closed monoidal category


Let π’ž be a monoidal category, with tensor productPlanetmathPlanetmathPlanetmath βŠ—. Then we say that

  • β€’

    π’ž is closed, or left closed, if the functorMathworldPlanetmath AβŠ—- on π’ž has a right adjoint [A,-]l

  • β€’

    π’ž is right closed if the functor -βŠ—B on π’ž has a right adjoint [B,-]r

  • β€’

    π’ž is biclosed if it is both left closed and right closed.

A biclosed symmetric monoidal category is also known as a symmetric monoidal closed categoryMathworldPlanetmath. In a symmetric monoidal closed category, AβŠ—Bβ‰…BβŠ—A, so [A,B]lβ‰…[A,B]r. In this case, we denote the right adjoint by [A,B].

Some examples:

  • β€’

    Any cartesian closed category is symmetric monoidal closed.

  • β€’

    In particular, as a category with finite products is symmetricPlanetmathPlanetmath monoidal, it is biclosed iff it is cartesian closed.

  • β€’

    An example of a biclosed monoidal category that is not symmetric monoidal is the category of bimodules over a non-commutative ring. The right adjoint of A×- is [A,-]l, where [A,B]l is the collection of all left R-linear bimodule homomorphisms from A to B, while the right adjoint of -×A is [A,-]r, where [A,B]r is the collection of all right R-linear bimodule homomorphisms from A to B. Unless R is commutativePlanetmathPlanetmath, [A,B]l≠[A,B]r in general.

more to come…

Title closed monoidal category
Canonical name ClosedMonoidalCategory
Date of creation 2013-03-22 18:30:25
Last modified on 2013-03-22 18:30:25
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 81-00
Classification msc 18-00
Classification msc 18D10
Related topic IndexOfCategories
Defines left closed
Defines right closed
Defines biclosed
Defines symmetric monoidal closed