# 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 $A,C$ 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.