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Let $[1]^n=\{0<1\}^n$ equipped with the product order. I consider the small category $\widehat{\square}$ of the category of partially ordered sets generated by the coface maps $\delta^\epsilon_i:[1]^{ …

Let $n\geq 1$. Let $[n]=\{0<1\}^n$ equipped with the product order. Let $f:[n]\to [n]$ be a strictly increasing map. When $f$ is bijective, there exists a permutation $\sigma$ of $\{1,\dots,n\}$ such …

$[n]$ is the set $\{0,1\}^n$ equipped with the product order $(\epsilon_1,\dots,\epsilon_n) \leq (\eta_1,\dots,\eta_n)$ if and only if $\forall i=1,\dots,n$, $\epsilon_i \leq \eta_i$. Let $$d((\epsilo …

I have found a way published in a recent preprint (https://doi.org/10.48550/arXiv.2209.02667). Theorem: Let $n\geq 1$. Let $f=(f_1,\dots,f_n):[n]\to [n]$ be a stricly increasing map. Then there is th …

The naive idea has to be slightly modified. The point is not to sort out all terms (it is a wrong intuition), but only where the variables $x_i$ are "alone". For example, the map $$(x_1,x_2,x_3,x_4)\m …