monoidal category with diagonals



Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory

Category theory



An arbitrary monoidal category (C,)(C,\otimes) does not admit maps xxxx\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.


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

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


  • 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 XΔX×XXXX\stackrel{\Delta}{\to} X\times X \to X \wedge X, where X×XXXX\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 Δ x:xxx\Delta_x\colon x\to x\otimes x for all xx, such as in the Kleisli category for a strong monad on a cartesian category, or in any Freyd category, then the ff for which (ff)Δ x=Δ yf(f\otimes f)\circ \Delta_x = \Delta_y \circ f are called “copyable”.

Last revised on April 18, 2019 at 08:16:39. See the history of this page for a list of all contributions to it.