Homotopy theory

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



Paths and cylinders

Homotopy groups

Basic facts




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 XX, – or rather: the union of all based path space objects for all basepoints xX 0x \in X_0 – similar to, but a little smaller than, the model X I× XX 0X^I \times_X X_0, which is discussed for instance at factorization lemma:

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

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

This is a much smaller model for the cone. In fact Δ[n]Δ[0]=Δ[n+1]\Delta[n]\star \Delta[0] = \Delta[n+1] is just the (n+1)(n+1)-simplex. On the other hand, the above pushout-construction produces simplicial sets with many (n+1)(n+1)-simplices, the one that one “expects”, but glued to others with some degenerate edges. Accordingly, there is, for n1n \geq 1, a proper inclusion

Δ[n]Δ[1](Δ[n]×Δ[1]) Δ[n]Δ[0]. \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× XX 0X^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

In components

Concretely, the décalage construction is the following.


For XX a simplicial set, the décalage Dec 0XsSetDec_0\, X \in sSet of XX, is the simplicial set obtained by shifting every dimension down by one, ‘forgetting’ the last face and degeneracy of XX in each dimension:

  • (Dec 0X) nX n+1(Dec_0 \, X)_n \coloneqq X_{n+1};

  • d k n,Dec 0Xd k n+1,Xd_k^{n,Dec_0 X} \coloneqq d^{n+1,X}_{k};

  • s k n,Dec 0Xs k n+1,Xs_k^{n,Dec_0 X} \coloneqq s^{n+1,X}_{k}.


  • (Dec 0X) nX n+1(Dec^0 \, X)_n \coloneqq X_{n+1};

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

  • s k n,Dec 0Xs k+1 n+1,Xs_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:


Write σ:Δ a×Δ aΔ a\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

σ *:sSet assSet a \sigma^* \colon sSet_a \to ssSet_a

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

σ *:sSetssSet. \sigma^* : sSet \to ssSet \,.

In terms of this the plain décalage is the functor induced from the restriction σ(,[0]):ΔΔ\sigma(-,[0]) : \Delta \to \Delta, of ordinal sum with 00, i.e.

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

In terms of cones

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



:sSet×sSetssSet \Box : sSet \times sSet \to ssSet

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

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

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

σ !(XY)=XY. \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:


The left adjoint to Dec 0:sSetsSetDec_0 : sSet \to sSet is

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

In particular for SsSetS \in sSet connected we have

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

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


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


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

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

So this exhibits the nn-cells of Dec 0XDec_0 X as being the cones of nn-simplices in XX.

Morphisms out of the décalage


For XsSetX \in sSet its décalage Dec 0XDec_0 X comes with two canonical morphisms out of it

Dec 0X Fib X W constX 0. \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 Δ[n]Δ[n]Δ[0]\Delta[n] \hookrightarrow \Delta[n]\star \Delta[0];

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

Or in terms of components, as discussed above,

  • the horizontal morphism is given by d last:Dec 0YYd_{last} \colon Dec_0 Y \to Y, hence in degree nn by the remaining face map d n+1:X n+1X nd_{n+1} \colon X_{n+1} \to X_n;

  • the vertical morphism is given in degree 0 by s 0:X 1X 0s_0 \colon X_1 \to X_0 and in every higher degree similarly by s 0s 0s 0s_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.


Fibration resolution

We discuss here how Dec 0XXDec_0 X \to X is a resolution of constX 0Xconst X_0 \to X by a Kan fibration.


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

  • the morphism d 0:Dec 0XconstX 0d_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 0X 1s_0 : X_0 \to X_1, and so on.

If XX is a Kan complex, then

  • the morphism d last:Dec 0XXd_{last} : Dec_0 X \to X is a Kan fibration;

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

For the second, notice that by remark the lifting problem

Λ n[n] Dec 0X Δ[n] X \array{ \Lambda^n[n] &\to& Dec_0 X \\ \downarrow && \downarrow \\ \Delta[n] &\to& X }

is equivalent to the lifting problem

(Λ n[n]Δ[0]) Λ i[n]Δ[n] X Δ[n]Δ[0] *. \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)(n+1)-horn inclusion Λ[n+1]Δ[n+1]\Lambda[n+1] \to \Delta[n+1]. So a lift exists if XX is a Kan complex.


By the above, Dec 0XDec_0 X is the disjoint union of over quasi-categories

Dec 0X= xX 0X /x. Dec_0 X = \coprod_{x \in X_0} X_{/x} \,.

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


For XX a Kan complex, the décalage morphism Dec 0XXDec_0 X \to X is a Kan fibration resolution of the inclusion constX 0Xconst X_0 \to X of the set of 0-cells of XX, regarded as a discrete simplicial set:

there is a commuting diagram

constX 0 Dec 0X X X, \array{ const X_0 &\stackrel{\simeq}{\to}& Dec_0 X \\ \downarrow && \downarrow \\ X &\to& X } \,,


  • the top morphism

    • is given in degree nn by the nn-fold degeneracy map s 0s 0s 0s_0 \circ s_0 \circ \cdots s_0;

    • is a weak homotopy equivalence;

  • the right vertical morphism

    • is given in degree nn by d n+1:X n+1X nd_{n+1} : X_{n+1} \to X_n

    • is a Kan fibration.


The inclusion constX 0Xconst X_0 \to X presents a canonical effective epimorphism in an (∞,1)-category in ∞Grpd into XX, 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 constX 0Xconst X_0 \to X along any morphism f:AXf : A \to X is presented by the ordinary pullback of any Kan fibration resolution, hence in particular of the décalage projection:

f *Dec 0XA× X hconstX 0. 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 (Δ a,+=σ,0=[1])(\Delta_a, + = \sigma, 0 = [-1]). This monoidal category carries a canonical monoid, namely the terminal object 1=[0]1 = [0] with its unique monoid structure. By duality, the monoidal category Δ a op\Delta_a^{op} has a comonoid, also denoted [0][0].

As is the case for any comonoid in a monoidal category, the comonoid [0][0] induces a comonad D 0=()+[0]=σ(,[0])D_0 = (-) + [0] = \sigma(-, [0]) on Δ a op\Delta_a^{op}. And, as is the case for any 22-functor, exponentiation Set Set^{-} as a 22-functor (say of the form cat opCatcat^{op} \to Cat, from small categories to locally small categories) takes the comonad D 0:Δ a opΔ a opD_0: \Delta_a^{op} \to \Delta_a^{op} in cat opcat^{op} to a comonad Set D 0:Set Δ a opSet Δ a opSet^{D_0}: Set^{\Delta_a^{op}} \to Set^{\Delta_a^{op}} in CatCat. This comonad, mapping F:Δ a opSetF: \Delta_a^{op} \to Set to FD 0:Δ a opSetF \circ D_0: \Delta_a^{op} \to Set, is precisely the décalage comonad Dec 0:SSetSSetDec_0: SSet \to SSet.

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

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

Consider then the monoidal product

σ:Δ a op×Δ a opΔ a op \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 Δ a op[Δ a op,Δ a op]\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

[Δ a op,Δ a op][SSet,SSet][\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

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

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

Dec 0:Set Δ a opSet Δ a op 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:Dec 0Idd_{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 PP on TopTop whose value at a space XX is a pullback

PX X I eval 0 |X| i X \array{ P X & \to & X^I \\ \downarrow & & \downarrow \mathrlap{eval_0} \\ |X| & \stackrel{i}{\to} & X }

where ii is the set-theoretic identity inclusion of XX equipped with the discrete topology. Thus we have

PX= x 0XP(X,x 0), P X \;=\; \sum_{x_0 \in X} P(X, x_0),

the sum over all possible basepoints x 0x_0 of path spaces based at x 0x_0. The analogy is made precise by a canonical isomorphism

Dec 0SingSingP, Dec_0 \circ Sing \;\cong\; Sing \circ P \,,

where SingcoloTopSet Δ opSing \;\colo\; Top \to Set^{\Delta^{op}} is the singular simplicial complex-functor.

A PP-coalgebra partitions XX 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 YY. See there for more details.


For simplicial classifying spaces

A central application is the special case where X=W¯GX = \overline{W} G is the simplicial classifying space of a simplicial group GG (see at simplicial principal bundle). In this case Dec 0W¯GDec_0 \overline{W} G, called WGW 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 WG=Dec 0(W¯G)W G \,=\, Dec^0(\overline{W}G) (Def. ).

For simplicial groups

The case of Dec 0GDec_0 G for GG a simplicial group is important in the simplicial theory of algebraic models for homotopy n-types.

In this case the morphism d last:Dec 0GGd_{last} : Dec_0\, G \to G, is an epimorphism. Taking the kernel of this and then applying π 0\pi_0, yields a crossed module constructed from the Moore complex of GG

NG 1/d 2(NG 2)NG 0,N G_1/d_2(NG_2)\to N G_0,

which has kernel π 1(G)\pi_1(G) and cokernel π 0(G)\pi_0(G). This crossed module represents the homotopy 2-type of GG. Applying the décalage twice leads to a crossed square which represents the 3-type of GG, … and so on.


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)


  • 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:


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 nn-types is given in

An application in the theory of stacks is discussed in

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

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