nLab
residual

Idea

Residuals are “exponential objects for a monoidal category”: they are compatible with the monoidal product \otimes. Because \otimes is (in general) not symmetric, we distinguish a left residual and a right residual. In a symmetric monoidal category, these are the same.

Recall the definition of an exponential object for a cartesian category:

Let A,CA,C be two objects of the cartesian category. An exponential object is an object C AC^A with data ev,ε\mathit{ev},\varepsilon such that for every object BB, there is a one-to-one mapping between morphisms

f:A×BC f : A \times B \to C

and morphisms

ε[f]:BC A. \varepsilon[f] : B \to C^A \; .

The backwards direction is given by ev:A×C AA\mathit{ev} : A \times C^A \to A, namely f=ev(A×ε[f])f = \mathit{ev} \circ (A \times \varepsilon[f]). Going back and forth is also required to be the identity: for g:BC Ag : B \to C^A, we must have g=ε[ev(A×g)]g = \varepsilon[ev \circ (A \times g)].

Definition

Let A,CA,C be two objects of a monoidal category.

A left residual of CC by AA is an object A\CA{\backslash}C together with an evaluation map lev:A(A\C)C\mathit{\lev} : A \otimes (A{\backslash}C) \to C and for objects BB a transformation from morphisms

f:ABC f : A \otimes B \to C

to morphisms

λ[f]:BA\C \lambda[f] : B \to A{\backslash}C

such that f=lev(Aλ[f])f = \mathit{\lev} \circ (A \otimes \lambda[f]), and g=λ[lev(Ag)]g = \lambda[\mathit{\lev} \circ (A \otimes g)] for every morphism g:BA\Cg : B \to A{\backslash}C.

A right residual of CC by AA is an object C/AC{/}A together with an evaluation map rev:(C/A)AC\mathit{rev} : (C{/}A) \otimes A \to C and for objects BB a transformation from morphisms

f:BAC f : B \otimes A \to C

to morphisms

ρ[f]:BC/A \rho[f] : B \to C{/}A

such that f=rev(ρ[f]A)f = \mathit{rev} \circ (\rho[f] \otimes A), and g=ρ[rev(gA)]g = \rho[\mathit{rev} \circ (g \otimes A)] for every morphism g:BC/Ag : B \to C{/}A.

Mnemonic for the notation: C/AC{/}A looks like dividing CC by AA on the right, and A\CA{\backslash}C looks like dividing CC by AA on the left.

Properties

Left and right residuals are unique up to isomosphism.

Examples

  • In every cartesian category, the exponential objects are left and right residuals.
  • Any monoidal closed category has all right residuals.

References

Last revised on February 15, 2017 at 08:26:59. See the history of this page for a list of all contributions to it.