nLab augmented simplicial set

Idea

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 additionally 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 is 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

$$X_\bullet \to const \lim_{\to} X_\bullet$$

into the constant simplicial object.

Definitions

Denote by $\Delta_+$ (also denoted $\Delta_a$) the augmented simplex category, which may be defined as the full subcategory of Cat on the 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, $sSet_+$, is the presheaf category

$$[\Delta_+^{op}, Set] \,.$$

There is a canonical inclusion $\Delta \hookrightarrow \Delta_+$ of the ordinary simplex category and the restriction of an augmented simplicial set along this inclusion is a simplicial set. This gives a forgetful functor

$$U : sSet_+ \to sSet \,.$$

A cartoon of an augmented simplicial set, showing just the face maps, looks like

$$X_\bullet = \left( \cdots X_2 \stackrel{\to}{\stackrel{\to}{\to}} X_1 \stackrel{\overset{d^0_1}{\to}}{\underset{d^0_0}{\to}} X_0 \stackrel{d^{-1}}{\to} X_{-1} \right)$$

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

$$d^{-1}\circ d^0_1 = d^{-1}\circ d^0_0 \,.$$

It follows that $d^{-1}$ coequalizes every pair of composites of face maps $X_n \to X_0$, so that an augmented simplicial set $X_\bullet$ is equivalent to a morphism of ordinary simplicial sets

$$U(X_\bullet) \to const X_{-1} \,.$$

We say that $U(X_\bullet)$ is augmented over $X_{-1}$.

More explicitly, an augmented simplicial set consists of

• for each integer $n \geq -1$, a set $X_n$ (so an infinite sequence $(X_{-1},X_0,X_1,X_2,\ldots)$ of sets),
• for each integer $n \geq -1$ and each natural number $m \leq n + 1$, a face map $d^n_m\colon X_{n+1} \to X_n$,
• for each integer $n \geq -1$ and each natural number $m \leq n$, a degeneracy map $s^n_m\colon X_n \to X_{n+1}$,
• satisfying the simplicial identities.

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:

• for each natural number $n$, a set $X_n$ (so an infinite sequence $(X_0,X_1,X_2,\ldots)$ of sets),
• for each natural number $n$ and each natural number $m \leq n$, a face map $d^n_m\colon X_{n+1} \to X_n$,
• for each natural number $n$ and each natural number $m \lt n$, a degeneracy map $s^n_m\colon X_n \to X_{n+1}$,
• satisfying the simplicial identities.

While this numbering is very nice for augmented simplicial sets, it is not standard, and it can be easily misunderstood, so we do not use it in this article.

Properties

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:

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Examples

• 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 equipped with a canonical 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 $[C^{op}, Set]$. Moreover, for each object $V \in C$, its component

  $$C(U)(V) \to X(V)$$

  is an augmented simplicial set.

  More generally, this applies to hypercovers.

Related concepts

• augmentation, augmentation ideal

• augmented algebra