n-group (category theory)

One of the principal examples of higher groups come from the homotopy types of Eilenberg–MacLane spaces ${\displaystyle K(A,n)}$ since they are the fundamental building blocks for constructing higher groups, and homotopy types in general. For instance, every group ${\displaystyle G}$ can be turned into an Eilenberg-Maclane space ${\displaystyle K(G,1)}$ through a simplicial construction,[1] and it behaves functorially. This construction gives an equivalence between groups and 1-groups. Note that some authors write ${\displaystyle K(G,1)}$ as ${\displaystyle BG}$, and for an abelian group ${\displaystyle A}$, ${\displaystyle K(A,n)}$ is written as ${\displaystyle B^{n}A}$.

The definition and many properties of 2-groups are already known. 2-groups can be described using crossed modules and their classifying spaces. Essentially, these are given by a quadruple ${\displaystyle (\pi _{1},\pi _{2},t,\omega )}$ where ${\displaystyle \pi _{1},\pi _{2}}$ are groups with ${\displaystyle \pi _{2}}$ abelian,

${\displaystyle t:\pi _{1}\to {\text{Aut}}(\pi _{2})}$

a group morphism, and ${\displaystyle \omega \in H^{3}(B\pi _{1},\pi _{2})}$ a cohomology class. These groups can be encoded as homotopy ${\displaystyle 2}$-types ${\displaystyle X}$ with ${\displaystyle \pi _{1}(X)=\pi _{1}}$ and ${\displaystyle \pi _{2}(X)=\pi _{2}}$, with the action coming from the action of ${\displaystyle \pi _{1}(X)}$ on higher homotopy groups, and ${\displaystyle \omega }$ coming from the Postnikov tower since there is a fibration

${\displaystyle B^{2}\pi _{2}\to X\to B\pi _{1}}$

coming from a map ${\displaystyle B\pi _{1}\to B^{3}\pi _{2}}$. Note that this idea can be used to construct other higher groups with group data having trivial middle groups ${\displaystyle \pi _{1},e,\ldots ,e,\pi _{n}}$, where the fibration sequence is now

${\displaystyle B^{n}\pi _{n}\to X\to B\pi _{1}}$

coming from a map ${\displaystyle B\pi _{1}\to B^{n+1}\pi _{n}}$ whose homotopy class is an element of ${\displaystyle H^{n+1}(B\pi _{1},\pi _{n})}$.

Another interesting and accessible class of examples which requires homotopy theoretic methods, not accessible to strict groupoids, comes from looking at homotopy 3-types of groups.[2] Essential, these are given by a triple of groups ${\displaystyle (\pi _{1},\pi _{2},\pi _{3})}$ with only the first group being non-abelian, and some additional homotopy theoretic data from the Postnikov tower. If we take this 3-group as a homotopy 3-type ${\displaystyle X}$, the existence of universal covers gives us a homotopy type ${\displaystyle {\hat {X}}\to X}$ which fits into a fibration sequence

${\displaystyle {\hat {X}}\to X\to B\pi _{1}}$

giving a homotopy ${\displaystyle {\hat {X}}}$ type with ${\displaystyle \pi _{1}}$ trivial on which ${\displaystyle \pi _{1}}$ acts on. These can be understood explicitly using the previous model of ${\displaystyle 2}$-groups, shifted up by degree (called delooping). Explicitly, ${\displaystyle {\hat {X}}}$ fits into a postnikov tower with associated Serre fibration

${\displaystyle B^{3}\pi _{3}\to {\hat {X}}\to B^{2}\pi _{2}}$

giving where the ${\displaystyle B^{3}\pi _{3}}$-bundle ${\displaystyle {\hat {X}}\to B^{2}\pi _{2}}$ comes from a map ${\displaystyle B^{2}\pi _{2}\to B^{4}\pi _{3}}$, giving a cohomology class in ${\displaystyle H^{4}(B^{2}\pi _{2},\pi _{3})}$. Then, ${\displaystyle X}$ can be reconstructed using a homotopy quotient ${\displaystyle {\hat {X}}//\pi _{1}\simeq X}$.

The previous construction gives the general idea of how to consider higher groups in general. For an n group with groups ${\displaystyle \pi _{1},\pi _{2},\ldots ,\pi _{n}}$ with the latter bunch being abelian, we can consider the associated homotopy type ${\displaystyle X}$ and first consider the universal cover ${\displaystyle {\hat {X}}\to X}$. Then, this is a space with trivial ${\displaystyle \pi _{1}({\hat {X}})=0}$, making it easier to construct the rest of the homotopy type using the postnikov tower. Then, the homotopy quotient ${\displaystyle {\hat {X}}//\pi _{1}}$ gives a reconstruction of ${\displaystyle X}$, showing the data of an ${\displaystyle n}$-group is a higher group, or Simple space, with trivial ${\displaystyle \pi _{1}}$ such that a group ${\displaystyle G}$ acts on it homotopy theoretically. This observation is reflected in the fact that homotopy types are not realized by simplicial groups, but simplicial groupoids[3]^{pg 295} since the groupoid structure models the homotopy quotient ${\displaystyle -//\pi _{1}}$.

Going through the construction of a 4-group ${\displaystyle X}$ is instructive because it gives the general idea for how to construct the groups in general. For simplicity, let's assume ${\displaystyle \pi _{1}=e}$ is trivial, so the non-trivial groups are ${\displaystyle \pi _{2},\pi _{3},\pi _{4}}$. This gives a postnikov tower

${\displaystyle X\to X_{3}\to B^{2}\pi _{2}\to *}$

where the first non-trivial map ${\displaystyle X_{3}\to B^{2}\pi _{2}}$ is a fibration with fiber ${\displaystyle B^{3}\pi _{3}}$. Again, this is classified by a cohomology class in ${\displaystyle H^{4}(B^{2}\pi _{2},\pi _{3})}$. Now, to construct ${\displaystyle X}$ from ${\displaystyle X_{3}}$, there is an associated fibration

${\displaystyle B^{4}\pi _{4}\to X\to X_{3}}$

given by a homotopy class ${\displaystyle [X_{3},B^{5}\pi _{4}]\cong H^{5}(X_{3},\pi _{4})}$. In principal[4] this cohomology group should be computable using the previous fibration ${\displaystyle B^{3}\pi _{3}\to X_{3}\to B^{2}\pi _{2}}$ with the Serre spectral sequence with the correct coefficients, namely ${\displaystyle \pi _{4}}$. Doing this recursively, say for a ${\displaystyle 5}$-group, would require several spectral sequence computations, at worse ${\displaystyle n!}$ many spectral sequence computations for an ${\displaystyle n}$-group.

• Algebraic invariants for homotopy types
• On Algebraic Models for Homotopy 3-Types
• Computing Homotopy Types Using Crossed N-Cubes of Groups
• Weak units and homotopy 3-types
• Algebraic models for homotopy n-types - musings by Tim porter discussing the pitfalls of modelling homotopy n-types with n-cubes

• Determination of the Second Homology and Cohomology Groups of a Space by Means of Homotopy Invariants
• The third cohomology group classifies crossed module extensions
• On the second cohomology group of a simplicial group
• Group cohomology with coefficients in a crossed-module

Note this is (slightly) distinct from the previous section, because it is about taking cohomology over a space ${\displaystyle X}$ with values in a higher group ${\displaystyle \mathbb {G} _{\bullet }}$, giving higher cohomology groups ${\displaystyle \mathbb {H} ^{*}(X,\mathbb {G} _{\bullet })}$. If we are considering ${\displaystyle X}$ as a homotopy type and assuming the homotopy hypothesis, then these are the same cohomology groups.

• Cohomology with coefficients in stacks
• Cohomology with values in a sheaf of crossed groups over a site