nLab powering

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.

---

Idea

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

$$\big[v_1,[v_2,v_3]\big] \;\simeq\; \big[v_2,[v_1,v_3]\big]$$

for all objects $v_i \in V$ (prop.). If we regard $V$ as a $V$-enriched category we write $V(v_1,v_2) \mathrel{:=} [v_1,v_2]$ and this reads

$$V\big(v_1,V(v_2,v_3)\big) \;\simeq\; V\big(v_2,V(v_1,v_3)\big) \,.$$

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

$$C\big(c_1, \pitchfork(v,c_2)\big) \;\simeq\; V\big(v, C(c_1,c_2)\big) \,.$$

Definition

Definition

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

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

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

We say that $C$ is powered or cotensored over $V$ if all such power objects exist.

Remark

Powers are frequently called cotensors and a $V$-category having all powers is called cotensored, while the word “power” is reserved for the case $V=$ 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 limit: in particular, where the domain is the unit $V$-category. 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

• $V$ itself is always powered over itself, with $\pitchfork(v_1,v_2) \mathrel{:=} [v_1,v_2]$.

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

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

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

  The defining natural isomorphism

  $$Hom_C(c_1,\pitchfork(S,c_2))\simeq Hom_{Set}(S,Hom_C(c_1,c_2))$$

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

• In a 2-category $\mathcal{K}$ (seen as a $\mathbf{Cat}$-enriched category), powers by the walking arrow $\downarrow$ are ways to internalize ‘generalized arrows’ of a given object $A:\mathcal{K}$. Specifically, ${A^\downarrow} := {\downarrow \pitchfork A}$, called the object of arrows of $A$ is, when it exists, an object such that:

  $$\mathcal{K}(X, A^\downarrow) \simeq \mathbf{Cat}(\downarrow, \mathcal{K}(X,A)) = \mathcal{K}(X,A)^\downarrow.$$

  Thus generalized elements of $A^\downarrow$ correspond to 2-cells between generalized elements of $A$, explaining why $A^\downarrow$ can be considered a ‘view from the inside’ of the internal structure of $A$.

Powering of $\infty$-toposes over $\infty$-groupoids

We discuss how the powering of $\infty$-toposes over $Grpd_\infty$ 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 $\mathbf{H}$ be an $\infty$-topos

• with terminal geometric morphism denoted

  (1)$$\mathbf{H} \underoverset {\underset{\Gamma}{\longrightarrow}} {\overset{LConst}{\longleftarrow}} {\;\;\;\;\bot\;\;\;\;} Grp_\infty \,,$$

  where the inverse image constructs locally constant $\infty$-stacks,

• and with its internal hom (mapping stack) adjunction denoted

  (2)$$\mathbf{H} \underoverset {\underset{Maps(X,-)}{\longrightarrow}} { \overset{ (-) \times X }{\longleftarrow} } {\;\;\;\; \bot \;\;\;\;} \mathbf{H}$$

  for $X \,\in\, \mathbf{H}$.

  Notice that this construction is also $\infty$-functorial in the first argument: $Maps\big( X \xrightarrow{f} Y ,\, A \big)$ is the morphism which under the $\infty$-Yoneda lemma over $\mathbf{H}$ (which is large but locally small, so that the lemma does apply) corresponds to

$$\mathbf{H} \big( (-) ,\, Maps(X,A) \big) \;\simeq\; \mathbf{H} \big( (-) \times X ,\, A \big) \xrightarrow{ \mathbf{H} \big( (-) \times f ,\, A \big) } \mathbf{H} \big( (-) \times Y ,\, A \big) \;\simeq\; \mathbf{H} \big( (-) ,\, Maps(X,A) \big) \,.$$

By definition, for any $S \in Grpd_\infty$ and $X \in \mathbf{H}$ the powering] is the (∞,1)-limit over the diagram constant on $X$

$$X^K \,=\, {\lim_\leftarrow}_K X$$

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

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

Remark

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

Proposition

The powering of $\mathbf{H}$ over $Grpd_\infty$ is given by the mapping stack out of the locally constant $\infty$-stacks:

$$\array{ Grpd_\infty^{op} \times \mathbf{H} & \overset{ LConst^{op} \times \mathrm{id} }{\longrightarrow} & \mathbf{H}^{op} \times \mathbf{H} & \overset{Maps(-,-)}{\longrightarrow} & \mathbf{H} }$$

in that this operation has the following properties:

1. For all $X,\,A \,\in\, \mathbf{H}$ and $S \,\in\, Grpd_\infty$ we have a natural equivalence

   $$\mathbf{H} \Big( X ,\, Maps \big( LConst(S) ,\, A \big) \Big) \;\; \simeq \;\; Grpd_\infty \Big( S ,\, \mathbf{H} \big( X ,\, A \big) \Big)$$

2. In its first argument the operation

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

      (3)$$Maps \big( LConst(\ast) ,\, X \big) \;\; \simeq \;\; X$$

   2. sends $\infty$-colimits to $\infty$-limits:

      (4)$$Maps \Big( \underset{ \longrightarrow }{\lim} \, LConst\big(S_\bullet\big) ,\, X \Big) \;\; \simeq \;\; \underset{ \longleftarrow }{\lim} \, Maps \Big( LConst\big(S_\bullet\big) ,\, X \Big) \,,$$

   where all equivalences shown are natural.

Proof

For the first statement to be proven, consider the following sequence of natural equivalences:

$$\begin{array}{lll} \mathbf{H} \Big( X ,\, Maps \big( LConst(S) ,\, A \big) \Big) & \;\simeq\; \mathbf{H} \big( X \times LConst(S) ,\, A \big) & \text{(2)} \\ & \;\simeq\; \mathbf{H} \Big( LConst(S) ,\, Maps \big( X ,\, A \big) \Big) & \text{(2)} \\ & \;\simeq\; Grpd_\infty \Big( S ,\, \Gamma \, Maps \big( X ,\, A \big) \Big) & \text{ (1) } \\ & \;\simeq\; Grpd_\infty \bigg( S ,\, \mathbf{H} \Big( \ast_{\mathbf{H}} ,\, Maps \big( X ,\, A \big) \Big) \bigg) & \text{by}\;\text{<a href="https://ncatlab.org/nlab/show/terminal+geometric+morphism#DirectImageOfTerminalGeometricMoprhismIsHomOutOfTerminalObject">this Prop.</a>} \\ & \;\simeq\; Grpd_\infty \Big( S ,\, \mathbf{H} \big( \ast_{\mathbf{H}} \times X ,\, A \big) \Big) & \text{(2)} \\ & \;\simeq\; Grpd_\infty \Big( S ,\, \mathbf{H} \big( X ,\, A \big) \Big) \end{array}$$

For the second statement, recall that hom-functors preserve limits in that there are natural equivalences of the form

(5)$$\mathbf{H} \Big( \underset{\underset{i}{\longrightarrow}}{\lim} \,, X_i ,\, \underset{\underset{j}{\longleftarrow}}{\lim} \,, A_j \Big) \;\; \simeq \;\; \underset{\underset{i}{\longleftarrow}}{\lim} \, \underset{\underset{j}{\longleftarrow}}{\lim} \, \mathbf{H} \Big( X_i ,\, A_j \Big) \,,$$

and that $\infty$-toposes have universal colimits, in particular that the product operation is a left adjoint (2) and hence preserves colimits:

(6)$$(-) \,\times\, \underset{{\longrightarrow}}{\lim} \, S_\bullet \;\; \simeq \;\; \underset{{\longrightarrow}}{\lim} \, \big( (-) \,\times\, S_\bullet \big) \,.$$

With this, we get the following sequences of natural equivalences:

$$\begin{array}{lll} & \mathbf{H} \bigg( (-) ,\, Maps \Big( \underset{\longrightarrow}{\lim} \, LConst(S_\bullet) ,\, X \Big) \bigg) \\ & \;\simeq\; \mathbf{H} \Big( (-) \times \underset{\longrightarrow}{\lim} \, LConst(S_\bullet) ,\, X \Big) & \text{ (2) } \\ & \;\simeq\; \mathbf{H} \Big( \underset{\longrightarrow}{\lim} \big( (-) \times LConst(S_\bullet) \big) ,\, X \Big) & \text{ (6) } \\ & \;\simeq\; \underset{\longleftarrow}{\lim} \, \mathbf{H} \big( (-) \times LConst(S_\bullet) ,\, X \big) & \text{ (5) } \\ & \;\simeq\; \underset{\longleftarrow}{\lim} \, \mathbf{H} \Big( (-) ,\, Maps \big( LConst(S_\bullet) ,\, X \big) \Big) & \text{ (2) } \\ & \;\simeq\; \mathbf{H} \Big( (-) ,\, \underset{\longleftarrow}{\lim} \, Maps \big( LConst(S_\bullet) ,\, X \big) \Big) & \text{ (5) } \,. \end{array}$$

This implies (4) by the $\infty$-Yoneda lemma over $\mathbf{H}$ (which is large but locally small, so that the lemma does apply).

Finally (3) is immediate from the fact that $LConst$ preserves the terminal object, by definition:

$$Maps \big( LConst(\ast) ,\, X \big) \;\simeq\; Maps \big( \ast_{\mathbf{H}} ,\, X \big) \;\simeq\; X \,.$$

Related concepts

• tensored and cotensored category

• copower, (∞,1)-copower

• pullback-power

References

Textbook accounts:

• Max Kelly, section 3.7 of: Basic concepts of enriched category theory (tac ,pdf)

• Francis Borceux, Vol 2, Section 6.5 of Handbook of Categorical Algebra, Cambridge University Press (1994)

• Emily Riehl, §3.7 in: Categorical Homotopy Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107261457, pdf]