TITLE: What do we mean by contractible for simplicial objects in a category?
QUESTION [5 upvotes]: EDIT: removed cruft from this question.
Recall that extra degeneracies for an augmented simplicial set $X$ are maps $s_0\colon X_n \to X_{n+1}$ for $n=-1,0,1,2,\ldots$ which satisfy the usual simplicial identities with respect to the existing $d_i$, $s_i$. This definition clearly works for simplicial objects in pretty much an arbitrary category.
A simplicial set, augmented by $X_{-1} = \ast$, with extra degeneracies is contractible. In fact, given enough structure on an ambient category $C$, one can sensibly talk about homotopy of simplicial objects in $C$ (for example, one can say when $sC$ is a category with cofibrant objects, and has a notion of homotopy of maps). [EDIT: By this I mean there is some sort of model structure around relative to which we can talk about homotopy]
So my question is this: is it reasonable to think of simplicial objects in $C$ with extra degeneracies as being contractible for any category $C$ with terminal object? Certainly, ignoring size issues, we can think of such things as being contractible after we embed them in the category of simplicial sets in $Pre(C)$, if not some smaller (co)completion category.
Secondarily, can I get away with saying a simplicial object with extra degeneracies "is a contractible simplicial object?"

REPLY [3 votes]: Actually the issue is not that simple. Here is a talk by Mike Barr on showing there are three different notions of 'contractible' for simplicial objects in a category that do not coincide even for simplicial sets. They even give explicit examples.

Michael Barr (joint with John F. Kennison, R. Raphael), Contractible simplicial objects, talk 9 October 2018 in the Logic and Categories seminar, McGill University, abstract, slides.

They find "Strong extra degeneracies" implies "Extra degeneracies" implies "Homotopic to a constant simplicial object", and these are strict implications. See the slides for precise definitions. Beware that the 'extra degeneracies' given in my question are not necessarily the same as Barr et al's definition.