nLab
powering

Contents

This entry is about the formal dual to tensoring in the generality of category theory. For the different concept of cotensor product of comodules see there.


Context

Enriched category theory

Limits and colimits

Contents

Idea

In a closed symmetric monoidal category VV the internal hom [,]:V op×VV[-,-] : V^{op} \times V \to V satisfies the natural isomorphism

[v 1,[v 2,v 3]][v 2,[v 1,v 3]] [v_1,[v_2,v_3]] \simeq [v_2,[v_1,v_3]]

for all objects v iVv_i \in V (prop.). If we regard VV as a VV-enriched category we write V(v 1,v 2):=[v 1,v 2]V(v_1,v_2) := [v_1,v_2] and this reads

V(v 1,V(v 2,v 3))V(v 2,V(v 1,v 3)). V(v_1,V(v_2,v_3)) \simeq V(v_2,V(v_1,v_3)) \,.

If we now pass more generally to any VV-enriched category CC then we still have the enriched hom object functor C(,):C op×CVC(-,-) : C^{op} \times C \to V. One says that CC is powered over VV if it is in addition equipped also with a mixed operation :V op×CC\pitchfork : V^{op} \times C \to C such that (v,c)\pitchfork(v,c) behaves as if it were a hom of the object vVv \in V into the object cCc \in C in that it satisfies the natural isomorphism

C(c 1,(v,c 2))V(v,C(c 1,c 2)). C(c_1,\pitchfork(v,c_2)) \simeq V(v,C(c_1,c_2)) \,.

Definition

Definition

Let VV be a closed monoidal category. In a VV-enriched category CC, the power of an object yCy\in C by an object vVv\in V is an object (v,y)C\pitchfork(v,y) \in C with a natural isomorphism

C(x,(v,y))V(v,C(x,y)) C(x, \pitchfork(v,y)) \cong V(v, C(x,y))

where C(,)C(-,-) is the VV-valued hom of CC and V(,)V(-,-) is the internal hom of VV.

We say that CC is powered or cotensored over VV if all such power objects exist.

Remark

Powers are frequently called cotensors and a VV-category having all powers is called cotensored, while the word “power” is reserved for the case V=V= Set. However, there seems to be no good reason for making this distinction. Moreover, the word “tensor” is fairly overused, and unfortunate since a tensor (= a copower) is a colimit, while a cotensor (= power) is a limit.

Properties

  • Powers are a special sort of weighted limits. Conversely, all weighted limits can be constructed from powers together with conical limits. The dual colimit notion of a power is a copower.

Examples

In 1-category theory

  • VV itself is always powered over itself, with (v 1,v 2):=[v 1,v 2]\pitchfork(v_1,v_2) := [v_1,v_2].

  • Every locally small category CC (V=(Set,×)V = (Set,\times) ) with all products is powered over Set: the powering operation

    (S,c):= sSc \pitchfork(S,c) := \prod_{s\in S} c

    of an object cc by a set SS forms the |S||S|-fold cartesian product of cc with itself, where |S||S| is the cardinality of SS.

    The defining natural isomorphism

    Hom C(c 1,(S,c 2))Hom Set(S,Hom C(c 1,c 2)) Hom_C(c_1,\pitchfork(S,c_2))\simeq Hom_{Set}(S,Hom_C(c_1,c_2))

    is effectively the definition of the product (see limit).

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 proogfs 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 \,.

References

Textbook accounts:

Last revised on October 18, 2021 at 06:07:51. See the history of this page for a list of all contributions to it.