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One of the most famous examples of higher Artin stacks is the stack of perfect complexes. I recall here the basic stuff:

We fix a function $b: \mathbb{Z} \rightarrow \mathbb{N}$ which is zero outside of finitely many values, and we let $\operatorname{Perf}^{\leq b} $ be the higher stack whose value on a scheme $X$ is the $\infty$ -category of perfect complexes $C$ over $X$ such that at any point $x \in X$, we have $h^{i}\left(C_{x}\right) \leq b(i) .$ So $\operatorname{Perf}^{\leq b} $ is an $n$ -stack, where $n$ is the length of the interval on which $b$ is nonzero.

It is proven that the Buchsbaum–Eisenbud schemes (or similarly the varieties of complexes) provide an atlas for it, so this is stack turns to be the quotient of these varieties of complexes by the equivalence relation identifying two complexes which are quasi-isomorphic.

My question is basically: what are geometrically the $n$-(auto)morphisms in this stack?

I guess that we have that chain complexes as objects, chain maps as $1$-morphisms, chain homotopies as $2$-morphisms, chain homotopies between chain homotopies as 3-morphisms (and so on...)

EDIT 1

It seems according to this, this and this paper, each point of the stack would be a complex $C$ with automorphism and higher automorphisms groups (in which the $n$-(auto)morphims that I wonder about live in) equal to:

$\pi_{1}\left(\operatorname{Perf}^{\leq b} , C\right) \simeq A u t(C) \quad \pi_{i}\left(\operatorname{Perf}^{\leq b} ,C\right) \simeq E x t^{1-i}(C,C)\: n > i>1$

which reflects the fact that "morphisms between complexes of amplitude in $[a, b]$ has homotopies and higher homotopies up to degree $n-1$, or equivalently that the $\infty$-category of complexes of amplitude in $[a, b]$ has $(n-1)$ -truncated mapping spaces."

Now according to the following defnition:

I think we get $E x t^{j}(C,C)=\operatorname {Hom}\left ( C , C[j] \right ), 0>j>-n$.

Why these maps between shifted complexes econde higher automorphisms of $E$? There should be some $-j$-simplex around there but I don't know how to see them (my knowledge of Homotopical Algebra is not big)

EDIT 2

A good explanation of the higher homotopy groups in terms of the $\operatorname{Ext}$ groups is already here. Anyway, it would be interesting to see how these homotopy groups reflect themselves in the higher automorphims of the objects that provide the atlas to the stack, i.e. the varieties of complexes.

One of the most famous examples of higher Artin stacks is the stack of perfect complexes. I recall here the basic stuff:

We fix a function $b: \mathbb{Z} \rightarrow \mathbb{N}$ which is zero outside of finitely many values, and we let $\operatorname{Perf}^{\leq b} $ be the higher stack whose value on a scheme $X$ is the $\infty$ -category of perfect complexes $C$ over $X$ such that at any point $x \in X$, we have $h^{i}\left(C_{x}\right) \leq b(i) .$ So $\operatorname{Perf}^{\leq b} $ is an $n$ -stack, where $n$ is the length of the interval on which $b$ is nonzero.

It is proven that the Buchsbaum–Eisenbud schemes (or similarly the varieties of complexes) provide an atlas for it, so this is stack turns to be the quotient of these varieties of complexes by the equivalence relation identifying two complexes which are quasi-isomorphic.

My question is basically: what are geometrically the $n$-(auto)morphisms in this stack?

I guess that we have that chain complexes as objects, chain maps as $1$-morphisms, chain homotopies as $2$-morphisms, chain homotopies between chain homotopies as 3-morphisms (and so on...)

EDIT 1

It seems according to this, this and this paper, each point of the stack would be a complex $C$ with automorphism and higher automorphisms groups (in which the $n$-(auto)morphims that I wonder about live in) equal to:

$\pi_{1}\left(\operatorname{Perf}^{\leq b} , C\right) \simeq A u t(C) \quad \pi_{i}\left(\operatorname{Perf}^{\leq b} ,C\right) \simeq E x t^{1-i}(C,C)\: n > i>1$

which reflects the fact that "morphisms between complexes of amplitude in $[a, b]$ has homotopies and higher homotopies up to degree $n-1$, or equivalently that the $\infty$-category of complexes of amplitude in $[a, b]$ has $(n-1)$ -truncated mapping spaces."

Now according to the following defnition:

I think we get $E x t^{j}(C,C)=\operatorname {Hom}\left ( C , C[j] \right ), 0>j>-n$.

Why these maps between shifted complexes econde higher automorphisms of $E$? There should be some $-j$-simplex around there but I don't know how to see them (my knowledge of Homotopical Algebra is not big)

EDIT 2

A good explanation of the higher homotopy groups in terms of the $\operatorname{Ext}$ groups is already here. Anyway, it would be interesting to see how these homotopy groups reflect themselves in the higher automorphims of the objects that provide the atlas to the stack, i.e. the varieties of complexes.

One of the most famous examples of higher Artin stacks is the stack of perfect complexes. I recall here the basic stuff:

We fix a function $b: \mathbb{Z} \rightarrow \mathbb{N}$ which is zero outside of finitely many values, and we let $\operatorname{Perf}^{\leq b} $ be the higher stack whose value on a scheme $X$ is the $\infty$ -category of perfect complexes $C$ over $X$ such that at any point $x \in X$, we have $h^{i}\left(C_{x}\right) \leq b(i) .$ So $\operatorname{Perf}^{\leq b} $ is an $n$ -stack, where $n$ is the length of the interval on which $b$ is nonzero.

It is proven that the Buchsbaum–Eisenbud schemes (or similarly the varieties of complexes) provide an atlas for it, so this is stack turns to be the quotient of these varieties of complexes by the equivalence relation identifying two complexes which are quasi-isomorphic.

My question is basically: what are geometrically the $n$-(auto)morphisms in this stack?

I guess that we have that chain complexes as objects, chain maps as $1$-morphisms, chain homotopies as $2$-morphisms, chain homotopies between chain homotopies as 3-morphisms (and so on...)

EDIT: It seems according to this, this and this paper, each point of the stack would be a complex $C$ with automorphism and higher automorphisms groups (in which the $n$-(auto)morphims that I wonder about live in) equal to:

$\pi_{1}\left(\operatorname{Perf}^{\leq b} , C\right) \simeq A u t(C) \quad \pi_{i}\left(\operatorname{Perf}^{\leq b} ,C\right) \simeq E x t^{1-i}(C,C)\: n > i>1$

which reflects the fact that "morphisms between complexes of amplitude in $[a, b]$ has homotopies and higher homotopies up to degree $n-1$, or equivalently that the $\infty$-category of complexes of amplitude in $[a, b]$ has $(n-1)$ -truncated mapping spaces."

Now according to the following defnition:

I think we get $E x t^{j}(C,C)=\operatorname {Hom}\left ( C , C[j] \right ), 0>j>-n$.

Why these maps between shifted complexes econde higher automorphisms of $E$? There should be some $-j$-simplex around there but I don't know how to see them (my knowledge of Homotopical Algebra is not big)

One of the most famous examples of higher Artin stacks is the stack of perfect complexes. I recall here the basic stuff:

We fix a function $b: \mathbb{Z} \rightarrow \mathbb{N}$ which is zero outside of finitely many values, and we let $\operatorname{Perf}^{\leq b} $ be the higher stack whose value on a scheme $X$ is the $\infty$ -category of perfect complexes $C$ over $X$ such that at any point $x \in X$, we have $h^{i}\left(C_{x}\right) \leq b(i) .$ So $\operatorname{Perf}^{\leq b} $ is an $n$ -stack, where $n$ is the length of the interval on which $b$ is nonzero.

It is proven that the Buchsbaum–Eisenbud schemes (or similarly the varieties of complexes) provide an atlas for it, so this is stack turns to be the quotient of these varieties of complexes by the equivalence relation identifying two complexes which are quasi-isomorphic.

My question is basically: what are geometrically the $n$-(auto)morphisms in this stack?

I guess that we have that chain complexes as objects, chain maps as $1$-morphisms, chain homotopies as $2$-morphisms, chain homotopies between chain homotopies as 3-morphisms (and so on...)

EDIT 1 

It seems according to this, this and this paper, each point of the stack would be a complex $C$ with automorphism and higher automorphisms groups (in which the $n$-(auto)morphims that I wonder about live in) equal to:

$\pi_{1}\left(\operatorname{Perf}^{\leq b} , C\right) \simeq A u t(C) \quad \pi_{i}\left(\operatorname{Perf}^{\leq b} ,C\right) \simeq E x t^{1-i}(C,C)\: n > i>1$

which reflects the fact that "morphisms between complexes of amplitude in $[a, b]$ has homotopies and higher homotopies up to degree $n-1$, or equivalently that the $\infty$-category of complexes of amplitude in $[a, b]$ has $(n-1)$ -truncated mapping spaces."

Now according to the following defnition:

I think we get $E x t^{j}(C,C)=\operatorname {Hom}\left ( C , C[j] \right ), 0>j>-n$.

Why these maps between shifted complexes econde higher automorphisms of $E$? There should be some $-j$-simplex around there but I don't know how to see them (my knowledge of Homotopical Algebra is not big)

EDIT 2

A good explanation of the higher homotopy groups in terms of the $\operatorname{Ext}$ groups is already here. Anyway, it would be interesting to see how these homotopy groups reflect themselves in the higher automorphims of the objects that provide the atlas to the stack, i.e. the varieties of complexes.

One of the most famous examples of higher Artin stacks is the stack of perfect complexes. I recall here the basic stuff:

We fix a function $b: \mathbb{Z} \rightarrow \mathbb{N}$ which is zero outside of finitely many values, and we let $\operatorname{Perf}^{\leq b} $ be the higher stack whose value on a scheme $X$ is the $\infty$ -category of perfect complexes $C$ over $X$ such that at any point $x \in X$, we have $h^{i}\left(C_{x}\right) \leq b(i) .$ So $\operatorname{Perf}^{\leq b} $ is an $n$ -stack, where $n$ is the length of the interval on which $b$ is nonzero.

It is proven that the Buchsbaum–Eisenbud schemes (or similarly the varieties of complexes) provide an atlas for it, so this is stack turns to be the quotient of these varieties of complexes by the equivalence relation identifying two complexes which are quasi-isomorphic.

My question is basically: what are geometrically the $n$-(auto)morphisms in this stack?

I guess that we have that chain complexes as objects, chain maps as $1$-morphisms, chain homotopies as $2$-morphisms, chain homotopies between chain homotopies as 3-morphisms (and so on...)

EDIT: It seems according to this, this and this paper, each point of the stack would be a complex $C$ with automorphism and higher automorphisms groups (in which the $n$-(auto)morphims that I wonder about live in) equal to:

$\pi_{1}\left(\operatorname{Perf}^{\leq b} , C\right) \simeq A u t(C) \quad \pi_{i}\left(\operatorname{Perf}^{\leq b} ,C\right) \simeq E x t^{1-i}(C,C)\: n > i>1$

which reflects the fact that "morphisms between complexes of amplitude in $[a, b]$ has homotopies and higher homotopies up to degree $n-1$, or equivalently that the $\infty$-category of complexes of amplitude in $[a, b]$ has $(n-1)$ -truncated mapping spaces." (cause according to this $ E x t^{1-i}(C,C)=Hom\left ( H^{i}(C) , H^{1}(C)\right )$ can be nonzero only for $1<i<n$)

My understanding is that these $Hom\left ( H^{i}(C) , H^{1}(C)\right ), i>1$ should correspond to higher mapping spaces but I don't know the exact form of them and specially I don't know how this reflects in terms of the $n$-atlas (the varieties of complexes)

One of the most famous examples of higher Artin stacks is the stack of perfect complexes. I recall here the basic stuff:

We fix a function $b: \mathbb{Z} \rightarrow \mathbb{N}$ which is zero outside of finitely many values, and we let $\operatorname{Perf}^{\leq b} $ be the higher stack whose value on a scheme $X$ is the $\infty$ -category of perfect complexes $C$ over $X$ such that at any point $x \in X$, we have $h^{i}\left(C_{x}\right) \leq b(i) .$ So $\operatorname{Perf}^{\leq b} $ is an $n$ -stack, where $n$ is the length of the interval on which $b$ is nonzero.

It is proven that the Buchsbaum–Eisenbud schemes (or similarly the varieties of complexes) provide an atlas for it, so this is stack turns to be the quotient of these varieties of complexes by the equivalence relation identifying two complexes which are quasi-isomorphic.

My question is basically: what are geometrically the $n$-(auto)morphisms in this stack?

I guess that we have that chain complexes as objects, chain maps as $1$-morphisms, chain homotopies as $2$-morphisms, chain homotopies between chain homotopies as 3-morphisms (and so on...)

EDIT: It seems according to this, this and this paper, each point of the stack would be a complex $C$ with automorphism and higher automorphisms groups (in which the $n$-(auto)morphims that I wonder about live in) equal to:

$\pi_{1}\left(\operatorname{Perf}^{\leq b} , C\right) \simeq A u t(C) \quad \pi_{i}\left(\operatorname{Perf}^{\leq b} ,C\right) \simeq E x t^{1-i}(C,C)\: n > i>1$

which reflects the fact that "morphisms between complexes of amplitude in $[a, b]$ has homotopies and higher homotopies up to degree $n-1$, or equivalently that the $\infty$-category of complexes of amplitude in $[a, b]$ has $(n-1)$ -truncated mapping spaces."

Now according to the following defnition:

I think we get $E x t^{j}(C,C)=\operatorname {Hom}\left ( C , C[j] \right ), 0>j>-n$.

Why these maps between shifted complexes econde higher automorphisms of $E$? There should be some $-j$-simplex around there but I don't know how to see them (my knowledge of Homotopical Algebra is not big)