nLab residual

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.

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

$$f : A \times B \to C$$

and morphisms

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

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

Definition

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

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

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

to morphisms

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

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

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

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

to morphisms

$$\rho[f] : B \to C{/}A$$

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

Mnemonic for the notation: $C{/}A$ looks like dividing $C$ by $A$ on the right, and $A{\backslash}C$ looks like dividing $C$ by $A$ 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.

Related concepts

• exponential object
• internal hom
• monoidal closed category

References

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

• Paul-André Melliès and Noam Zeilberger, Functors Are Type Refinement Systems, in Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL ‘15, 3–16. New York, NY, USA, 2015. ACM. doi:10.1145/2676726.2676970.