nLab
powered and copowered category

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Contents

Definition

Throughout, let 𝒱\mathcal{V} be a BΓ©nabou cosmos for enriched category theory.

Definition

(tensoring and cotensoring)

Let π’ž\mathcal{C} be a 𝒱\mathcal{V}-enriched category.

  1. A powering (or cotensoring) of π’ž\mathcal{C} over 𝒱\mathcal{V} is

    1. a functor

      [βˆ’,βˆ’]:𝒱 opΓ—π’žβŸΆπ’ž [-,-] \;\colon\; \mathcal{V}^{op} \times \mathcal{C} \longrightarrow \mathcal{C}
    2. for each vβˆˆπ’±v \in \mathcal{V} a natural isomorphism of the form

      (1)𝒱(v,π’ž(c 1,c 2))β‰ƒπ’ž(c 1,[v,c 2]) \mathcal{V}(v, \mathcal{C}(c_1,c_2) ) \;\simeq\; \mathcal{C}(c_1,[v,c_2])
  2. A copowering (or tensoring) of π’ž\mathcal{C} over 𝒱\mathcal{V} is

    1. a functor

      (βˆ’)βŠ—(βˆ’):π’±Γ—π’žβŸΆπ’ž (-)\otimes(-) \;\colon\; \mathcal{V} \times \mathcal{C} \longrightarrow \mathcal{C}
    2. for each vβˆˆπ’±v \in \mathcal{V} a natural isomorphism of the form

      (2)π’ž(vβŠ—c 1,c 2)≃𝒱(v,π’ž(c 1,c 2)) \mathcal{C}(v \otimes c_1, c_2) \;\simeq\; \mathcal{V}(v, \mathcal{C}(c_1,c_2) )

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

Properties

Proposition

(tensoring left adjoint to cotensoring)

If π’ž\mathcal{C} is both tensored and cotensored over 𝒱\mathcal{V} (Def. ), then for fixed vβˆˆπ’±v \in \mathcal{V} the operations of tensoring with vv and of cotensoring with 𝒱\mathcal{V} form a pair of adjoint functors

π’žAAβŠ₯AA⟢[v,βˆ’]⟡vβŠ—(βˆ’)π’ž \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):

π’ž(vβŠ—c 1,c 2)≃𝒱(v,π’ž(c 1,c 2))β‰ƒπ’ž(c 1,[v,c 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

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

If π’ž\mathcal{C} is both tensored and cotensored over 𝒱\mathcal{V} then

  1. an initial object βˆ…\emptyset of the underlying category of π’ž\mathcal{C} is also enriched initial, in that the hom-object out of it is the terminal object *\ast of 𝒱\mathcal{V}

    π’ž(βˆ…,c)≃* \mathcal{C}(\emptyset, c) \;\simeq\; \ast
  2. a terminal object *\ast of the underlying category of π’ž\mathcal{C} is also enriched terminal, in that the hom-object into it is the terminal object of 𝒱\mathcal{V}:

    π’ž(c,*)≃* \mathcal{C}(c, \ast) \;\simeq\; \ast
Proof

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

By prop. , 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

vβŠ—βˆ…β‰ƒβˆ… v \otimes \emptyset \;\simeq\; \emptyset

for all vβˆˆπ’±v \in \mathcal{V}, in particular for βˆ…βˆˆπ’±\emptyset \in \mathcal{V}. Therefore the natural isomorphism (2) implies for all vβˆˆπ’±v \in \mathcal{V} that

π’ž(βˆ…,c)β‰ƒπ’ž(βˆ…βŠ—βˆ…,c)≃𝒱(βˆ…,π’ž(βˆ…,c))≃* \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 𝒱(βˆ’,βˆ’)=[βˆ’,βˆ’]\mathcal{V}(-,-) = [-,-] in 𝒱\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.

Examples

Example

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

Example

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

Last revised on August 1, 2018 at 08:28:12. See the history of this page for a list of all contributions to it.