Definition 4.5.1.1.1 (00FV): Smash Products of Pointed Sets

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Table of Contents
Part 1: Sets
Chapter 4: Tensor Products of Pointed Sets
Section 4.5: The Smash Product of Pointed Sets
Subsection 4.5.1: Foundations
Definition 4.5.1.1.1: Smash Products of Pointed Sets (cite)

Statistics Cite

Definition 4.5.1.1.1 ▶ Smash Products of Pointed Sets

The smash product of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$^[1] is the pointed set $X\wedge Y$^[2] satisfying the bijection

\[ \mathsf{Sets}_{*}\webleft (X\wedge Y,Z\webright ) \cong \textup{Hom}^{\otimes }_{\mathsf{Sets}_{*}}\webleft (X\times Y,Z\webright ), \]

naturally in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.

Footnotes

[1] Further Terminology: In the context of monoids with zero as models for $\mathbb {F}_{1}$-algebras, the smash product $X\wedge Y$ is also called the tensor product of $\mathbb {F}_{1}$-modules of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ or the tensor product of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ over $\mathbb {F}_{1}$.

[2] Further Notation: In the context of monoids with zero as models for $\mathbb {F}_{1}$-algebras, the smash product $X\wedge Y$ is also denoted $X\otimes _{\mathbb {F}_{1}}Y$.

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