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Don’t worry about the appearance of the page. It looks good and there are those amongst us who spend ’idle’ moments looking through recent entries and, so as to learn new insights into things, clean up the editing, e.g. theorem environments etc. (When this is done to one of ’my’ pages I look back and say to myself.. that is how I should have done it! then next time, I most likely have forgotten! do the same thing and feel a bit silly afterwards.)
It’s good practice to provide a link to the page in the Forum discussion, so people can go directly to it: special Delta-space. One related page we already have is Segal space, a notion which omits the contractibility of $X_0$ but is otherwise the same.
I’m not really fond of the "special" terminology (since it gives no indication of in what way the object is "special"), and "$\Delta$-space" has the problem that some people now use "$\Delta$-set" to mean a semi-simplicial set, i.e. a simplicial set without degeneracies. On the other hand, I don’t know of a good replacement.
It must be admitted that “semi-simplicial” in the sense of Mike’s #3 would be perfectly reasonable terminology, were it not for the naming history pointed out by Jim (which looks weird in retrospect – why would adding degeneracies make anything “semi”? more like removing them would do that). “Facial” is not such a bad substitute IMO.
Thanks Mike.
Do you think it´s good to merge the two pages?
I don’t think “special delta space” is a good name but I’m not sure how to call it. Special simplicial space is probably better but still has the problem you mentioned . Perhaps “Segal space with a contractible space of objects”? I would feel better with a shorter name. By the way, the nLab definition of a Segal space is different than the one of Rezk (a model for the homotopy theory of homotopy theory).
@Jim: While I have always felt along the lines that Todd said (e.g. “semigroup”, “semicategory” all mean lacking identities, so consistency demands that “semi-simplicial” mean lacking degeneracies), I used to feel sufficiently guilty about the ahistoricity not to use “semi-simplicial” in that way. But then I went back and read the original paper. Quoting some more context from the paper you cited:
Originally ([5]), Delta sets were referred to as semi-simplicial complexes, and, once the degeneracy operations we are about to discuss were discovered, the term complete semi- simplicial complex (c.s.s. set, for short) was introduced. Over time, with Delta sets becoming of less interest, “complete semi-simplicial” was abbreviated back to “semi-simplicial”
In other words, “semi-simplicial” originally referred to the objects without degeneracies, with the adjective “complete” added when degeneracies were present. So using “semi-simplicial” to mean “without degeneracies” is actually more historically faithful. (I believe the original purpose of the adjective “semi-” was to distinguish these objects from simplicial complexes, not to indicate anything about degeneracies.) Of course, there was also the interlude later on during which “semi-simplicial” meant what we now call “simplicial”, but given that it is never used that way today, I think there’s no reason to prevent ourselves from using it in the more logical way.
@MatanP (is that your given name or your family name?): I don’t think the pages should be merged, as the two serve different purposes: special $\Delta$-spaces are a model for $\infty$-monoids, while Segal spaces are a model for internal categories in $\infty$-groupoids. But they should link to each other. (The nLab definition of Segal space is equivalent to Rezk’s; use 2-out-of-3 and stuff.)
What about something like “reduced Segal space”?
Mike: I like your name suggestion. I’ll edit the entry accordingly. My given name is Matan and my surname is Prezma.
In some places I saw people use “restricted simplicial object” to mean without degeneracies (so I used it as well).
ok. Changed the entry name to reduced Segal space with the obvious adaptations and linked it to Segal space.
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