A semi-simplicial set is like a simplicial set, but without degeneracy maps: it is a sequence of sets together with functions called face maps between them which encode that an element in has “faces” (boundary segments) which are elements in .
More generally, for any other category, a functor is a semi-simplicial object in .
The original paper
defined both (what we now call) semi-simplicial sets, under the name semi-simplicial complexes, and (what we now call) simplicial sets, under the name complete semi-simplicial complexes. The motivation for the name “semi-simplicial” was that a semi-simplicial set is like a simplicial complex, but lacks the property that a simplex is uniquely determined by its vertices. Then they added the degeneracies and a corresponding adjective “complete.”
Over time it became clear that “complete semi-simplicial complexes” were much more important and useful than the non-complete ones. This seems to have led first to the omission of the adjective “complete,” and then the omission of the prefix “semi” (and at some point the replacement of “complex” by “set”), resulting in the current name simplicial sets.
Anyone more knowledgable about the history, please correct/improve the preceeding paragraph.
The concept is essentially the same as that of -set, as used by Rourke and Sanderson. Their motivation was from geometric topology.
On the other hand, in other contexts the prefix “semi-” is used to denote absence of identities (such as a semigroup (which is, admittedly, missing more than identities relative to a group) or a semicategory). Thus if we start from the modern name “simplicial sets” it makes independent sense to refer to their degeneracy-less variant as “semi-simplicial sets.” This is coincidentally in line with the original terminology of Eilenberg and Zilber, but not of course with the intermediate usage of “semi-simplicial set” for what we now call a “simplicial set.”
C. P. Rourke, and B. J. Sanderson, Δ-Sets I: Homotopy Theory. The Quarterly Journal of Mathematics 22: 321–338 (1971) (PDF)
S. Buoncristiano, C.P. Rourke, and B. J. Sanderson, A geometric approach to Homology Theory, LMS Lect. Notes 18, (1976)
James E. McClure, On semisimplicial sets satisfying the Kan condition (ArXiv).
A discussion of a model structure on semi-simplicial sets is in
See also the references at semi-simplicial object.
Wikipedia, Delta sets