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)HΓLConstGrp , \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)HMaps(X,)()×XH \mathbf{H} \underoverset {\underset{Maps(X,-)}{\longrightarrow}} { \overset{ (-) \times X }{\longleftarrow} } {\;\;\;\; \bot \;\;\;\;} \mathbf{H}

    for XHX \,\in\, \mathbf{H}.

    Notice that this construction is also \infty -functorial in the first argument: Maps(XfY,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 SGrpd S \in Grpd_\infty and XHX \in \mathbf{H} the powering] is the (∞,1)-limit over the diagram constant on XX

X K=lim KX X^K \,=\, {\lim_\leftarrow}_K X

while the tensoring is the (∞,1)-colimit over the diagram constant on XX

KX=lim KX. K \cdot X \,=\, {\lim_{\to}}_K X \,.

Remark

Under Isbell duality, the powering operations on homotopy types XX corresponds to higher order Hochschild cohomology of suitable algebras of functions on XX, as discussed there.

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:

  1. For all X,AHX,\,A \,\in\, \mathbf{H} and SGrpd 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)
  2. In its first argument the operation

    1. sends the terminal object (the point) to the identity:

      (3)Maps(LConst(*),X)X Maps \big( LConst(\ast) ,\, X \big) \;\; \simeq \;\; X
    2. sends \infty -colimits to \infty -limits:

      (4)Maps(limLConst(S ),X)limMaps(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,ΓMaps(X,A)) (1) Grpd (S,H(* H,Maps(X,A))) bythis 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(limi,X i,limj,A j)limilimjH(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)()×limS 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(limLConst(S ),X)) H(()×limLConst(S ),X) (2) H(lim(()×LConst(S )),X) (6) limH(()×LConst(S ),X) (5) limH((),Maps(LConst(S ),X)) (2) H((),limMaps(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 LConstLConst 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.