nLab
decalage

Definition

The décalage $\mathrm{Dec}Y$ of a simplicial set $Y$, is the simplicial set obtained by shifting every dimension down by one, ‘forgetting’ the last face and degeneracy of $Y$ in each dimension:

• $(\mathrm{Dec}Y{)}_n=Y_{n+1}$;
• $d_k^{n,\mathrm{Dec}Y}=d_k^{n+1,Y}$;
• $s_k^{n,\mathrm{Dec}Y}=s_k^{n+1,Y}$.

This simplicial set comes with a natural projection, $d_{\mathrm{last}}:\mathrm{Dec}Y\to Y$, given by the ‘left over’ face map. Moreover this map gives a homotopy equivalence

$$\mathrm{Dec}Y\simeq \mathrm{const}(Y_0),$$Dec\,Y \simeq const(Y_0),

between $\mathrm{Dec}Y$ and the constant simplicial set on $Y_0$.

The same construction works in other contexts, as is easily seen. The case of $\mathrm{Dec}G$ for $G$ a simplicial group is important in the simplicial theory of algebraic models for homotopy n-types.

The map $d_{\mathrm{last}}:\mathrm{Dec}Y\to Y$, is a Kan fibration, and in particular, in the simplicial group case, $d_{\mathrm{last}}:\mathrm{Dec}G\to G$, is an epimorphism. Taking the kernel of this and then applying ${\pi }_0$, yields a crossed module constructed from the Moore complex of $G$

$$N{G}_1/d_2({\mathrm{NG}}_2)\to N{G}_0,$$N G_1/d_2(NG_2)\to N G_0,

which has kernel ${\pi }_1(G)$ and cokernel ${\pi }_0(G)$.

Decalage comonad

Decalage also has an abstract category theoretic description as follows. The simplex category, as a monoidal category $(\Delta ,+,0)$ equipped with the monoid $1$, is the “walking monoid”, i.e., is initial among monoidal categories equipped with a monoid. Therefore ${\Delta }^{\mathrm{op}}$ is the walking comonoid; as a result, there is a comonad

$$-+1:{\Delta }^{\mathrm{op}}\to {\Delta }^{\mathrm{op}}$$- + 1: \Delta^{op} \to \Delta^{op}

which induces a comonad on simplicial sets whose underlying functor is precisely decalage:

$$\mathrm{Dec}:{\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}\to {\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}$$Dec: Set^{\Delta^{op}} \to Set^{\Delta^{op}}

The map $d_{\mathrm{last}}:\mathrm{Dec}\to \mathrm{Id}$ is the counit of this comonad. The comonad itself is analogous to a kind of unbiased path space comonad $P$ on $\mathrm{Top}$ whose value at a space $X$ is a pullback

$$\begin{array}{ccc}PX& \to & X^I\\ \downarrow & & \downarrow {\mathrm{eval}}_0\\ \mid X\mid & \stackrel{i}{\to }& X\end{array}$$\array{ P X & \to & X^I \\ \downarrow & & \downarrow eval_0 \\ |X| & \stackrel{i}{\to} & X }

where $i$ is the set-theoretic identity inclusion of $X$ equipped with the discrete topology. Thus we have

$$PX=\sum _{x_0\in X}P(X,x_0),$$P X = \sum_{x_0 \in X} P(X, x_0),

the sum over all possible basepoints $x_0$ of path spaces based at $x_0$. The analogy is made precise by a canonical isomorphism

$$\mathrm{Dec}\circ S\cong S\circ P$$Dec \circ S \cong S \circ P

where $S:\mathrm{Top}\to {\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}$ is simplicial singularization.

A $P$-coalgebra partitions $X$ into path components and exhibits contractibility of each component. Similarly, a coalgebra of the decelage comonad exhibits the acyclicity of the underlying simplicial set.

Total Décalage

Using either the simplicial comonadic resolution? generated by the above comonad or directly using ordinal sum, we get a bisimplicial set known as the total décalage of $Y$. See there for more details.

Literature

A reasonably full discussion of the décalage can be found in Luc Illusie’s thesis,

• L. Illusie, 1971, Complexe cotangent et deformations I, volume 239 of Lecture Notes in Maths , Springer-Verlag. and 1972, Complexe cotangent et deformations II, volume 283 of Lecture Notes in Maths , Springer-Verlag.

It is used in Duskin’s Memoir,

• John Duskin, 1975, Simplicial methods and the interpretation of “triple” cohomology, number 163 in Mem. Amer. Math. Soc., 3, Amer. Math. Soc.

The notion of decalage is widely appearing since the paper introducing the method of cohomological descent in Hodge theory:

• Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77.

An application in theory of stacks is in

• Anders Kock, The stack quotient of a groupoid, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 44 no. 2 (2003), p. 85–104 numdam

See also

• P. J. Ehlers, Algebraic Homotopy in Simplicially Enriched Groupoids, 1993, University of Wales Bangor, available here, (see also the reference below and the Tim Porter’s Menagerie notes

• Danny Stevenson, Décalage and Kan’s simplicial loop group functor (arXiv:1112.0474)