**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**

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**Homotopy groups**

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**Theorems**

A homotopy $2$-type is a view of a space where we consider its properties only up to the $2$nd homotopy group $\pi_2$. To make this precise, we look at maps that ‘see’ invariants in dimensions 0,1, and 2. These are the 2-equivalences:

A continuous map $X \to Y$ is a **homotopy $2$-equivalence** if it induces isomorphisms on $\pi_i$ for $i = 0, 1, 2$ at each basepoint. Two spaces share the same **homotopy $2$-type** if they are linked by a zig-zag chain of homotopy $2$-equivalences.

For any ‘nice’ space $X$, you can kill its homotopy groups in higher dimensions by attaching cells, thus constructing a new space $Y$ so that the inclusion of $X$ into $Y$ is a homotopy $2$-equivalence; up to (weak) homotopy equivalence, the result is the same for any space with the same homotopy $2$-type. Accordingly, a **homotopy $2$-type** may alternatively be defined as a space with trivial $\pi_i$ for $i \gt 2$, or as the unique (weak) homotopy type of such a space, or as its fundamental $\infty$-groupoid (which will be a $2$-groupoid).

See the general discussion in homotopy n-type.

Homotopy $2$-types can be classified by various different types of algebraic data.

Homotopy $2$-types can be classified up to weak homotopy type by crossed modules of groupoids. These are the $2$-truncated versions of crossed complexes. Such a $C$ consists of a morphism

$\delta: C_2 \to C_1$

of groupoids with object set $C_0$ such that $C_2$ is totally disconnected, i.e. is a family of groups $C_2(x), x \in C_0$. Further the groupoid $C_1$ operates on this family of groups so that (using right operations) if $a: x \to y$ in $C_1$ and $u \in C_2(x)$ then $u^a \in C(y)$; and the usual rules for an operation are satisfied, namely $(uv)^a=u^a v^a$, $u^1=u$, $(u^a)^b=u^{a b}$ when these are defined. Further the two basic crossed module rules hold:

CM1) $\delta(u^a)= a^{-1} (\delta u) a$;

CM2) $v^{-1} u v = u ^{\delta v}$;

for all $a \in C_1, u,v \in C_2$ when the rules make sense.

Such a crossed module may be extended to a crossed complex $sk^2 C$ by adding trivial elements in dimensions higher than 2. Hence there is a simplicial nerve $N^\Delta C$ which in dimension $n$ is

$Crs(\Pi (\Delta^n_*), sk^2 C).$

The geometric realisation of this is the classifying space $BC$. Its first and second homotopy groups at $x \in C_0$ are the cokernel and kernel of $\delta: C_2(x) \to C_1(x,x)$. Its components are those of the groupoid $C_1$. All other homotopy groups are trivial.

If $X$ is a CW-complex then there is a bijection of homotopy classes

$[X,BC] \cong [\Pi X_*, sk^2 C],$

and hence there is a map $X \to B(cotr^2 \Pi X_*)$ inducing isomorphisms of homotopy groups in dimensions 1 and 2.

Here the cotruncation $cotr^n D$ of a general crossed complex $D$ agree with $D$ up to dimension $(n-1)$, is $Cok \delta_{n+1}$ in dimension $n$, and is trivial in higher dimensions.

It is in this sense that crossed modules of groupoids classify weak homotopy $2$-types.

The category $Crs^2$ of such crossed modules of groupoids is equivalent to that of strict 2-groupoids. Further, $Crs^2$ is monoidal closed:

$Crs^2(C \otimes D, E) \cong Crs^2(C, CRS^2(D,E))$

and with a unit interval object $I$ so that (left) homotopies are determined as morphisms $Crs^2(I \otimes D,E)$ or as elements of $CRS^2(D,E)_1$.

As a crossed module give rise to an internal groupoid in the category of groups (or groupoids), we can take the nerve of that structure and get a simplicial group (or simplicially enriched groupoid). From a simplicial group(oid), $G$, one can define a simplicial set called the classifying space $\overline{W}G$ of the simplicial group, $G$, for which construction see simplicial group. We thus can start with a crossed module $C$ form a simplicial group and then take $\overline{W}$ of that to get another model of $\mathcal{B}C$.

With respect to the standard homotopy theory-structure on 2-groupoids (2-truncated infinity-groupoids) these are equivalent to homotopy 2-types. See at *homotopy hypothesis* for more on this.

see

- Antonio Martínez Cegarra; Benjamın A. Heredia ; Josué Remedios,
*Double groupoids and homotopy 2-types*Appl. Categ. Struct. 20, No. 4, 323-378 (2012), see also arXiv:1003.3820.

homotopy level | n-truncation | homotopy theory | higher category theory | higher topos theory | homotopy type theory |
---|---|---|---|---|---|

h-level 0 | (-2)-truncated | contractible space | (-2)-groupoid | true/unit type/contractible type | |

h-level 1 | (-1)-truncated | contractible-if-inhabited | (-1)-groupoid/truth value | (0,1)-sheaf/ideal | mere proposition/h-proposition |

h-level 2 | 0-truncated | homotopy 0-type | 0-groupoid/set | sheaf | h-set |

h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid |

h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | (3,1)-sheaf/2-stack | h-2-groupoid |

h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | (4,1)-sheaf/3-stack | h-3-groupoid |

h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf/n-stack | h-$n$-groupoid |

h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |

Last revised on January 19, 2023 at 11:41:25. See the history of this page for a list of all contributions to it.