Re: terminology for simplicial sets

[Richard Garner <richard.garner@mq.edu.au>]

Dear Peter,

Though I do not have a direct answer to your question the following
seems at least relevant. By a semi-simplicial set, we mean a presheaf
on Delta_f, the lluf subcategory of Delta spanned by the face
operators (I think the terminology here is nowadays standard). The
inclusion i: Delta_f -> Delta induces by left Kan extension a functor
Lan_i from the category of semi-simplicial sets to the category of
simplicial sets, which is faithful and, though not full, at least full
on isomorphisms. Now a simplicial set satisfies the condition you name
just when it lies in the replete image of this functor. This suggests
that one might reasonably call a simplicial set satisfying your
condition "semi-simplicial", and a map between two such
"semi-simplicial" when it maps non-degenerate simplices to
non-degenerate simplices. Another way of looking at it is that the
semi-simplicial objects are those admitting coalgebra structure for
the comonad (i^* o Lan_i) on simplicial sets; since Lan_i is full on
isomorphisms, such structure will be unique up to unique isomorphism
when it exists. The semi-simplicial maps between such objects are
those which are coalgebra homomorphisms for some (and hence every)
choice of coalgebra structure on their domain and codomain.

Richard

On 19 October 2010 21:54, Prof. Peter Johnstone
<P.T.Johnstone@dpmms.cam.ac.uk> wrote:
> In something I've been thinking about recently, the condition
> on a simplicial set that all faces of non-degenerate simplices
> are non-degenerate seems to play a significant role. Does anyone
> know whether this condition has been considered previously, and
> if so whether it has a standard name?
>
> The condition is of course satisfied by those simplicial sets
> which are derived from simplicial complexes in the standard way,
> but it's more general: it allows the possibility that two
> (formally) different faces of a non-degenerate simplex might
> coincide, as long as they're not degenerate.
>
> Peter Johnstone
>

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