nLab decalage

Idea

In simplicial homotopy theory, décalage refers to a well-behaved model of path space objects for simplicial sets:

If you take a simplicial set and ‘throw away’ the last face and degeneracy, and relabel, shifting everything down one ‘notch’, you get a new simplicial set. This is what is called the décalage of a simplicial set.

It is a model for the path space object of $X$, – or rather: the union of all based path space objects for all basepoints $x \in X_0$ – similar to, but a little smaller than, the model $X^I \times_X X_0$, which is discussed for instance at factorization lemma:

In the latter case an $n$-cell in the path space is a morphism to $X$ from the simplicial cone over the $n$-simplex modeled as the pushout $(\Delta[n] \times \Delta[1]) \coprod_{\Delta[n]} \Delta[0]$. This is the simplicial set obtained by forming the simplicial cylinder over $\Delta[n]$ and then contracting one end to the point.

Contrary to that, an $n$-simplex in the décalage of $X$ is a morphism to $X$ from the cone over $\Delta[n]$ modeled simply by the join of simplicial sets $\Delta[n] \star \Delta[0]$.

This is a much smaller model for the cone. In fact $\Delta[n]\star \Delta[0] = \Delta[n+1]$ is just the $(n+1)$-simplex. On the other hand, the above pushout-construction produces simplicial sets with many $(n+1)$-simplices, the one that one “expects”, but glued to others with some degenerate edges. Accordingly, there is, for $n \geq 1$, a proper inclusion

$$\Delta[n] \star \Delta[1] \hookrightarrow (\Delta[n] \times \Delta[1]) \coprod_{\Delta[n]} \Delta[0] \,.$$

As a result, the décalage construction is often more convenient than forming $X^I \times_X X_0$.

Definition

The plain definition of the décalage of a simplicial set is very simple, stated below in

• In components

However, in order to appreciate and handle this definition, it is useful to understand it as a special case of total décalage, stated below in

• As a restriction of total décalage .

From this one sees more manifestly that the décalage of a simplicial set is built from cones in the original simplicial set. This we discuss below in

• In terms of cones.

In this last formulation it is clearest what the two canonical morphisms out of the décalage of a simplicial set mean. These we define in

• Morphisms out of décalage.

In components

Concretely, the décalage construction is the following.

Definition

For $X$ a simplicial set, the décalage $Dec_0\, X \in sSet$ of $X$, is the simplicial set obtained by shifting every dimension down by one, ‘forgetting’ the last face and degeneracy of $X$ in each dimension:

• $(Dec_0 \, X)_n \coloneqq X_{n+1}$;

• $d_k^{n,Dec_0 X} \coloneqq d^{n+1,X}_{k}$;

• $s_k^{n,Dec_0 X} \coloneqq s^{n+1,X}_{k}$.

Analogously:

• $(Dec^0 \, X)_n \coloneqq X_{n+1}$;

• $d_k^{n,Dec^0 X} \coloneqq d^{n+1,X}_{k+1}$;

• $s_k^{n,Dec^0 X} \coloneqq s^{n+1,X}_{k+1}$.

(Illusie 72, review in Stevenson 11, Def. 2)

As a restriction of total décalage

It is often useful to understand this as a special case of the total décalage construction:

Definition

Write $\sigma \colon \Delta_a \times \Delta_a \to \Delta_a$ for the ordinal sum operation on the augmented simplex category. The total décalage functor is precompositon with this

$$\sigma^* \colon sSet_a \to ssSet_a$$

or rather its restriction from augmented simplicial sets to just simplicial sets/bisimplicial sets.

$$\sigma^* : sSet \to ssSet \,.$$

In terms of this the plain décalage is the functor induced from the restriction $\sigma(-,[0]) : \Delta \to \Delta$, of ordinal sum with $0$, i.e.

$$Dec_0 X := (\sigma(-,[0]))^* X \,.$$

In terms of cones

The perspective from total décalage makes fairly manifest that décalage forms cones in $X$, as we discuss now. To this end, notice the relation of total décalage to join of simplicial sets:

Definition

Write

$$\Box : sSet \times sSet \to ssSet$$

for the box product functor that takes $X,Y \in sSet$ to the bisimplicial set

$$(X \Box Y) : ([k],[l]) \mapsto X_k \times X_l \,.$$

Lemma

If $X, Y \in$ sSet are connected, then their join of simplicial sets $X \star Y$ is expressed by the left adjoint to total décalage as

$$\sigma_!(X \Box Y) = X \star Y \,.$$

This appears as (Stevenson 12, lemma 2.1).

It follows that the left adjoint of plain décalage forms joins with the 0-simplex:

Corollary

The left adjoint to $Dec_0 : sSet \to sSet$ is

$$C := \sigma_!((-) \Box \Delta[0]) \,.$$

In particular for $S \in sSet$ connected we have

$$C(S) = S \star \Delta[0] \,.$$

This appears as (Stevenson 12, cor. 2.1).

Remark

The join of simplicial sets with the 0-simplex $X \star \Delta[0]$ forms a simplicial model for the cone over $X$.

Corollary

By adjunction we have for all $n \in \mathbb{N}$

$$(Dec_0 X)_n = Hom_{sSet}( \Delta[n] \star \Delta[0], X) \,.$$

So this exhibits the $n$-cells of $Dec_0 X$ as being the cones of $n$-simplices in $X$.

Morphisms out of the décalage

Proposition

For $X \in sSet$ its décalage $Dec_0 X$ comes with two canonical morphisms out of it

$$\array{ Dec_0 X &\overset{\in Fib}{\longrightarrow}& X \\ \big\downarrow {}^{\mathrlap{\in \mathrm{W}}} \\ const X_0 \mathrlap{\,.} }$$

Here, in terms of the description above of décalage by cones:

• the horizontal morphism is induced from the canonical inclusion $\Delta[n] \hookrightarrow \Delta[n]\star \Delta[0]$;

• the vertical morphism is given by the canonical inclusion $\Delta[0] \hookrightarrow \Delta[n]\star \Delta[0]$.

Or in terms of components, as discussed above,

• the horizontal morphism is given by $d_{last} \colon Dec_0 Y \to Y$, hence in degree $n$ by the remaining face map $d_{n+1} \colon X_{n+1} \to X_n$;

• the vertical morphism is given in degree 0 by $s_0 \colon X_1 \to X_0$ and in every higher degree similarly by $s_0 \circ s_0 \circ \cdots \circ s_0$.

(review in Stevenson 11, Def. 2)

These morphisms are a Kan fibration and a simplicial weak equivalence, respectively, by the following discussion.

Properties

Fibration resolution

We discuss here how $Dec_0 X \to X$ is a resolution of $const X_0 \to X$ by a Kan fibration.

Proposition

For $X$ a simplicial set, the two morphisms from prop. have the following properties.

• the morphism $d_0 : Dec_0 X \to const X_0$ is a weak homotopy equivalence, in fact a deformation retract; a weak inverse is given by the morphism which in degree 0 is the degeneracy $s_0 : X_0 \to X_1$, and so on.

If $X$ is a Kan complex, then

• the morphism $d_{last} : Dec_0 X \to X$ is a Kan fibration;

Proof

The first statement is classical, it appears for instance as (Stevenson 11, lemma 5).

For the second, notice that by remark the lifting problem

$$\array{ \Lambda^n[n] &\to& Dec_0 X \\ \downarrow && \downarrow \\ \Delta[n] &\to& X }$$

is equivalent to the lifting problem

$$\array{ (\Lambda^n[n] \star \Delta[0]) \coprod_{\Lambda^i[n]} \Delta[n] &\to& X \\ \downarrow && \downarrow \\ \Delta[n] \star \Delta[0] &\to& * } \,.$$

Here the left morphism is an anodyne morphism, in fact is an $(n+1)$-horn inclusion $\Lambda[n+1] \to \Delta[n+1]$. So a lift exists if $X$ is a Kan complex.

Remark

By the above, $Dec_0 X$ is the disjoint union of over quasi-categories

$$Dec_0 X = \coprod_{x \in X_0} X_{/x} \,.$$

For each of these the statement that the projection $X_{/x} \to X$ is a Kan fibration if $X$ is a Kan complex, and moreover that it is a a right fibration if $X$ is a quasi-category, is (Joyal, theorem 3.19), reproduced also as (HTT, prop. 2.1.2.1). Notice that left/right fibrations into a Kan complex are automatically Kan fibrations (by the discussion at Left fibration in ∞-groupoids).

Corollary

For $X$ a Kan complex, the décalage morphism $Dec_0 X \to X$ is a Kan fibration resolution of the inclusion $const X_0 \to X$ of the set of 0-cells of $X$, regarded as a discrete simplicial set:

there is a commuting diagram

$$\array{ const X_0 &\stackrel{\simeq}{\to}& Dec_0 X \\ \downarrow && \downarrow \\ X &\to& X } \,,$$

where

• the top morphism

  • is given in degree $n$ by the $n$-fold degeneracy map $s_0 \circ s_0 \circ \cdots s_0$;

  • is a weak homotopy equivalence;

• the right vertical morphism

  • is given in degree $n$ by $d_{n+1} : X_{n+1} \to X_n$

  • is a Kan fibration.

Remark

The inclusion $const X_0 \to X$ presents a canonical effective epimorphism in an (∞,1)-category in ∞Grpd into $X$, out of a 0-truncated object. By the above, the décalage is a natural fibration resolution of this canonical “atlas”.

This is useful for instance in the discussion of homotopy pullbacks of this effective epimorphism: by the discussion there the homotopy pullback of $const X_0 \to X$ along any morphism $f : A \to X$ is presented by the ordinary pullback of any Kan fibration resolution, hence in particular of the décalage projection:

$$f^* Dec_0 X \simeq A \times_X^{h} const X_0 \,.$$

Décalage comonad

Décalage also has an abstract category theoretic description as follows. The augmented simplex category with the ordinal sum operation is a monoidal category $(\Delta_a, + = \sigma, 0 = [-1])$. This monoidal category carries a canonical monoid, namely the terminal object $1 = [0]$ with its unique monoid structure. By duality, the monoidal category $\Delta_a^{op}$ has a comonoid, also denoted $[0]$.

As is the case for any comonoid in a monoidal category, the comonoid $[0]$ induces a comonad $D_0 = (-) + [0] = \sigma(-, [0])$ on $\Delta_a^{op}$. And, as is the case for any $2$-functor, exponentiation $Set^{-}$ as a $2$-functor (say of the form $cat^{op} \to Cat$, from small categories to locally small categories) takes the comonad $D_0: \Delta_a^{op} \to \Delta_a^{op}$ in $cat^{op}$ to a comonad $Set^{D_0}: Set^{\Delta_a^{op}} \to Set^{\Delta_a^{op}}$ in $Cat$. This comonad, mapping $F: \Delta_a^{op} \to Set$ to $F \circ D_0: \Delta_a^{op} \to Set$, is precisely the décalage comonad $Dec_0: SSet \to SSet$.

(By similar reasoning, there is a second comonad $D^0 = [0] + (-) = \sigma([0], -)$ on $\Delta_a^{op}$, which in turn induces a second comonad $Set^{D^0}: Set^{\Delta_a^{op}} \to Set^{\Delta_a^{op}}$. This second décalage comonad is denoted by Stevenson as $Dec^0: SSet \to SSet$.)

There are tautologically equivalent formulations. One formulation invokes the fact that $\Delta_a$ together with the terminal monoid $[0]$ constitute the “walking monoid”, i.e., $\Delta_a$ is initial among monoidal categories equipped with a monoid. Similarly, $\Delta_a^{op}$ is the walking comonoid: by initiality, strict monoidal functors $\Delta_a^{op}\to [C, C]$ are precisely in correspondence with comonoids in the endofunctor category $[C, C]$ (as a monoidal category under endofunctor composition), that is to say, with comonads on $C$.

Consider then the monoidal product

$$\sigma : \Delta_a^{op} \times \Delta_a^{op} \to \Delta_a^{op}$$

Analogous to Cayley embeddings of monoids into endofunction monoids, either way of currying this product produces a strict monoidal functor $\Delta_a^{op}\to [\Delta_a^{op},\Delta_a^op]$ into an endofunctor category. By applying 2-functoriality as above, there is additionally a strict monoidal functor

$$[\Delta_a^{op}, \Delta_a^{op}] \to [\mathbf{SSet},\mathbf{SSet}]$$

given by precomposition. Composing these two strict monoidal functors, there is a strict monoidal functor

$$\Delta_a^{op}\to [\mathbf{SSet}, \mathbf{SSet}].$$

Hence by the “walking” correspondence, the value of $[0]$ under this monoidal functor is a comonad on simplicial sets whose underlying functor is décalage:

$$Dec_0 \colon Set^{\Delta_a^{op}} \to Set^{\Delta_a^{op}}$$

(Tautologically, though, this is merely an elaborate way to rephrase the earlier description of this comonad.)

The map $d_{last} \colon Dec_0 \to Id$ is the counit of the comonad. The comonad itself is analogous to a kind of unbased path space object comonad $P$ on $Top$ whose value at a space $X$ is a pullback

$$\array{ P X & \to & X^I \\ \downarrow & & \downarrow \mathrlap{eval_0} \\ |X| & \stackrel{i}{\to} & X }$$

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

$$P X \;=\; \sum_{x_0 \in X} P(X, x_0),$$

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

$$Dec_0 \circ Sing \;\cong\; Sing \circ P \,,$$

where $Sing \;\colon\; Top \to Set^{\Delta^{op}}$ is the singular simplicial complex-functor.

A $P$-coalgebra partitions $X$ into path components and exhibits contractibility of each component. Similarly, a coalgebra of the décalage 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.

Examples

For simplicial classifying spaces

A central application is the special case where $X = \overline{W} G$ is the simplicial classifying space of a simplicial group $G$ (see at simplicial principal bundle). In this case $Dec_0 \overline{W} G$, called $W G$, is a standard model for the universal simplicial principal bundle.

Or rather, with the conventions used at simplicial classifying spaces (which are those of Goerss & Jardine, p. 269) we have $W G \,=\, Dec^0(\overline{W}G)$ (Def. ).

For simplicial groups

The case of $Dec_0 G$ for $G$ a simplicial group is important in the simplicial theory of algebraic models for homotopy n-types.

In this case the morphism $d_{last} : Dec_0\, G \to G$, is an epimorphism. Taking the kernel of this and then applying $\pi_0$, yields a crossed module constructed from the Moore complex of $G$

$$N G_1/d_2(NG_2)\to N G_0,$$

which has kernel $\pi_1(G)$ and cokernel $\pi_0(G)$. This crossed module represents the homotopy 2-type of $G$. Applying the décalage twice leads to a crossed square which represents the 3-type of $G$, … and so on.

Related concepts

• bar construction

• diagonal of a bisimplicial set

• simplicial classifying space

References

Original sources are

• Luc Illusie, Complexe cotangent et déformations I, volume 239 of Lecture Notes in Maths , Springer-Verlag. and 1972, Complexe cotangent et déformations II, volume 283 of Lecture Notes in Maths , Springer-Verlag (1971)

and

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

The notion of décalage has been widely used 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.

Reviews are in

• Phil Ehlers, Algebraic Homotopy in Simplicially Enriched Groupoids, 1993, University of Wales Bangor, (pdf here)

A detailed account of various technical aspects:

• Danny Stevenson, around Lemma 15 of: Décalage and Kan’s simplicial loop group functor, Theory and Applications of Categories, Vol. 26, 2012, No. 28, pp 768-787 (arXiv:1112.0474, tac:26-28)

and

• Danny Stevenson, Section 2.2 of Classifying theory for simplicial parametrized groups (arXiv:1203.2461)

Closely related technical results are in section 3 of

• André Joyal, The theory of quasi-categories and its applications , lectures at CRM Barcelona (2008)

The link with simplicial groups and algebraic models of homotopy $n$-types is given in

• Tim Porter, n-types of simplicial groups and crossed n-cubes, Topology, 32, (1993), 5–24.

• Tim Porter, The crossed menagerie

An application in the theory of stacks:

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

On the adjunction unit of total décalage:

• Viktoriya Ozornova, Martina Rovelli, The unit of the total décalage adjunction, Journal of Homotopy and Related Structures 15 (2020) 333–349 [doi.org/10.1007/s40062-020-00257-1]