 2-group upgrade of $Q$.
$$\uparrow \downarrow$$
non-abelian cohomology $H^2(Q,\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1),\omega)$, i.e. 2-cocycles $(\alpha,\chi)$ up to 2-coboundaries
- $(\alpha,\chi)$ being a 2-cocycle means $\alpha: Q \to \{1\text{-morphisms}\}, \chi: Q^2 \to \{2\text{-morphisms}\}$ with $\chi_{p,q}: \alpha_{p} \alpha_{q} \Rightarrow \alpha_{pq}$, $\alpha_q$ is in the equivalence class $\omega(q)$, and the 2-coycle condition holds: $\chi_{xy,z}\circ\chi_{x,y} = \chi_{x,yz}\circ(\alpha_x(\chi_{y,z}))\circ\text{associator}$ as 2-morphisms $(\alpha_x\circ\alpha_y)\circ\alpha_z \Rightarrow \alpha_{xyz}$
- A 2-coboundary $(\alpha,\chi) \to (\alpha',\chi')$ is map $\lambda: Q\to\{2\text{-morphisms}\}$ such that $\lambda_x: \alpha_x \Rightarrow \alpha'_x$ with $\lambda_{xy}\circ\chi_{x,y} = \chi'_{x,y}\circ(\lambda_x\lambda_y)$ as 2-morphism $\alpha_x \circ \alpha_x \Rightarrow \alpha'_{xy}$.
And finally: How does one prove all of that? If I have not made any huge mistakes, the classical proofs from Schreier theory all carry over if one replaces all equations by the appropriate commutative diagrams.
There is even a regular action of the group $H^2(Q,Z(\mathcal{C}_1)^\times)$ on the set $H^2(Q,\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1),\omega)$ if it is non-empty given by $[\zeta] . [\alpha,\chi] := [\alpha, \zeta.\!\chi]$ where $\zeta.\!\chi$ means the 2-morphism $\alpha_p\circ\alpha_q \overset{\chi_{p,q}}{\Longrightarrow} \alpha_{pq} \overset{1_{\alpha_{pq}} \ast \zeta_{p,q}}{\Longrightarrow} \alpha_{pq}$
This begs the question: What's a "graded category"? A graded category is a category $\mathcal{C}$ enriched in $(\mathcal{V},\otimes)$ that is equipped with a decomposition $\mathcal{C}(x,y) = \bigoplus_{q\in Q} \mathcal{C}_q(x,y)$ such that $\operatorname{id}_x \in \mathcal{C}_1(x,x)$ and composition of morphisms decomposes accordingly as $\circ: \mathcal{C}_p \otimes \mathcal{C}_q \to \mathcal{C}_{pq}$.
In particular: Considering only the morphisms in $\mathcal{C}_1$ gives us another category (with the same set of objects).
A graded category is crossed if every $\mathcal{C}_q(x,x)$ contains an isomorphism.
Every group extension $1\to K\to G\to Q\to 1$ gives a $Q$-grading on the one-object category $G$ if we let $(\mathcal{V},\oplus,\otimes)$ be $(\mathsf{Set},\sqcup,\times)$, namely the decomposition into cosets of $K$. These gradings are always crossed.
A graded $k$-algebra $A=\bigoplus_{q\in Q}$ gives a $Q$-grading on the one-object category $A$ if we let $(\mathcal{V},\oplus,\otimes)$ be $(k\mathsf{-mod},\oplus,\otimes)$. These gradings are crossed iff they are crossed in the usual sense, i.e. if all $A_q$ contain at least one unit.
A crossed $Q$-graded category induces an outer action $Q\to\operatorname{Out}_{\mathcal{V}\mathsf{-cat}}(\mathcal{C}_1)$ by choosing 1-isomorphisms $u_{q,x}\in \mathcal{C}_q(x,x)$ for every $x\in Ob(\mathcal{C})$ and $q\in Q$ and mapping $q$ to the equivalence class $[\alpha_q]\in\operatorname{Out}(\mathcal{C}_1)$ of the "conjugation"-functor $\alpha_q: \mathcal{C}_1\to\mathcal{C}_1, (x\xleftarrow{f}y) \mapsto (x \xleftarrow{u_{q,x} \circ f \circ u_{q,y}^{-1}} y)$.
- For group extensions $1\to K\to G\to Q\to 1$ this is exactly the outer action $Q\to\operatorname{Out}(K)$ induced by conjugation.
- For crossed, $Q$-graded algebras this is also the outer action $Q\to\operatorname{Out}(A_1)$ induced by conjugation.
Now, what precisely does the theorem do? For every fixed group homomorphism $\omega: Q \to \pi_1(\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1))$ it establishes really two pairs bijection between the following three sets:
$Q$-graded, crossed categories extending $\mathcal{C}_1$ that induce $\omega$
$$\uparrow \downarrow$$
2-group morphisms $\mathscr{Q}\to\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1)$ up to equivalence that induce $\omega$ on the $\pi_1$-groups, where $\mathscr{Q}$ is the 2-group upgrade of $Q$.
$$\uparrow \downarrow$$
non-abelian cohomology $H^2(Q,\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1),\omega)$, i.e. 2-cocycles $(\alpha,\chi)$ up to 2-coboundaries
- $(\alpha,\chi)$ being a 2-cocycle means $\alpha: Q \to \{1\text{-morphisms}\}, \chi: Q^2 \to \{2\text{-morphisms}\}$ with $\chi_{p,q}: \alpha_{p} \alpha_{q} \Rightarrow \alpha_{pq}$, $\alpha_q$ is in the equivalence class $\omega(q)$, and the 2-coycle condition holds: $\chi_{xy,z}\circ\chi_{x,y} = \chi_{x,yz}\circ(\alpha_x(\chi_{y,z}))\circ\text{associator}$ as 2-morphisms $(\alpha_x\circ\alpha_y)\circ\alpha_z \Rightarrow \alpha_{xyz}$
- A 2-coboundary $(\alpha,\chi) \to (\alpha',\chi')$ is map $\lambda: Q\to\{2\text{-morphism}\}$$\lambda: Q\to\{2\text{-morphisms}\}$ such that $\lambda_x: \alpha_x \Rightarrow \alpha'_x$ with $\lambda_{xy}\circ\chi_{x,y} = \chi'_{x,y}\circ(\lambda_x\lambda_y)$ as 2-morphism $\alpha_x \circ \alpha_x \Rightarrow \alpha'_{xy}$.
And finally: How does one prove all of that? If I have not made any huge mistakes, the classical proofs from Schreier theory all carry over if one replaces all equations by the appropriate commutative diagrams.
This begs the question: What's a "graded category"? A graded category is a category $\mathcal{C}$ enriched in $(\mathcal{V},\otimes)$ that is equipped with a decomposition $\mathcal{C}(x,y) = \bigoplus_{q\in Q} \mathcal{C}_q(x,y)$ such that $\operatorname{id}_x \in \mathcal{C}_1(x,x)$ and composition of morphisms decomposes accordingly as $\circ: \mathcal{C}_p \otimes \mathcal{C}_q \to \mathcal{C}_{pq}$.
In particular: Considering only the morphisms in $\mathcal{C}_1$ gives us another category (with the same set of objects).
A graded category is crossed if every $\mathcal{C}_q(x,x)$ contains an isomorphism.
Every group extension $1\to K\to G\to Q\to 1$ gives a $Q$-grading on the one-object category $G$ if we let $(\mathcal{V},\oplus,\otimes)$ be $(\mathsf{Set},\sqcup,\times)$, namely the decomposition into cosets of $K$. These gradings are always crossed.
A graded $k$-algebra $A=\bigoplus_{q\in Q}$ gives a $Q$-grading on the one-object category $A$ if we let $(\mathcal{V},\oplus,\otimes)$ be $(k\mathsf{-mod},\oplus,\otimes)$. These gradings are crossed iff they are crossed in the usual sense, i.e. if all $A_q$ contain at least one unit.
A crossed $Q$-graded category induces an outer action $Q\to\operatorname{Out}_{\mathcal{V}\mathsf{-cat}}(\mathcal{C}_1)$ by choosing 1-isomorphisms $u_{q,x}\in \mathcal{C}_q(x,x)$ for every $x\in Ob(\mathcal{C})$ and $q\in Q$ and mapping $q$ to the equivalence class $[\alpha_q]\in\operatorname{Out}(\mathcal{C}_1)$ of the "conjugation"-functor $\alpha_q: \mathcal{C}_1\to\mathcal{C}_1, (x\xleftarrow{f}y) \mapsto (x \xleftarrow{u_{q,x} \circ f \circ u_{q,y}^{-1}} y)$.
- For group extensions $1\to K\to G\to Q\to 1$ this is exactly the outer action $Q\to\operatorname{Out}(K)$ induced by conjugation.
- For crossed, $Q$-graded algebras this is also the outer action $Q\to\operatorname{Out}(A_1)$ induced by conjugation.
Now, what precisely does the theorem do? For every fixed group homomorphism $\omega: Q \to \pi_1(\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1))$ it establishes really two pairs bijection between the following three sets:
$Q$-graded, crossed categories extending $\mathcal{C}_1$ that induce $\omega$
$$\uparrow \downarrow$$
2-group morphisms $\mathscr{Q}\to\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1)$ up to equivalence that induce $\omega$ on the $\pi_1$-groups, where $\mathscr{Q}$ is the 2-group upgrade of $Q$.
$$\uparrow \downarrow$$
non-abelian cohomology $H^2(Q,\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1),\omega)$, i.e. 2-cocycles $(\alpha,\chi)$ up to 2-coboundaries
- $(\alpha,\chi)$ being a 2-cocycle means $\alpha: Q \to \{1\text{-morphisms}\}, \chi: Q^2 \to \{2\text{-morphisms}\}$ with $\chi_{p,q}: \alpha_{p} \alpha_{q} \Rightarrow \alpha_{pq}$, $\alpha_q$ is in the equivalence class $\omega(q)$, and the 2-coycle condition holds: $\chi_{xy,z}\circ\chi_{x,y} = \chi_{x,yz}\circ(\alpha_x(\chi_{y,z}))\circ\text{associator}$ as 2-morphisms $(\alpha_x\circ\alpha_y)\circ\alpha_z \Rightarrow \alpha_{xyz}$
- A 2-coboundary $(\alpha,\chi) \to (\alpha',\chi')$ is map $\lambda: Q\to\{2\text{-morphism}\}$ such that $\lambda_{xy}\circ\chi_{x,y} = \chi'_{x,y}\circ(\lambda_x\lambda_y)$ as 2-morphism $\alpha_x \circ \alpha_x \Rightarrow \alpha'_{xy}$.
And finally: How does one prove all of that? If I have not made any huge mistakes, the classical proofs from Schreier theory all carry over if one replaces all equations by the appropriate commutative diagrams.
This begs the question: What's a "graded category"? A graded category is a category $\mathcal{C}$ enriched in $(\mathcal{V},\otimes)$ that is equipped with a decomposition $\mathcal{C}(x,y) = \bigoplus_{q\in Q} \mathcal{C}_q(x,y)$ such that $\operatorname{id}_x \in \mathcal{C}_1(x,x)$ and composition of morphisms decomposes accordingly as $\circ: \mathcal{C}_p \otimes \mathcal{C}_q \to \mathcal{C}_{pq}$.
In particular: Considering only the morphisms in $\mathcal{C}_1$ gives us another category (with the same set of objects).
A graded category is crossed if every $\mathcal{C}_q(x,x)$ contains an isomorphism.
Every group extension $1\to K\to G\to Q\to 1$ gives a $Q$-grading on the one-object category $G$ if we let $(\mathcal{V},\oplus,\otimes)$ be $(\mathsf{Set},\sqcup,\times)$, namely the decomposition into cosets of $K$. These gradings are always crossed.
A graded $k$-algebra $A=\bigoplus_{q\in Q}$ gives a $Q$-grading on the one-object category $A$ if we let $(\mathcal{V},\oplus,\otimes)$ be $(k\mathsf{-mod},\oplus,\otimes)$. These gradings are crossed iff they are crossed in the usual sense, i.e. if all $A_q$ contain at least one unit.
A crossed $Q$-graded category induces an outer action $Q\to\operatorname{Out}_{\mathcal{V}\mathsf{-cat}}(\mathcal{C}_1)$ by choosing 1-isomorphisms $u_{q,x}\in \mathcal{C}_q(x,x)$ for every $x\in Ob(\mathcal{C})$ and $q\in Q$ and mapping $q$ to the equivalence class $[\alpha_q]\in\operatorname{Out}(\mathcal{C}_1)$ of the "conjugation"-functor $\alpha_q: \mathcal{C}_1\to\mathcal{C}_1, (x\xleftarrow{f}y) \mapsto (x \xleftarrow{u_{q,x} \circ f \circ u_{q,y}^{-1}} y)$.
- For group extensions $1\to K\to G\to Q\to 1$ this is exactly the outer action $Q\to\operatorname{Out}(K)$ induced by conjugation.
- For crossed, $Q$-graded algebras this is also the outer action $Q\to\operatorname{Out}(A_1)$ induced by conjugation.
Now, what precisely does the theorem do? For every fixed group homomorphism $\omega: Q \to \pi_1(\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1))$ it establishes really two pairs bijection between the following three sets:
$Q$-graded, crossed categories extending $\mathcal{C}_1$ that induce $\omega$
$$\uparrow \downarrow$$
2-group morphisms $\mathscr{Q}\to\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1)$ up to equivalence that induce $\omega$ on the $\pi_1$-groups, where $\mathscr{Q}$ is the 2-group upgrade of $Q$.
$$\uparrow \downarrow$$
non-abelian cohomology $H^2(Q,\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1),\omega)$, i.e. 2-cocycles $(\alpha,\chi)$ up to 2-coboundaries
- $(\alpha,\chi)$ being a 2-cocycle means $\alpha: Q \to \{1\text{-morphisms}\}, \chi: Q^2 \to \{2\text{-morphisms}\}$ with $\chi_{p,q}: \alpha_{p} \alpha_{q} \Rightarrow \alpha_{pq}$, $\alpha_q$ is in the equivalence class $\omega(q)$, and the 2-coycle condition holds: $\chi_{xy,z}\circ\chi_{x,y} = \chi_{x,yz}\circ(\alpha_x(\chi_{y,z}))\circ\text{associator}$ as 2-morphisms $(\alpha_x\circ\alpha_y)\circ\alpha_z \Rightarrow \alpha_{xyz}$
- A 2-coboundary $(\alpha,\chi) \to (\alpha',\chi')$ is map $\lambda: Q\to\{2\text{-morphisms}\}$ such that $\lambda_x: \alpha_x \Rightarrow \alpha'_x$ with $\lambda_{xy}\circ\chi_{x,y} = \chi'_{x,y}\circ(\lambda_x\lambda_y)$ as 2-morphism $\alpha_x \circ \alpha_x \Rightarrow \alpha'_{xy}$.
And finally: How does one prove all of that? If I have not made any huge mistakes, the classical proofs from Schreier theory all carry over if one replaces all equations by the appropriate commutative diagrams.
In the meantime I found the right nlab pages to read... The answer seems to be 2-groups! Specifically automorphism 2-groups. I will write up what I have come to understand so far (though I have not checked every last detail of this):
What's a 2-group? Like an ordinary group is a category with 1 object in which every morphism is invertible, a (coherent) 2-group is a 2-category with only one object in which every 1- and 2-morphism is invertible (and we have picked a weak inverse of every 1-morphism).
Generic Example: For every 2-category $\mathscr{C}$ and every object $\mathcal{C}\in\mathscr{C}$ there is a automorphism-2-group $\operatorname{AUT}_\mathscr{C}(\mathcal{C})$ which has the self-equivalences $\mathcal{C}\to\mathcal{C}$ as 1-morphisms and natural isomorphisms between those as 2-morphisms.
This includes both the case of groups by letting $\mathscr{C}$ be the 2-category of categories and the case of $k$-algebra by letting $\mathscr{C}$ be the 2-category of $k$-linear categories.
A group $K$ is a category with one object and $\operatorname{AUT}_\mathsf{cat}(K)$ is a 2-group with $\operatorname{Aut}_{\mathsf{Grp}}(K)$ as its set of 1-morphisms and for every $h\in K$ a 2-morphism $\alpha \to \kappa_h\circ\alpha$ where $\kappa_h$ is the conjugation with $h$.
A $k$-algebra is a $k$-linear category with one object and we get its automorphism-2-group $\operatorname{AUT}_{k\mathsf{-cat}}(A)$ which similary consists of $\operatorname{Aut}_{k\mathsf{-Alg}}(A)$ and $A^\times$ as its sets of 1- and 2-morphisms respectively.
Other example: Any ordinary group $Q$ can "upgraded" to a 2-group $\mathscr{Q}$ by letting 1-morphisms be elements of $Q$ just as usual and letting 2-morphisms be only identities.
Conversely, any 2-group $\mathscr{G}$ naturally defines an ordinary group $\pi_1(\mathscr{G})$ by considering 1-morphisms up to equivalence.
In our examples
- $\pi_1(\operatorname{AUT}_{\mathsf{cat}}(K)) = \operatorname{Out}(K)$,
- $\pi_1(\operatorname{AUT}_{k\mathsf{-cat}}(A)) = \operatorname{Out}(A)$,
- $\pi_1(\mathscr{Q}) = Q$ I will write $\operatorname{Out}_\mathscr{C}(\mathcal{C})$ as shorthand for $\pi_1(\operatorname{AUT}_\mathscr{C}(\mathcal{C}))$
In addition, any 2-group $\mathscr{G}$ also defines an abelian group $\pi_2(\mathscr{G})$ consisting of the 2-isomorphisms of the identity 1-morphism (which is abelian by an Eckmann-Hilton argument) and $\pi_1$ naturally acts on $\pi_2$ by conjugation.
- In the generic example, $\pi_2(\operatorname{AUT}_\mathscr{C}(\mathcal{C})))$ is the unit group of the commutative monoid $Z(\mathcal{C})$ on which $\operatorname{Out}_\mathscr{C}(\mathcal{C})$ acts by "conjugation"
- $\pi_2(\operatorname{AUT}_\mathsf{cat}(K)) = Z(K)$ with the natural action by $\operatorname{Out}(K)$.
- Similary, $\pi_2(\operatorname{AUT}_{k\mathsf{-cat}}(A)) = Z(A)^\times$ with the natural action by $\operatorname{Out}(A)$.
- And trivially $\pi_2(\mathscr{Q}) = 1$
Where do extensions and gradings come in? A common generalisation of both theorem in my question is the following:
Let $Q$ be a group,$(\mathcal{V},\oplus,\otimes)$ be a bimonoidal category and $\mathscr{C}:=\mathcal{V}\mathsf{-cat}$ be the 2-category of categories enriched in $(\mathcal{V},\otimes)$.
Then: For every group homomorphism $\omega: Q\to\operatorname{Out}_\mathscr{C}(\mathcal{C}_1)$, the non-abelian cohomology $H^2(Q,\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1),\omega)$ is in canonical bijection with "crossed, $Q$-graded categories" $\mathcal{C}$ that have $\mathcal{C}_1$ as their degree-1-part and induced outer action $\omega$.
Furthermore: These two sets are non-empty iff a certain obstruction $o(\omega)\in H^3(Q,Z(\mathcal{C}_1)^\times)$ vanishes.
This begs the question: What's a "graded category"? A graded category is a category $\mathcal{C}$ enriched in $(\mathcal{V},\otimes)$ that is equipped with a decomposition $\mathcal{C}(x,y) = \bigoplus_{q\in Q} \mathcal{C}_q(x,y)$ such that $\operatorname{id}_x \in \mathcal{C}_1(x,x)$ and composition of morphisms decomposes accordingly as $\circ: \mathcal{C}_p \otimes \mathcal{C}_q \to \mathcal{C}_{pq}$.
In particular: Considering only the morphisms in $\mathcal{C}_1$ gives us another category (with the same set of objects).
A graded category is crossed if every $\mathcal{C}_q(x,x)$ contains an isomorphism.
Every group extension $1\to K\to G\to Q\to 1$ gives a $Q$-grading on the one-object category $G$ if we let $(\mathcal{V},\oplus,\otimes)$ be $(\mathsf{Set},\sqcup,\times)$, namely the decomposition into cosets of $K$. These gradings are always crossed.
A graded $k$-algebra $A=\bigoplus_{q\in Q}$ gives a $Q$-grading on the one-object category $A$ if we let $(\mathcal{V},\oplus,\otimes)$ be $(k\mathsf{-mod},\oplus,\otimes)$. These gradings are crossed iff they are crossed in the usual sense, i.e. if all $A_q$ contain at least one unit.
A crossed $Q$-graded category induces an outer action $Q\to\operatorname{Out}_{\mathcal{V}\mathsf{-cat}}(\mathcal{C}_1)$ by choosing 1-isomorphisms $u_{q,x}\in \mathcal{C}_q(x,x)$ for every $x\in Ob(\mathcal{C})$ and $q\in Q$ and mapping $q$ to the equivalence class $[\alpha_q]\in\operatorname{Out}(\mathcal{C}_1)$ of the "conjugation"-functor $\alpha_q: \mathcal{C}_1\to\mathcal{C}_1, (x\xleftarrow{f}y) \mapsto (x \xleftarrow{u_{q,x} \circ f \circ u_{q,y}^{-1}} y)$.
- For group extensions $1\to K\to G\to Q\to 1$ this is exactly the outer action $Q\to\operatorname{Out}(K)$ induced by conjugation.
- For crossed, $Q$-graded algebras this is also the outer action $Q\to\operatorname{Out}(A_1)$ induced by conjugation.
Now, what precisely does the theorem do? For every fixed group homomorphism $\omega: Q \to \pi_1(\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1))$ it establishes really two pairs bijection between the following three sets:
$Q$-graded, crossed categories extending $\mathcal{C}_1$ that induce $\omega$
$$\uparrow \downarrow$$
2-group morphisms $\mathscr{Q}\to\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1)$ up to equivalence that induce $\omega$ on the $\pi_1$-groups, where $\mathscr{Q}$ is the 2-group upgrade of $Q$.
$$\uparrow \downarrow$$
non-abelian cohomology $H^2(Q,\operatorname{AUT}_\mathscr{C}(\mathcal{C}_1),\omega)$, i.e. 2-cocycles $(\alpha,\chi)$ up to 2-coboundaries
- $(\alpha,\chi)$ being a 2-cocycle means $\alpha: Q \to \{1\text{-morphisms}\}, \chi: Q^2 \to \{2\text{-morphisms}\}$ with $\chi_{p,q}: \alpha_{p} \alpha_{q} \Rightarrow \alpha_{pq}$, $\alpha_q$ is in the equivalence class $\omega(q)$, and the 2-coycle condition holds: $\chi_{xy,z}\circ\chi_{x,y} = \chi_{x,yz}\circ(\alpha_x(\chi_{y,z}))\circ\text{associator}$ as 2-morphisms $(\alpha_x\circ\alpha_y)\circ\alpha_z \Rightarrow \alpha_{xyz}$
- A 2-coboundary $(\alpha,\chi) \to (\alpha',\chi')$ is map $\lambda: Q\to\{2\text{-morphism}\}$ such that $\lambda_{xy}\circ\chi_{x,y} = \chi'_{x,y}\circ(\lambda_x\lambda_y)$ as 2-morphism $\alpha_x \circ \alpha_x \Rightarrow \alpha'_{xy}$.
And finally: How does one prove all of that? If I have not made any huge mistakes, the classical proofs from Schreier theory all carry over if one replaces all equations by the appropriate commutative diagrams.