I have a sequence $X_0,X_1,\ldots$ of Abelian groups, along with face maps $d_0,\ldots, d_n: X_{n-1}\to X_n$ which satisfy the standard cosimplicial identities EXCEPT at the very bottom, where in $\mathrm{Hom}(X_0, X_2)$, I have
$$d_2d_0 \ne d_0d_1$$
However, I have one extra map $d_\infty:X_1\to X_2$ which satisfies the following relations in $\mathrm{Hom}(X_0,X_2)$ $$ d_\infty d_0 = d_2d_0 $$ $$ d_\infty d_1 = d_0d_1 $$ and the following relations in $\mathrm{Hom}(X_1,X_3)$ $$ d_0d_\infty = d_1d_\infty $$ $$ d_2d_\infty = d_3d_\infty $$ By hand, I can build a chain complex out of $X_\cdot$, adding a $d_\infty$ to the usual cosimplicial differential. The relations above imply that $d^2=0$.
Is this familiar to anyone?
At first I had hoped that I just had the "wrong" face maps and if I changed around my face maps $X_1\to X_2$ in a clever enough way that this would become a regular (semi)-cosimplicial Abelian group. But I don't see it.
Another thought I had is that I have many other $d_\infty$ maps that I just haven't seen yet because I don't know the relations that they should satisfy and so am not looking in the right place.
I also noticed that $d_\infty$ satisfies the relations I'd expect from the map $[1]=\lbrace 0,1\rbrace$ to $[2]=\lbrace 0,1,2\rbrace$ which is the constant map to $1$. But this seems not to lead anywhere because of the failure of the face identities at the bottom; i.e., the maps $X_0\to X_1$ and $X_1\to X_2$ don't act like regular injections of ordered sets.