simplicial homotopy


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Homotopy theory



A simplicial homotopy is a homotopy in the classical model structure on simplicial sets. It can also be defined combinatorially; in that form one can define a homotopy 2-cell between morphisms of simplicial objects in any category CC.

Definition via cylinders

SSet has a cylinder functor given by cartesian product with the standard 1-simplex, I:=Δ[1]I := \Delta[1]. (In fact, one can define simplicial cylinders, Δ[1]X\Delta[1]\odot X, more generally, for example for XX being a simplicial object in an cocomplete category CC,(see below).)

Therefore for f,g:XYf,g : X \to Y two morphisms of simplicial sets, a homotopy η:fg\eta : f \Rightarrow g is a morphism η:X×Δ[1]Y\eta : X \times \Delta[1] \to Y such that the diagram

XX×Δ[0] Id×δ 1 X×Δ[1] Id×δ 0 X×Δ[0]X f η g Y \array{ X \simeq X\times \Delta[0] &\stackrel{Id \times \delta^1}{\to}& X \times \Delta[1] & \stackrel{Id \times \delta^0}{\leftarrow}& X \times \Delta[0] \simeq X \\ & {}_f\searrow &\downarrow^\eta& \swarrow_{g} \\ && Y }


Remark: Since in the standard model structure on simplicial sets every simplicial set is cofibrant, this indeed defines left homotopies.

Combinatorial definition

Given morphisms f,g,:XYf,g,:X\to Y of simplicial objects in any category CC, a simplicial homotopy is a family of morphisms, h i:X nY n+1h_i:X_n\to Y_{n+1}, i=0,,ni= 0,\ldots,n of CC, such that d 0h 0=f nd_0 h_0 = f_n, d n+1h n=g nd_{n+1} h_n = g_n and

d ih j={h j1d i, i<j d ih i1, i=j0 h jd i1, i>j+1. d_i h_j = \left\lbrace\array{ h_{j-1}d_i, & i\lt j \\ d_i h_{i-1}, &i=j\neq 0\\ h_j d_{i-1}, & i\gt j+1. }\right.
s ih j={h j+1s i, ij h js i1, i>j. s_i h_j = \left\lbrace\array{ h_{j+1} s_i, & i\le j\\ h_j s_{i-1}, & i\gt j. }\right.

Remark / warning on conventions

The above formulae give one of the, at least, two forms of the combinatorial specfication of a homotopy between ff and gg. (When trying to construct a specific homotopy using a combinatorial form, check which convention is being used!) The two forms correspond to different conventions such as saying that this is a homotopy from gg to ff, or reversing the labelling of the h ih_i.


It is fairly easy to prove that the combinatorial definition of homotopy agrees with the one via the cylinder both for simplicial sets and for simplicial objects in any finitely cocomplete category, CC. This uses the fact that the category of simplicial objects in a cocomplete category, CC, has copowers with finite simplicial sets and hence in particular with Δ[1]\Delta[1]. (As there are explicit formulae for the construction of copowers …)

Tim: With only my own resources available, I was unable to find them so was hoping someone kind would come up with them. They derive from the coend formulae​/Kan extension formulae using some combinatorics to discuss the indexing sets. I think Quillen gave some form of them, but have not got a copy of HA. I needed them recently and could not find them in any of the usual sources, and did not manage to work them out using the Kan extension idea either (Help please anyone). We could do with those formulae or with a reference to them at least.

In the case of the category of (not necessarily abelian) groups, the combinatorial definition equals the one via cylinder only if the role of “cylinder” for a group GG is played by a simplicial object in the category of groups which in degree nn equals the free product of (n+2)(n+2) copies of GG, indexed by the set Δ[1] n\Delta[1]_n (noted by Swan and quoted in exercise 8.3.5 of Weibel: Homological algebra).

Tim Porter: Perhaps we need an explicit description of copowers in simplicial objects also. I pointed out in an edit above that the combinatorial description is much more general than just for simplicial objects in an abelian category.

Can specific references to Swan be given, anyone?

Zoran Škoda: I agree that one should talk about copowers etc.



Precisely when YY is a Kan complex, the relation

(fg)(simplicialhomotopyfg:XY) (f \sim g) \Leftrightarrow (\exists simplicial homotopy f \Rightarrow g : X \to Y )

is an equivalence relation.


Since Kan complexes are precisely the fibrant objects with respect to the standard model structure on simplicial sets this follows from general statements about homotopy in model categories.

The following is a direct proof.

We first show that the homotopy between points x,y:Δ[0]Yx,y : \Delta[0] \to Y is an equivalence relation when YY is a Kan complex.

We identify in the following xx and yy with vertices in the image of these maps.

  • -reflexivity- For every vertex xY 0x \in Y_0, the degenerate 1-simplex s 0xS 1s_0 x \in S_1 has, by the simplicial identities, 0-faces d 0s 0x=xd_0 s_0 x = x and d 1s 0x=xd_1 s_0 x = x.

    (d 1s 0x)s 0x(d 0s 0x) (d_1 s_0 x) \stackrel{s_0 x}{\to} (d_0 s_0 x)

    Therefore the morphism η:Δ[0]×Δ[1]Y\eta : \Delta[0] \times \Delta[1] \to Y that takes the unique non-degenerate 1-simplex in Δ[1]\Delta[1] to s 0xs_0 x constitutes a homotopy from xx to itself.

  • -transitivity- let v 2:xyv_2 : x \to y and v 0:yzv_0 : y \to z in Y 1Y_1 be 1-cells. Together they determine a map from the horn Λ 1 2\Lambda^2_1 to YY,

    (v 2,v 2):Λ 1 2Y. (v_2, v_2) : \Lambda^2_1 \to Y \,.

    By the Kan complex property there is an extension θ\theta of this morphism through the 2-simplex Delta 2Delta^2

    Λ 1 2 (v 0,v 2) Y θ Δ[2]. \array{ \Lambda^2_1 &\stackrel{(v_0,v_2)}{\to}& Y \\ \downarrow & \nearrow_{\theta} \\ \Delta[2] } \,.

    If we again identify θ\theta with its image (the image of its unique non-degenerate 2-cell) in Y 2Y_2, then using the simplicial identities we find

    (d 0d 2θ)=(d 1d 0θ) d 2θ θ d 0θ (d 1d 2θ)=(d 1d 1θ) d 1θ (d 0d 1θ)=(d 0d 1θ) \array{ && (d_0 d_2 \theta) = (d_1 d_0 \theta) \\ & {}^{d_2 \theta }\nearrow & \Downarrow \theta & \searrow^{d_0 \theta} \\ (d_1 d_2 \theta) = (d_1 d_1 \theta) && \stackrel{d_1 \theta}{\to} && (d_0 d_1 \theta) = (d_0 d_1 \theta) }

    that the 1-cell boundary bit d 1θd_1 \theta in turn has 0-cell boundaries

    d 0d 1θ=d 0d 0θ=z d_0 d_1 \theta = d_0 d_0 \theta = z


    d 1d 1θ=d 1d 2θ=x. d_1 d_1 \theta = d_1 d_2 \theta = x \,.

    This means that d 1θd_1 \theta is a homotopy xzx \to z.

  • -symmetry- In a similar manner, suppose that v 2:xyv_2 : x \to y is a 1-cell in Y 1Y_1 that constitutes a homotopy from xx to yy. Let v 1:=s 0xv_1 := s_0 x be the degenerate 1-cell on xx. Since d 1v 1=d 1v 2d_1 v_1 = d_1 v_2 together they define a map Λ 0 2v 1,v 2Y\Lambda^2_0 \stackrel{v_1, v_2}{\to} Y which by the Kan property of YY we may extend to a map θ\theta'

    Λ 0 2 v 1,v 2 Y θ Δ[2] \array{ \Lambda^2_0 &\stackrel{v_1, v_2}{\to}& Y \\ \downarrow & \nearrow_{\theta'} \\ \Delta[2] }

    on the full 2-simplex.

    Now the 1-cell boundary d 0θd_0 \theta' has, using the simplicial identities, 0-cell boundaries

    d 0d 0θ=d 0d 1θ=x d_0 d_0 \theta' = d_0 d_1 \theta' = x


    d 1d 0θ=d 0d 2θ=y d_1 d_0 \theta' = d_0 d_2 \theta' = y

    and hence yields a homotopy yxy \to x. So being homotopic is a symmetric relation on vertices in a Kan complex.

Finally we use the fact that SSet is a cartesian closed category to deduce from this statements about vertices the corresponding statement for all map:

a morphism f:XYf : X \to Y is the Hom-adjunct of a morphism f¯:Δ[0][X,Y]\bar f : \Delta[0] \to [X,Y], and a homotopy η:X×Δ[1]Y\eta : X \times \Delta[1] \to Y is the adjunct of a morphism η¯:Δ[1][X,Y]\bar \eta : \Delta[1] \to [X,Y]. Therefore homotopies η:fg\eta : f \Rightarrow g are in bijection with homotopies η¯:f¯g¯\bar \eta : \bar f \to \bar g.


Let AA be an abelian category and f,g:XYf,g : X\to Y homotopic morphisms of simplicial objects in AA. Then the induced maps f *,g *:N(X)N(Y)f_*, g_* : N(X)\to N(Y) of their normalized chain complexes are chain homotopic.


Let AA be an abelian category. The morphisms of simplicial objects (variant: of unbounded chain complexes) in AA, which are homotopic to zero, form an ideal. More precisely

  • being homotopic is an equivalence relation on the class of morphism,

  • f 10f_1\sim 0 and f 20f_2\sim 0 implies f 1+f 20f_1+f_2 \sim 0,

  • if fhf\circ h (resp. hfh\circ f) exists and if f0f\sim 0 then fh0f\circ h\sim 0 (resp. hf0h\circ f\sim 0).


  • Goerss, Jardine, Simplicial homotopy theory (ps)

Cylinder based homotopy is also discussed extensively in

  • K. H. Kamps and T. Porter, Abstract Homotopy and Simple Homotopy Theory, World Scientific Publishing Co. Inc., River Edge, NJ.
Revised on January 26, 2014 12:48:14 by Anonymous Coward (