# nLab monoidal category with diagonals

contents

### Context

#### Monoidal categories

monoidal categories

category theory

# contents

## Idea

An arbitrary monoidal category $(C,\otimes)$ does not admit maps $x\to x\otimes x$, unlike the case of a cartesian monoidal category, where the monoidal product is the categorical product. A monoidal category with diagonals is a monoidal category with the extra structure of a consistent system of such diagonal maps.

## Definition

A consistent system of diagonal maps $\Delta_x\colon x\to x\otimes x$ as $x$ varies through the objects of a monoidal category $(C,\otimes,I)$ should be natural, so that $(f\otimes f)\circ \Delta_x = \Delta_y \circ f$, for any $f\colon x\to y$. Hence such a system is a natural transformation from the identity functor on $C$ to the composite $C \to C\times C \stackrel{\otimes}{\to} C$.

Another desirable property is that the diagonal map $\Delta_I\colon I\to I\otimes I$ on the tensor unit $I$ is the inverse of the left unitor $\ell_I\colon I\otimes I \stackrel{\sim}{\to} I$ (which is the same as the right unitor $r_I$).

## Examples

• Any cartesian monoidal category has a canonical structure of diagonal maps.

• The category of pointed sets with smash product is a monoidal category with diagonals, taking the diagonal maps to be the composites $X\stackrel{\Delta}{\to} X\times X \to X \wedge X$, where $X\times X \to X\wedge X$ is the defining quotient map for the smash product.

The stronger notion of relevance monoidal category is discussed in

• K. Dosen and Z. Petric, Relevant Categories and Partial Functions, Publications de l’Institut Mathématique, Nouvelle Série, Vol. 82(96), pp. 17–23 (2007) arXiv:math/0504133

When a premonoidal category comes equipped with a morphism $\Delta_x\colon x\to x\otimes x$ for all $x$, such as in the Kleisli category for a strong monad on a cartesian category, or in any Freyd category, then the $f$ for which $(f\otimes f)\circ \Delta_x = \Delta_y \circ f$ are called “copyable”.