nLab
powering of ∞-toposes over ∞-groupoids -- section
Powering of -toposes over -groupoids
Powering of ∞ \infty -toposes over ∞ \infty -groupoids
We discuss how the powering of
∞
\infty
-toposes over
Grpd ∞
Grpd_\infty
is given by forming mapping stacks out of locally constant
∞
\infty
-stacks . All of the following formulas and their proofs hold verbatim also for Grothendieck toposes , as they just use general abstract properties.
Let H \mathbf{H} be an
∞
\infty
-topos
with terminal geometric morphism denoted
(1) H ⊥ ⟶ Γ ⟵ LConst Grp ∞ ,
\mathbf{H}
\underoverset
{\underset{\Gamma}{\longrightarrow}}
{\overset{LConst}{\longleftarrow}}
{\;\;\;\;\bot\;\;\;\;}
Grp_\infty
\,,
where the inverse image constructs locally constant
∞
\infty
-stacks ,
and with its internal hom (mapping stack ) adjunction denoted
(2) H ⊥ ⟶ Maps ( X , − ) ⟵ ( − ) × X H
\mathbf{H}
\underoverset
{\underset{Maps(X,-)}{\longrightarrow}}
{ \overset{ (-) \times X }{\longleftarrow} }
{\;\;\;\; \bot \;\;\;\;}
\mathbf{H}
for X ∈ H X \,\in\, \mathbf{H} .
Notice that this construction is also
∞
\infty
-functorial in the first argument: Maps ( X → f Y , A ) Maps\big( X \xrightarrow{f} Y ,\, A \big) is the morphism which under the
∞
\infty
-Yoneda lemma over H \mathbf{H} (which is large but locally small, so that the lemma does apply) corresponds to
H ( ( − ) , Maps ( X , A ) ) ≃ H ( ( − ) × X , A ) → H ( ( − ) × f , A ) H ( ( − ) × Y , A ) ≃ H ( ( − ) , Maps ( X , A ) ) .
\mathbf{H}
\big(
(-)
,\,
Maps(X,A)
\big)
\;\simeq\;
\mathbf{H}
\big(
(-) \times X
,\,
A
\big)
\xrightarrow{
\mathbf{H}
\big(
(-) \times f
,\,
A
\big)
}
\mathbf{H}
\big(
(-) \times Y
,\,
A
\big)
\;\simeq\;
\mathbf{H}
\big(
(-)
,\,
Maps(X,A)
\big)
\,.
By definition, for any S ∈ Grpd ∞ S \in Grpd_\infty and X ∈ H X \in \mathbf{H} the powering ] is the (∞,1)-limit over the diagram constant on X X
X K = lim ← K X
X^K \,=\, {\lim_\leftarrow}_K X
while the tensoring is the (∞,1)-colimit over the diagram constant on X X
K ⋅ X = lim → K X .
K \cdot X \,=\, {\lim_{\to}}_K X
\,.
Proposition
The powering of H \mathbf{H} over
Grpd ∞
Grpd_\infty
is given by the mapping stack out of the locally constant
∞
\infty
-stacks :
Grpd ∞ op × H ⟶ LConst op × id H op × H ⟶ Maps ( − , − ) H
\array{
Grpd_\infty^{op}
\times
\mathbf{H}
&
\overset{ LConst^{op} \times \mathrm{id} }{\longrightarrow}
&
\mathbf{H}^{op}
\times
\mathbf{H}
&
\overset{Maps(-,-)}{\longrightarrow}
&
\mathbf{H}
}
in that this operation has the following properties:
For all X , A ∈ H X,\,A \,\in\, \mathbf{H} and S ∈ Grpd ∞ S \,\in\, Grpd_\infty we have a natural equivalence
H ( X , Maps ( LConst ( S ) , A ) ) ≃ Grpd ∞ ( S , H ( X , A ) )
\mathbf{H}
\Big(
X
,\,
Maps
\big(
LConst(S)
,\,
A
\big)
\Big)
\;\;
\simeq
\;\;
Grpd_\infty
\Big(
S
,\,
\mathbf{H}
\big(
X
,\,
A
\big)
\Big)
In its first argument the operation
sends the terminal object (the point ) to the identity:
(3) Maps ( LConst ( * ) , X ) ≃ X
Maps
\big(
LConst(\ast)
,\,
X
\big)
\;\;
\simeq
\;\;
X
sends
∞
\infty
-colimits to
∞
\infty
-limits :
(4) Maps ( lim ⟶ LConst ( S • ) , X ) ≃ lim ⟵ Maps ( LConst ( S • ) , X ) ,
Maps
\Big(
\underset{
\longrightarrow
}{\lim}
\,
LConst\big(S_\bullet\big)
,\,
X
\Big)
\;\;
\simeq
\;\;
\underset{
\longleftarrow
}{\lim}
\,
Maps
\Big(
LConst\big(S_\bullet\big)
,\,
X
\Big)
\,,
where all equivalences shown are natural .
Proof
For the first statement to be proven, consider the following sequence of natural equivalences :
H ( X , Maps ( LConst ( S ) , A ) ) ≃ H ( X × LConst ( S ) , A ) (2) ≃ H ( LConst ( S ) , Maps ( X , A ) ) (2) ≃ Grpd ∞ ( S , Γ Maps ( X , A ) ) (1) ≃ Grpd ∞ ( S , H ( * H , Maps ( X , A ) ) ) by this Prop. ≃ Grpd ∞ ( S , H ( * H × X , A ) ) (2) ≃ Grpd ∞ ( S , H ( X , A ) )
\begin{array}{lll}
\mathbf{H}
\Big(
X
,\,
Maps
\big(
LConst(S)
,\,
A
\big)
\Big)
&
\;\simeq\;
\mathbf{H}
\big(
X
\times
LConst(S)
,\,
A
\big)
&
\text{(2) }
\\
& \;\simeq\;
\mathbf{H}
\Big(
LConst(S)
,\,
Maps
\big(
X
,\,
A
\big)
\Big)
&
\text{(2) }
\\
& \;\simeq\;
Grpd_\infty
\Big(
S
,\,
\Gamma
\,
Maps
\big(
X
,\,
A
\big)
\Big)
&
\text{ (1) }
\\
& \;\simeq\;
Grpd_\infty
\bigg(
S
,\,
\mathbf{H}
\Big(
\ast_{\mathbf{H}}
,\,
Maps
\big(
X
,\,
A
\big)
\Big)
\bigg)
&
\text{by}\;\text{<a href="https://ncatlab.org/nlab/show/terminal+geometric+morphism#DirectImageOfTerminalGeometricMoprhismIsHomOutOfTerminalObject">this Prop.</a>}
\\
& \;\simeq\;
Grpd_\infty
\Big(
S
,\,
\mathbf{H}
\big(
\ast_{\mathbf{H}}
\times
X
,\,
A
\big)
\Big)
&
\text{(2) }
\\
& \;\simeq\;
Grpd_\infty
\Big(
S
,\,
\mathbf{H}
\big(
X
,\,
A
\big)
\Big)
\end{array}
For the second statement, recall that hom-functors preserve limits in that there are natural equivalences of the form
(5) H ( lim ⟶ i , X i , lim ⟵ j , A j ) ≃ lim ⟵ i lim ⟵ j H ( X i , A j ) ,
\mathbf{H}
\Big(
\underset{\underset{i}{\longrightarrow}}{\lim}
\,,
X_i
,\,
\underset{\underset{j}{\longleftarrow}}{\lim}
\,,
A_j
\Big)
\;\;
\simeq
\;\;
\underset{\underset{i}{\longleftarrow}}{\lim}
\,
\underset{\underset{j}{\longleftarrow}}{\lim}
\,
\mathbf{H}
\Big(
X_i
,\,
A_j
\Big)
\,,
and that ∞ \infty -toposes have universal colimits , in particular that the product operation is a left adjoint (2) and hence preserves colimits :
(6) ( − ) × lim ⟶ S • ≃ lim ⟶ ( ( − ) × S • ) .
(-)
\,\times\,
\underset{{\longrightarrow}}{\lim} \, S_\bullet
\;\;
\simeq
\;\;
\underset{{\longrightarrow}}{\lim} \,
\big(
(-)
\,\times\,
S_\bullet
\big)
\,.
With this, we get the following sequences of natural equivalences :
H ( ( − ) , Maps ( lim ⟶ LConst ( S • ) , X ) ) ≃ H ( ( − ) × lim ⟶ LConst ( S • ) , X ) (2) ≃ H ( lim ⟶ ( ( − ) × LConst ( S • ) ) , X ) (6) ≃ lim ⟵ H ( ( − ) × LConst ( S • ) , X ) (5) ≃ lim ⟵ H ( ( − ) , Maps ( LConst ( S • ) , X ) ) (2) ≃ H ( ( − ) , lim ⟵ Maps ( LConst ( S • ) , X ) ) (5) .
\begin{array}{lll}
&
\mathbf{H}
\bigg(
(-)
,\,
Maps
\Big(
\underset{\longrightarrow}{\lim}
\,
LConst(S_\bullet)
,\,
X
\Big)
\bigg)
\\
& \;\simeq\;
\mathbf{H}
\Big(
(-)
\times
\underset{\longrightarrow}{\lim}
\,
LConst(S_\bullet)
,\,
X
\Big)
&
\text{ (2) }
\\
& \;\simeq\;
\mathbf{H}
\Big(
\underset{\longrightarrow}{\lim}
\big(
(-)
\times
LConst(S_\bullet)
\big)
,\,
X
\Big)
&
\text{ (6) }
\\
& \;\simeq\;
\underset{\longleftarrow}{\lim}
\,
\mathbf{H}
\big(
(-)
\times
LConst(S_\bullet)
,\,
X
\big)
&
\text{ (5) }
\\
& \;\simeq\;
\underset{\longleftarrow}{\lim}
\,
\mathbf{H}
\Big(
(-)
,\,
Maps
\big(
LConst(S_\bullet)
,\,
X
\big)
\Big)
&
\text{ (2) }
\\
& \;\simeq\;
\mathbf{H}
\Big(
(-)
,\,
\underset{\longleftarrow}{\lim}
\,
Maps
\big(
LConst(S_\bullet)
,\,
X
\big)
\Big)
&
\text{ (5) }
\,.
\end{array}
This implies (4) by the
∞
\infty
-Yoneda lemma over H \mathbf{H} (which is large but locally small, so that the lemma does apply).
Finally (3) is immediate from the fact that LConst LConst preserves the terminal object, by definition:
Maps ( LConst ( * ) , X ) ≃ Maps ( * H , X ) ≃ X .
Maps
\big(
LConst(\ast)
,\,
X
\big)
\;\simeq\;
Maps
\big(
\ast_{\mathbf{H}}
,\,
X
\big)
\;\simeq\;
X
\,.
Last revised on October 31, 2023 at 16:38:42.
See the history of this page for a list of all contributions to it.