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For $n\ge 0$, let $E_n$ be the set of all equivalence relations on $[n]:=\{0,\dotsc,n\}$. Now given two equivalence relations $R,R'\in E_n$, we build their join $$R\vee R' := \langle R\cup R'\rangle,$$ the smallest equivalence relation containing $R$ and $R'$. The following seems to be clear:

  1. $(E_n,\vee)$ is a bounded join-semilattice, i. e. it is a commutative monoid (with $1=\Delta$, the smallest equivalence relation) where each element is idempotent.

  2. Consider the cosimplicial maps $d_i:[n]\to [n+1]$ and $s_i:[n]\to [n-1]$ and define maps $D_i:E_n\to E_{n+1}$ and $S_i:E_n\to E_{n-1}$ by $$D_iR:=\langle d_i^{\times 2}R\rangle~~~\text{ and }~~~S_iR:=\langle s_i^{\times 2}R\rangle$$ Then they should satisfy the cosimplicial identities and moreover, $D_i(R\vee R')=D_iR\vee D_iR'$ and $S_i(R\vee R')=S_iR\vee S_iR'$.

All in all, this should give us a cosimplicial object in the category of bounded semilattices.

If I didn’t make any obvious mistake, this looks to me as a construction which must be well-known and written down somewhere, maybe in a different terminology. Thus, I assume that I missed something which does not work so easily. Does someone find a mistake or know the general construction from the literature?

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