Define mapping spaces for ruptured simplicial sets

Develop a rigorous construction of the mapping space Map(X, Y) for ruptured simplicial sets X and Y by specifying a canonical ruptured structure on the standard simplices Δ[n] and proving that this yields a well-defined object in the category of ruptured simplicial sets (rsSet), including verification that coherence is preserved and gap-witnessed horns are mapped appropriately.

Background

The paper extends simplicial sets to ruptured simplicial sets that track coherent simplices and gap-witnessed horns. Standard constructions like products are developed, but the mapping-space construction is only outlined.

The authors note that defining Map(𝒳, 𝒴) requires choosing a canonical ruptured structure on the standard simplices and proving that the resulting mapping space lives naturally in the category of ruptured simplicial sets, which is deferred to future work.