monoidal category with diagonals



Monoidal categories

monoidal categories

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In higher category theory

Category theory



A general monoidal category (C,)(C,\otimes) does not admit diagonal natural transformations of the form xxxx \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.


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 of the diagonal functor with the given monoidal product functor.

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).



Smash-monoidal diagonals


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

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

(2)X 1×X 2=(𝒳 1,x 1)×(𝒳 2,x 2)(𝒳 1×𝒳 2,(x 1,x 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 1X 1X 1×X 2X 1X 2 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)𝒳 Δ 𝒳 𝒳×𝒳 x (x,x) \array{ \mathcal{X} &\overset{ \Delta_{\mathcal{X}} }{\longrightarrow}& \mathcal{X} \times \mathcal{X} \\ x &\mapsto& (x,x) }

a suitable notion of monoidal diagonals:


[Smash monoidal diagonals]

For XPointedTopologicalSpacesX \,\in\, PointedTopologicalSpaces, let D X:XXXD_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 DD (Def. ) makes the symmetric monoidal category (1) of pointed topological spaces with smash product a monoidal category with diagonals, in that

  1. DD is a natural transformation;

  2. S 0D S 0S 0S 0S^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

Σ :PointedTopologicalSpacesHighlyStructuredSpectra \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)Σ XΣ (D X)(Σ X)(Σ X) \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.

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

(E,1 E,m E)SymmetricMonoids(Ho(Spectra),𝕊,) \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:

[Σ Xc iΣ n iE]E˜ n i(X) [c 1][c 2][Σ XΣ (D X)(Σ X)(Σ X)(c 1c 2)(Σ n 1E)(Σ n 2E)m EΣ n 1+n 2E]E˜ n 1+n 2(X). \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}


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 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 January 19, 2021 at 12:28:32. See the history of this page for a list of all contributions to it.