nLab monoidal category with diagonals

Idea

A general monoidal category $(C,\otimes)$ does not admit diagonal natural transformations of the form $x \longrightarrow x\otimes x$, unlike the case of a cartesian monoidal category (where the monoidal product is the Cartesian product). A monoidal category with diagonals is a monoidal category with the extra structure of a consistent system of such diagonal morphisms.

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 $C \to C \times C \stackrel{\otimes}{\to} C$ of the diagonal functor with the given monoidal product functor.

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

General

• Any cartesian monoidal category has a canonical structure of diagonal maps given by the actual diagonal morphisms.

• Given a cartesian monoidal category, the wide subcategory consisting of the monomorphisms is a monoidal category with the same product and unit. The diagonal morphism of the original category, always a monomorphism, plays the role of the diagonal maps in the subcategory. Now there are no projections, since, for instance, in Set, the projection maps are almost never injective.

Smash-monoidal diagonals

Write

(1)$$\big( PointedTopologicalSpaces, S^0, \wedge \big) \;\;\in\; SymmetricMonoidalCategories$$

• for the category of pointed topological spaces (with respect to some convenient category of topological spaces such as compactly generated topological spaces or D-topological spaces)

• regarded as a symmetric monoidal category with tensor product the smash product and unit the 0-sphere $S^0 \,=\, \ast_+$.

This category also has a Cartesian product, given on pointed spaces $X_i = (\mathcal{X}_i, x_i)$ with underlying $\mathcal{X}_i \in TopologicalSpaces$ by

(2)$$X_1 \times X_2 \;=\; (\mathcal{X}_1, x_1) \times (\mathcal{X}_2, x_2) \;\coloneqq\; \big( \mathcal{X}_1 \times \mathcal{X}_2 , (x_1, x_2) \big) \,.$$

But since this smash product is a non-trivial quotient of the Cartesian product

(3)$$X_1 \wedge X_1 \,\coloneqq\, \frac{X_1 \times X_2}{ X_1 \vee X_2 }$$

it is not itself cartesian, but just symmetric monoidal.

However, via the quotienting (3), it still inherits, from the diagonal morphisms on underlying topological spaces

(4)$$\array{ \mathcal{X} &\overset{ \Delta_{\mathcal{X}} }{\longrightarrow}& \mathcal{X} \times \mathcal{X} \\ x &\mapsto& (x,x) }$$

a suitable notion of monoidal diagonals:

Definition

[Smash monoidal diagonals]

For $X \,\in\, PointedTopologicalSpaces$, let $D_X \;\colon\; X \longrightarrow X \wedge X$ be the composite

of the Cartesian diagonal morphism (2) with the coprojection onto the defining quotient space (3).

It is immediate that:

Proposition

The smash monoidal diagonal $D$ (Def. ) makes the symmetric monoidal category (1) of pointed topological spaces with smash product a monoidal category with diagonals, in that

1. $D$ is a natural transformation;

2. $S^0 \overset{\;\;D_{S^0}\;\;}{\longrightarrow} S^0 \wedge S^0$ is an isomorphism.

While elementary in itself, this has the following profound consequence:

Remark

[Suspension spectra have diagonals]

Since the suspension spectrum-functor

$$\Sigma^\infty \;\colon\; PointedTopologicalSpaces \longrightarrow HighlyStructuredSpectra$$

is a strong monoidal functor from pointed topological spaces (1) to any standard category of highly structured spectra (by this Prop.) it follows that suspension spectra have monoidal diagonals, in the form of natural transformations

(5)$$\Sigma^\infty X \overset{ \;\; \Sigma^\infty(D_X) \;\; }{\longrightarrow} \big( \Sigma^\infty X \big) \wedge \big( \Sigma^\infty X \big)$$

to their respective symmetric smash product of spectra, which hence makes them into comonoid objects, namely coring spectra.

For example, given a Whitehead-generalized cohomology theory $\widetilde E$ represented by a ring spectrum

$$\big(E, 1^E, m^E \big) \;\; \in \; SymmetricMonoids \big( Ho(Spectra), \mathbb{S}, \wedge \big)$$

the smash-monoidal diagonal structure (5) on suspension spectra serves to define the cup product $(-)\cup (-)$ in the corresponding multiplicative cohomology theory structure:

$$\begin{aligned} & \big[ \Sigma^\infty X \overset{c_i}{\longrightarrow} \Sigma^{n_i} E \big] \,\in\, {\widetilde E}{}^{n_i}(X) \\ & \Rightarrow \;\; [c_1] \cup [c_2] \, \coloneqq \, \Big[ \Sigma^\infty X \overset{ \Sigma^\infty(D_X) }{\longrightarrow} \big( \Sigma^\infty X \big) \wedge \big( \Sigma^\infty X \big) \overset{ ( c_1 \wedge c_2 ) }{\longrightarrow} \big( \Sigma^{n_1} E \big) \wedge \big( \Sigma^{n_2} E \big) \overset{ m^E }{\longrightarrow} \Sigma^{n_1 + n_2}E \Big] \;\; \in \, {\widetilde E}{}^{n_1+n_2}(X) \,. \end{aligned}$$

Related concepts

• relevance monoidal category

• semicartesian monoidal category

• monoidal category

References

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: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”.

• C. Führmann. Varieties of effects, Proc. Fossacs 2002. Doi:10.1007/3-540-45931-6_11