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Where an ordinary simplicial set $X$ may be thought of a space made up of $n$-simplices $\sigma \in X_n$ for all $n \in \{0,1,2,3, \cdots \}$, an augmented simplicial set in addition has a set $X_{-1}$ of “$(-1)$-simplices” such that each 0-simplex has a single $(-1)$-dimensional face and such that the $(-1)$-dimensional faces of the two faces of any $1$-simplex coincide.
Equivalently this may be thought of as the data that encodes a morphism $X \to const X_{-1}$ in sSet between ordinary simplicial sets from an underlying simplicial set to a constant simplicial set with $X_{-1}$ as its set of $k$-simplices for all $k$.
It is in this latter form that augmented simplicial sets maybe mostly arise in practice, whereas the former incarnation offers a more succinct way of thinking about them.
For instance a major source of augmented simplicial objects are given by colimits or rather homotopy colimits over simplicial diagrams in a model category: for $X\colon \Delta^{op} \to C$ a simplicial object in a category with colimits, its colimit cocone may be thought of as a morphism
into the constant simplicial object.
Let $\Delta_+$ (also denoted $\Delta_a$) be the augmented simplex category, namely the category of finite ordinals and order-preserving maps between these. $\Delta_+$ is equivalent to the full subcategory of Cat on free categories over finite and possibly empty linear directed graphs, which are
$[-1] := \emptyset$;
$[0] := (0)$;
$[1] := (0 \to 1)$;
$[2] := (0 \to 1 \to 2)$;
and so on.
An augmented simplicial set $X$ is a presheaf on $\Delta_+$, and the category of augmented simplicial sets is the presheaf category
There is a canonical inclusion $\Delta \hookrightarrow \Delta_+$ of the ordinary simplex category and that the restriction of an augmented simplicial set along this inclusion is a simplicial set. This gives a forgetful functor
A cartoon of an augmented simplicial set, showing just the face maps, looks like
where the only new simplicial identity satisfied by the new face map in degree $-1$ is that it coequalizes the degree-$0$ face maps in that
It follows that $d^{-1}$ coequalizes in fact every pair of composites of face maps $X_n \to X_0$, so that equivalently an augmented simplicial set $X_\bullet$ is a morphism of ordinary simplicial sets
We say that that $U(X_\bullet)$ is augmented over $X_{-1}$.
More explicitly, an augmented simplicial set consists of
Above, we use the traditional system of numbering for a simplicial set. However, part of the motivation behind augmented simplicial sets is that this is a more sensible numbering system:
While this numbering is very nice for augmented simplicial sets, it is not standard and is can be easily misunderstood, so we don't use it in this article.
Anything that applies to simplicial sets should also apply to augmented simplicial sets, if one properly takes care of the negative thinking necessary to deal with $X_{-1}$.
Every augmented simplicial set has an underlying unaugmented simplicial set found by forgetting $X_{-1}$ (and $d^{-1}_0$). Conversely, every unaugmented simplicial set gives rise to a free augmented simplicial set by augmentation over $\pi_0(X)$ (the set of connected components of $X$) and a cofree augmented simplicial set by augmentation over the point (the singleton set). This defines adjunctions:
For $C$ a site, $\{U_i \to X\}$ a covering family we have the Cech nerve simplicial presheaf $C(U) \in [C^{op}, sSet]$. This comes canonically equipped with a morphism of simplicial presheaves to the one represented by $X\colon C(U) \to X$. This is an augmented simplicial object in the category of presheaves $[C^{op}, Set]$. Morover, over each object $V \in C$ its component
is an augmented simplicial set.
More generally this applies to hypercovers.