nLab
residual

Contents

This entry as written remains a partial duplicate of internal hom. See there for more. But lattices with internal homs are also known as residuated lattices.

Contents

Idea

Residuals, in the sense here, 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

The term “residual” for left/right internal homs is (non-standard and) used in

Last revised on September 9, 2021 at 10:41:31. See the history of this page for a list of all contributions to it.