ScalingStacks

Let us reformulate the definitions of the previous sections in terms of ${\mathbf{F}}_{2}$-vector spaces with a basis.

The functor ${\mathbf{F}}_{2}[-]$ gives an equivalence from the category of pointed sets to the category with objects ${\mathbf{F}}_{2}$-vector spaces with a basis and where maps are ${\mathbf{F}}_{2}$-linear maps sending a basis element to a basis element or $0$.

Under this equivalence, we have the following correspondences:

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  a coproduct of pointed spaces corresponds to a direct sum with basis the union of bases

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  a wedge product of pointed spaces corresponds to a tensor product with basis the product of bases

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  a $G$-graded pointed set corresponds to a $G$-graded ${\mathbf{F}}_{2}$-vector space with a basis consisting of homogeneous elements

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  a $G$-filtered pointed set corresponds to a $G$-filtered ${\mathbf{F}}_{2}$-vector space $V$, ie a family $\{V_{\geq g}\}_{g\in G}$ of subspaces of $V$ with $V_{\geq g}\subset V_{\geq g^{\prime}}$ if $g>g^{\prime}$, with a basis $B$ such that $B\cap V_{\geq g}$ is a basis of $V_{\geq g}$ for all $g\in G$ and such that given $v\in V\setminus\{0\}$, the set $\{g\in G\ |\ V_{\geq g}\neq 0\}$ is non-empty and has a maximal element

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  a differential pointed set corresponds to an ${\mathbf{F}}_{2}$-vector space with a basis together with a bounded differential.