nLab simplicial homotopy

Idea

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 $C$.

Definition via cylinders

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

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

$$\array{ X \simeq X\times \Delta[0] &\stackrel{Id \times \delta^1}{\to}& X \times \Delta[1] & \stackrel{Id \times \delta^0}{\leftarrow}& X \times \Delta[0] \simeq X \\ & {}_f\searrow &\downarrow^\eta& \swarrow_{g} \\ && Y }$$

commutes.

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

Combinatorial definition

Given morphisms $f,g,:X\to Y$ of simplicial objects in any category $C$, a simplicial homotopy is a family of morphisms, $h_i:X_n\to Y_{n+1}$, $i= 0,\ldots,n$ of $C$, such that $d_0 h_0 = f_n$, $d_{n+1} h_n = g_n$ and

$$d_i h_j = \left\lbrace\array{ h_{j-1}d_i, & i\lt j \\ d_i h_{i-1}, &i=j\neq 0\\ h_j d_{i-1}, & i\gt j+1. }\right.$$

$$s_i h_j = \left\lbrace\array{ h_{j+1} s_i, & i\le j\\ h_j s_{i-1}, & i\gt j. }\right.$$

Note that we did not explicitly say what $d_i h_j$ should be when $i = j + 1$, but if we look at the rule for $d_{j + 1} h_{j + 1}$, we see this requires $d_{j + 1} h_{j + 1} = d_{j + 1} h_j$, which is the relevant compatibility condition.

If we use the cylinder convention, this is equivalent to specifying a homotopy $g \Rightarrow f$. Define $\eta_0 = d_0h_0$, $\eta_{n+1}=d_{n+1}h_n$, and $\eta_j = d_j h_j = d_j h_{j-1}$ for all $1\leq j\leq n$. The map $\eta : X \times \Delta[1] \rightarrow Y$ as seen in the cylinder definition is then formed by the universal property of coproducts. In particular, the set $\Delta[1](n)$ has $n+2$ elements which can be presented as a string of length $n+1$ of the form $(0\cdots01\cdots1)$. The map $\eta_j$ is the restriction of $\eta$ to the string that contains exactly $j$ number of $0$‘s in it.

Remark / warning on conventions

The above formulae give one of the, at least, two forms of the combinatorial specification of a homotopy between $f$ and $g$. (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 $g$ to $f$, or reversing the labelling of the $h_i$.

Commentary

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, $C$. This uses the fact that the category of simplicial objects in a cocomplete category, $C$, has copowers with finite simplicial sets and hence in particular with $\Delta[1]$. For a simplicial set $K$ and a simplicial object $X_{\geq 0}$ in $C$, the tensoring is done levelwise using $(K \odot X)_n := K_n \odot_{\mathbf{Set}} X_n$. Here, $\odot_{\mathbf{Set}}$ denote the usual coproduct tensoring over sets for any category $C$.

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 $G$ is played by a simplicial object in the category of groups which in degree $n$ equals the free product of $(n+2)$ copies of $G$, indexed by the set $\Delta[1]_n$ (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.

Anonymous Coward: With exercise 8.3.5 of Weibel in mind, what is the notion of “cylinder” meant in the assertion “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” for a general finitely cocomplete category?

Tim Porter: I have transferred this question to the nForum where it will be easier for others to reply. In the meantime some indication is given in Kamps and Porter as referenced below. I myself do not quite understand your question as it is presently stated, but this may be that I am too near to the subject matter to see the difficulty.

Properties

Lemma

Precisely when $Y$ is a Kan complex, the relation

$$(f \sim g) \Leftrightarrow (\exists simplicial homotopy f \Rightarrow g : X \to Y )$$

is an equivalence relation.

Proof

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 $x,y : \Delta[0] \to Y$ is an equivalence relation when $Y$ is a Kan complex.

We identify in the following $x$ and $y$ with vertices in the image of these maps.

• -reflexivity- For every vertex $x \in Y_0$, the degenerate 1-simplex $s_0 x \in S_1$ has, by the simplicial identities, 0-faces $d_0 s_0 x = x$ and $d_1 s_0 x = x$.

  $$(d_1 s_0 x) \stackrel{s_0 x}{\to} (d_0 s_0 x)$$

  Therefore the morphism $\eta : \Delta[0] \times \Delta[1] \to Y$ that takes the unique non-degenerate 1-simplex in $\Delta[1]$ to $s_0 x$ constitutes a homotopy from $x$ to itself.

• -transitivity- let $v_2 : x \to y$ and $v_0 : y \to z$ in $Y_1$ be 1-cells. Together they determine a map from the horn $\Lambda^2_1$ to $Y$,

  $$(v_2, v_2) : \Lambda^2_1 \to Y \,.$$

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

  $$\array{ \Lambda^2_1 &\stackrel{(v_0,v_2)}{\to}& Y \\ \downarrow & \nearrow_{\theta} \\ \Delta[2] } \,.$$

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

  $$\array{ && (d_0 d_2 \theta) = (d_1 d_0 \theta) \\ & {}^{d_2 \theta }\nearrow & \Downarrow \theta & \searrow^{d_0 \theta} \\ (d_1 d_2 \theta) = (d_1 d_1 \theta) && \stackrel{d_1 \theta}{\to} && (d_0 d_1 \theta) = (d_0 d_1 \theta) }$$

  that the 1-cell boundary bit $d_1 \theta$ in turn has 0-cell boundaries

  $$d_0 d_1 \theta = d_0 d_0 \theta = z$$

  and

  $$d_1 d_1 \theta = d_1 d_2 \theta = x \,.$$

  This means that $d_1 \theta$ is a homotopy $x \to z$.

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

  $$\array{ \Lambda^2_0 &\stackrel{v_1, v_2}{\to}& Y \\ \downarrow & \nearrow_{\theta'} \\ \Delta[2] }$$

  on the full 2-simplex.

  Now the 1-cell boundary $d_0 \theta'$ has, using the simplicial identities, 0-cell boundaries

  $$d_0 d_0 \theta' = d_0 d_1 \theta' = x$$

  and

  $$d_1 d_0 \theta' = d_0 d_2 \theta' = y$$

  and hence yields a homotopy $y \to x$. 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 $f : X \to Y$ is the Hom-adjunct of a morphism $\bar f : \Delta[0] \to [X,Y]$, and a homotopy $\eta : X \times \Delta[1] \to Y$ is the adjunct of a morphism $\bar \eta : \Delta[1] \to [X,Y]$. Therefore homotopies $\eta : f \Rightarrow g$ are in bijection with homotopies $\bar \eta : \bar f \to \bar g$.

Proposition

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

Proposition

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

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

• $f_1\sim 0$ and $f_2\sim 0$ implies $f_1+f_2 \sim 0$,

• if $f\circ h$ (resp. $h\circ f$) exists and if $f\sim 0$ then $f\circ h\sim 0$ (resp. $h\circ f\sim 0$).

References

• Goerss, Jardine, Simplicial homotopy theory (pdf)

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.