Write
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
But since this smash product is a non-trivial quotient of the Cartesian product
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
a suitable notion of monoidal diagonals:
[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:
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
$D$ is a natural transformation;
$S^0 \overset{\;\;D_{S^0}\;\;}{\longrightarrow} S^0 \wedge S^0$ is an isomorphism.
While elementary in itself, this has the following profound consequence:
[Suspension spectra have diagonals]
Since the suspension spectrum-functor
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
to their respective symmetric smash product of spectra.
For example, given a Whitehead-generalized cohomology theory $\widetilde E$ represented by a ring spectrum
the smash-monoidal diagonal structure (5) on suspension spectra serves to define the cup product $(-)\cup (-)$ in the corresponding multiplicative cohomology theory structure:
Created on January 19, 2021 at 12:22:18. See the history of this page for a list of all contributions to it.