nLab
augmented simplicial set

Augmented simplicial sets

Idea

Where an ordinary simplicial set XX may be thought of a space made up of nn-simplices σX n\sigma \in X_n for all n{0,1,2,3,}n \in \{0,1,2,3, \cdots \}, an augmented simplicial set in addition has a set X 1X_{-1} of “(1)(-1)-simplices” such that each 0-simplex has a single (1)(-1)-dimensional face and such that the (1)(-1)-dimensional faces of the two faces of any 11-simplex coincide.

Equivalently this may be thought of as the data that encodes a morphism XconstX 1X \to const X_{-1} in sSet between ordinary simplicial sets from an underlying simplicial set to a constant simplicial set with X 1X_{-1} as its set of kk-simplices for all kk.

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:Δ opCX\colon \Delta^{op} \to C a simplicial object in a category with colimits, its colimit cocone may be thought of as a morphism

X constlim X X_\bullet \to const \lim_{\to} X_\bullet

into the constant simplicial object.

Definitions

Denote by Δ +\Delta_+ (also denoted Δ a\Delta_a) the augmented simplex category, which may be defined as the full subcategory of Cat on free categories over finite and possibly empty linear directed graphs, which are

  • [1]:=[-1] := \emptyset;

  • [0]:=(0)[0] := (0);

  • [1]:=(01)[1] := (0 \to 1);

  • [2]:=(012)[2] := (0 \to 1 \to 2);

  • and so on.

An augmented simplicial set XX is a presheaf on Δ +\Delta_+ and the category of augmented simplicial sets is the presheaf category

sSet +:=[Δ + op,Set]. sSet_+ := [\Delta_+^{op}, Set] \,.

There is a canonical inclusion ΔΔ +\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

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

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

X =(X 2X 1d 0 0d 1 0X 0d 1X 1) 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 1-1 is that it coequalizes the degree-00 face maps in that

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

It follows that d 1d^{-1} coequalizes in fact every pair of composites of face maps X nX 0X_n \to X_0, so that equivalently an augmented simplicial set X X_\bullet is a morphism of ordinary simplicial sets

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

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

More explicitly, an augmented simplicial set consists of

  • for each integer n1n \geq -1, a set X nX_n (so an infinite sequence (X 1,X 0,X 1,X 2,)(X_{-1},X_0,X_1,X_2,\ldots) of sets),
  • for each integer n1n \geq -1 and each natural number mn+1m \leq n + 1, a face map d m n:X n+1X nd^n_m\colon X_{n+1} \to X_n,
  • for each integer n1n \geq -1 and each natural number mnm \leq n, a degeneracy map s m n:X nX n+1s^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 nn, a set X nX_n (so an infinite sequence (X 0,X 1,X 2,)(X_0,X_1,X_2,\ldots) of sets),
  • for each natural number nn and each natural number mnm \leq n, a face map d m n:X n+1X nd^n_m\colon X_{n+1} \to X_n,
  • for each natural number nn and each natural number m<nm \lt n, a degeneracy map s m n:X nX n+1s^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 is can be easily misunderstood, so we don't 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 1X_{-1}.

Every augmented simplicial set has an underlying unaugmented simplicial set found by forgetting X 1X_{-1} (and d 0 1d^{-1}_0). Conversely, every unaugmented simplicial set gives rise to a free augmented simplicial set by augmentation over π 0(X)\pi_0(X) (the set of connected components of XX) and a cofree augmented simplicial set by augmentation over the point (the singleton set). This defines adjunctions:

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Examples

  • For CC a site, {U iX}\{U_i \to X\} a covering family we have the Cech nerve simplicial presheaf C(U)[C op,sSet]C(U) \in [C^{op}, sSet]. This comes canonically equipped with a morphism of simplicial presheaves to the one represented by X:C(U)XX\colon C(U) \to X. This is an augmented simplicial object in the category of presheaves [C op,Set][C^{op}, Set]. Morover, over each object VCV \in C its component

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

    is an augmented simplicial set.

    More generally this applies to hypercovers.

Revised on June 7, 2014 10:30:03 by Adeel Khan (77.9.241.174)