a "semi-simplicial complex" is what we would now call a functor $\Delta^\mathrm{op}_{\mathrm{injective}}\to \mathrm{Set}$, i.e., a "simplicial set without degeneracy operators". I think the "semi" was meant to distinguish it from the classical notion of a simplicial complex. The "complete" means you have degeneracy operators too.
I never really checked it, but I'm pretty sure that what Hatcher calls in his book a Δ-complex is exactly [the geometric realization of] a semisimplicial set (in modern terminology). I find his choice of terminology really unfortunate
Yes, those are the same thing. But it's not as if he invented that name whole cloth. Rourke and Sanderson worked with the semisimplicial sets under the name Delta-sets; passing to the geometric realization it is not unnatural to use the word 'complex' instead. Semisimplicial complex is something of a mouthful, and seems odd unless you're already explicitly comparing to simplicial complexes.
I guess complete is supposed to indicate you're not missing any important simplices?
I mean, I said it is unfortunate because it is different from the terminology used in the rest of the homotopy theory literature
I understand that Hatcher himself is not a homotopy theorist, and that explains a lot of his choices but this "from outside" perspective that makes his book out of sinc with the rest of homotopy theory is one of the biggest flaws of it
Not really htpy related, but why does Gerhard Paseman have at least three separate accounts?
Also, I think there are a lot of unfortunate things in Hatcher's book, like how he tries to set up singular homology without categories, functors, or natural transformations
