nLab powered and copowered category

1. Definition

Throughout, let $\mathcal{V}$ be a Bénabou cosmos for enriched category theory.

Definition 1.1. (tensoring and cotensoring)

Let $\mathcal{C}$ be a $\mathcal{V}$-enriched category.

1. A powering (or cotensoring) of $\mathcal{C}$ over $\mathcal{V}$ is

   1. a functor

      $$[-,-] \;\colon\; \mathcal{V}^{op} \times \mathcal{C} \longrightarrow \mathcal{C}$$

   2. for each $v \in \mathcal{V}$ a natural isomorphism of the form

      (1)$$\mathcal{V}(v, \mathcal{C}(c_1,c_2) ) \;\simeq\; \mathcal{C}(c_1,[v,c_2])$$

2. A copowering (or tensoring) of $\mathcal{C}$ over $\mathcal{V}$ is

   1. a functor

      $$(-)\otimes(-) \;\colon\; \mathcal{V} \times \mathcal{C} \longrightarrow \mathcal{C}$$

   2. for each $v \in \mathcal{V}$ a natural isomorphism of the form

      (2)$$\mathcal{C}(v \otimes c_1, c_2) \;\simeq\; \mathcal{V}(v, \mathcal{C}(c_1,c_2) )$$

If $\mathcal{C}$ is equipped with a (co-)powering it is called (co-)powered over $\mathcal{V}$.

2. Properties

Proposition 2.1. (tensoring left adjoint to cotensoring)

If $\mathcal{C}$ is both tensored and cotensored over $\mathcal{V}$ (Def. 1.1), then for fixed $v \in \mathcal{V}$ the operations of tensoring with $v$ and of cotensoring with $\mathcal{V}$ form a pair of adjoint functors

$$\mathcal{C} \underoverset {\underset{ [v,-] }{\longrightarrow}} {\overset{ v \otimes (-) }{\longleftarrow}} {\phantom{AA}\bot\phantom{AA}} \mathcal{C}$$

Proof. The hom-isomorphism characterizing the pair of adjoint functors is provided by the composition of the natural isomorphisms (1) and (2):

$$\array{ \mathcal{C}(v \otimes c_1, c_2) \;\simeq\; \mathcal{V}(v, \mathcal{C}(c_1,c_2) ) \;\simeq\; \mathcal{C}(c_1,[v,c_2]) }$$

  ▮

It follows that:

Proposition 2.2. (in tensored and cotensored categories initial/terminal objects are enriched initial/terminal)

If $\mathcal{C}$ is both tensored and cotensored over $\mathcal{V}$ then

1. an initial object $\emptyset$ of the underlying category of $\mathcal{C}$ is also enriched initial, in that the hom-object out of it is the terminal object $\ast$ of $\mathcal{V}$

   $$\mathcal{C}(\emptyset, c) \;\simeq\; \ast$$

2. a terminal object $\ast$ of the underlying category of $\mathcal{C}$ is also enriched terminal, in that the hom-object into it is the terminal object of $\mathcal{V}$:

   $$\mathcal{C}(c, \ast) \;\simeq\; \ast$$

Proof. We discuss the first claim, the second is formally dual.

By prop. 2.1, tensoring is a left adjoint. Since left adjoints preserve colimits, and since an initial object is the colimit over the empty diagram, it follows that

$$v \otimes \emptyset \;\simeq\; \emptyset$$

for all $v \in \mathcal{V}$, in particular for $\emptyset \in \mathcal{V}$. Therefore the natural isomorphism (2) implies for all $v \in \mathcal{V}$ that

$$\mathcal{C}(\emptyset, c) \;\simeq\; \mathcal{C}( \emptyset \otimes \emptyset, c ) \;\simeq\; \mathcal{V}( \emptyset, \mathcal{C}(\emptyset, c) ) \;\simeq\; \ast$$

where in the last step we used that the internal hom $\mathcal{V}(-,-) = [-,-]$ in $\mathcal{V}$ sends colimits in its first argument to limits (this prop.) and used that a terminal object is the limit over the empty diagram.  ▮

3. Examples

Example 3.1. The symmetric closed monoidal category $\mathcal{V}$ is tensored and cotensored over itself, with tensoring being its tensor product and powering being its internal hom.

Example 3.2. For $\mathcal{C}$ a small $\mathcal{V}$-enriched category, the $\mathcal{V}$-enriched category of enriched presheaves $[\mathcal{C}^{op}, \mathcal{V}]$ is tensored and cotensored (this Prop.)

4. Related concepts

• two-variable adjunction

• enriched model category

5. References

• G. Max Kelly, Adjunction for enriched categories, in: Reports of the Midwest Category Seminar III, Lecture Notes in Mathematics 106, Springer (1969) [doi:10.1007/BFb0059145]