homotopy

This page is about homotopy as a transformation. For homotopy sets in homotopy categories, see homotopy (as an operation).

**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 many categories $C$ in which one does homotopy theory, there is a notion of *homotopy* between morphisms, which is closely related to the 2-morphisms in higher category theory: a homotopy between two morphisms is a way in which they are equivalent.

If we regard such a category as a presentation of an $(\infty,1)$-category, then homotopies $f\sim g$ present the 2-cells $f\Rightarrow g$ in the resulting $(\infty,1)$-category.

For $f,g\colon X \longrightarrow Y$ two continuous functions between topological spaces $X,Y$, then a **left homotopy**

$\eta \colon f \,\Rightarrow_L\, g$

is a continuous function

$\eta \;\colon\; X \times I \longrightarrow Y$

out of the standard cylinder object over $X$: the product space of $X$ with the Euclidean closed interval $I \coloneqq [0,1]$, such that this fits into a commuting diagram of the form

$\array{
X
\\
{}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}}
\\
X \times I &\stackrel{\eta}{\longrightarrow}& Y
\\
{}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}}
\\
X
}
\,.$

(graphics grabbed from J. Tauber here)

Let $X$ be a topological space and let $x,y \in X$ be two of its points, regarded as functions $x,y \colon \ast \longrightarrow X$ from the point to $X$. Then a left homotopy, def. 1, between these two functions is a commuting diagram of the form

$\array{
\ast
\\
{}^{\mathllap{\delta_0}}\downarrow & \searrow^{\mathrlap{x}}
\\
I &\stackrel{\eta}{\longrightarrow}& Y
\\
{}^{\mathllap{\delta_1}}\uparrow & \nearrow_{\mathrlap{y}}
\\
\ast
}
\,.$

This is simply a continuous path in $X$ whose endpoints are $x$ and $y$.

If $C$ is enriched over Top, then a **homotopy** in $C$ between maps $f,g:X\,\rightrightarrows \,Y$ is a map $H:[0,1] \to C(X,Y)$ in $Top$ such that $H(0)=f$ and $H(1)=g$. In $Top$ itself this is the classical notion.

If $C$ has copowers, then an equivalent definition is a map $[0,1]\odot X\to Y$, while if it has powers, an equivalent definition is a map $X\to \pitchfork([0,1],Y)$.

There is a similar definition in a simplicially enriched category, replacing $[0,1]$ with the 1-simplex $\Delta^1$, with the caveat that in this case not all *simplicial homotopies* need be composable even if they match correctly. (This depends on whether or not all (2,1)-horns in the simplicial set, $C(X,Y)$, have fillers.) Likewise in a dg-category we can use the “chain complex interval” to get a notion of *chain homotopy*.

If $\mathcal{C}$ is a model category, it has an intrinsic notion of homotopy determined by its factorizations. For more on the following see at *homotopy in a model category*.

Let $\mathcal{C}$ be a model category and $X \in \mathcal{C}$ an object.

- A
**path object**$Path(X)$ for $X$ is a factorization of the diagonal $\nabla_X \colon X \to X \times X$ as

$\nabla_X
\;\colon\;
X \underoverset{\in W}{i}{\longrightarrow} Path(X) \overset{(p_0,p_1)}{\longrightarrow} X \times X
\,.$

where $X\to Path(X)$ is a weak equivalence. This is called a **good path object** if in addition $Path(X) \to X \times X$ is a fibration.

- A
**cylinder object**$Cyl(X)$ for $X$ is a factorization of the codiagonal (or “fold map”) $\Delta_X X \sqcup X \to X$ as

$\Delta_X
\;\colon\;
X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \underoverset{p}{\in W}{\longrightarrow} X
\,.$

where $Cyl(X) \to X$ is a weak equivalence. This is called a **good cylinder object** if in addition $X \sqcup X \to Cyl(X)$ is a cofibration.

By the factorization axioms every object in a model category has both a good path object and as well as a good cylinder object according to def. 2. But in some situations one is genuinely interested in using non-good such objects.

For instance in the classical model structure on topological spaces, the obvious object $X\times [0,1]$ is a cylinder object, but not a good cylinder unless $X$ itself is cofibrant (a cell complex in this case).

More generally, the path object $Path(X)$ of def. 2 is analogous to the powering $\pitchfork(I,X)$ with an interval object and the cyclinder object $Cyl(X)$ is analogous to the tensoring with a cylinder object $I\odot X$. In fact, if $\mathcal{C}$ is a $V$-enriched model category and $X$ is fibrant/cofibrant, then these powers and copowers are in fact examples of (good) path and cylinder objects if the interval object is sufficiently good.

Let $f,g \colon X \longrightarrow Y$ be two parallel morphisms in a model category.

- A
**left homotopy**$\eta \colon f \Rightarrow_L g$ is a morphism $\eta \colon Cyl(X) \longrightarrow Y$ from a cylinder object of $X$, def. 2, such that it makes this diagram commute:

$\array{
X &\longrightarrow& Cyl(X) &\longleftarrow& X
\\
& {}_{\mathllap{f}}\searrow &\downarrow^{\mathrlap{\eta}}& \swarrow_{\mathrlap{g}}
\\
&&
Y
}
\,.$

- A
**right homotopy**$\eta \colon f \Rightarrow_R g$ is a morphism $\eta \colon X \to Path(Y)$ to some path object of $X$, def. 2, such that this diagram commutes:

$\array{
&& X
\\
& {}^{\mathllap{f}}\swarrow & \downarrow^{\mathrlap{\eta}} & \searrow^{\mathrlap{g}}
\\
Y &\longleftarrow& Path(Y) &\longrightarrow& Y
}
\,.$

By remark 1 it follows that in a $Top$-enriched model category, any enriched homotopy between maps $X\to Y$ is a left homotopy if $X$ is cofibrant and a right homotopy if $Y$ is fibrant. Similar remarks hold for other enrichments.

For more see at *homotopy in a model category*.

Clearly the concepf of left homotopy in def. 2 only needs part of the model category axioms and thus makes sense more generally in suitable cofibration categories. Dually, the concepf of path ojects in def. 2 makes sense more generally in suitable fibration categories such as categories of fibrant objects in the sense of Brown.

Likewise if there is a cylinder functor, one gets functorially defined cylinder objects, etc.

homotopy | cohomology | homology | |
---|---|---|---|

$[S^n,-]$ | $[-,A]$ | $(-) \otimes A$ | |

category theory | covariant hom | contravariant hom | tensor product |

homological algebra | Ext | Ext | Tor |

enriched category theory | end | end | coend |

homotopy theory | derived hom space $\mathbb{R}Hom(S^n,-)$ | cocycles $\mathbb{R}Hom(-,A)$ | derived tensor product $(-) \otimes^{\mathbb{L}} A$ |

See the references at *homotopy theory* and at *model category*.

Revised on July 18, 2017 00:42:14
by Urs Schreiber
(84.61.249.8)