nLab smash-monoidal diagonals -- section

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:

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