homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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
As a result, the décalage construction is often more convenient than forming $X^I \times_X X_0$.
The plain definition of the décalage of a simplicial set is very simple, stated below in
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
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 this last formulation it is clearest what the two canonical morphisms out of the décalage of a simplicial set mean. These we define in
Concretely, the décalage construction is the following.
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)
It is often useful to understand this as a special case of the total décalage construction:
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
or rather its restriction from augmented simplicial sets to just simplicial sets/bisimplicial sets.
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.
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:
Write
for the box product functor that takes $X,Y \in sSet$ to the bisimplicial set
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
This appears as (Stevenson 12, lemma 2.1).
It follows that the left adjoint of plain décalage forms joins with the 0-simplex:
The left adjoint to $Dec_0 : sSet \to sSet$ is
In particular for $S \in sSet$ connected we have
This appears as (Stevenson 12, cor. 2.1).
The join of simplicial sets with the 0-simplex $X \star \Delta[0]$ forms a simplicial model for the cone over $X$.
By adjunction we have for all $n \in \mathbb{N}$
So this exhibits the $n$-cells of $Dec_0 X$ as being the cones of $n$-simplices in $X$.
For $X \in sSet$ its décalage $Dec_0 X$ comes with two canonical morphisms out of it
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.
We discuss here how $Dec_0 X \to X$ is a resolution of $const X_0 \to X$ by a Kan fibration.
For $X$ a simplicial set, the two morphisms from prop. have the following properties.
If $X$ is a Kan complex, then
The first statement is classical, it appears for instance as (Stevenson 11, lemma 5).
For the second, notice that by remark the lifting problem
is equivalent to the lifting problem
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.
By the above, $Dec_0 X$ is the disjoint union of over quasi-categories
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).
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
where
the top morphism
is given in degree $n$ by the $n$-fold degeneracy map $s_0 \circ s_0 \circ \cdots s_0$;
the right vertical morphism
is given in degree $n$ by $d_{n+1} : X_{n+1} \to X_n$
is a Kan fibration.
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:
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
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
given by precomposition. Composing these two strict monoidal functors, there is a strict monoidal functor
Hence by the “walking” correspondence, the value of $[0]$ under this monoidal functor is a comonad on simplicial sets whose underlying functor is décalage:
(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
where $i$ is the set-theoretic identity inclusion of $X$ equipped with the discrete topology. Thus we have
the sum over all possible basepoints $x_0$ of path spaces based at $x_0$. The analogy is made precise by a canonical isomorphism
where $Sing \;\colo\; Top \to Set^{\Delta^{op}}$ 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.
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.
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. ).
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 $G$
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.
Original sources are
and
The notion of décalage has been widely used since the paper introducing the method of cohomological descent in Hodge theory:
Reviews are in
A detailed account of various technical aspects:
and
Closely related technical results are in section 3 of
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.
An application in the theory of stacks is discussed in
Last revised on July 10, 2021 at 11:46:31. See the history of this page for a list of all contributions to it.