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Crossed complex over groupoid $C_{1}$ is a sequence

$...\to C_{n}{\xrightarrow[{}]{\delta _{n}}}C_{n-1}\to ...{\xrightarrow[{}]{\delta _{3}}}C_{2}{\xrightarrow[{}]{\delta _{2}}}C_{1}$

such that

(1) $C_{n}$ is a totally disconnected groupoid for $n\geq 2$ with the same set of objects as $C_{1}$, namely $C_{0}$.

(2) For $n\geq 2$ there is an action of the groupoid $C_{1}$ on the right on each $C_{n}$.

(3) For $n\geq 2$ arrows $\delta _{n}\colon C_{n}\to C_{n-1}$ are morphisms of groupoids over $C_{0}$ and preserves the action of $C_{1}$.

(4) This data satisfies next two axioms:

(CX1) $\delta _{n-1}\delta _{n}=0$ for $n\geq 3$.

(CX2) Image ${\text{im}}\delta _{2}$ acts by conjugation on $C_{2}$ and trivially on $C_{n}$ for $n\geq 3$.

Category of crossed complexes

A morphism of crossed complexes $f\colon C\to D$ is a family of morphisms of groupoids $f_{n}\colon C_{n}\to D_{n}$ all inducing the same map of vertices $f_{0}\colon C_{0}\to D_{0}$, and compatible with the boundary maps and the actions of $C_{1}$ and $D_{1}$.

Crossed complexes together with morphisms form a category denoted by ${\text{Crs}}$.

Examples

(1) 1-truncated crossed complex is just a groupoid.

(2) 2-truncated crossed complex $C_{2}{\xrightarrow[{}]{\delta _{2}}}C_{1}$, where $C_{2}$ and $C_{1}$ are groups, is a crossed module.

(3) Let $X_{*}$ be a filtered space. Combine all base points $x\in X_{0}$ to get fundamental groupoids $\pi _{n}(X_{n},X_{n-1},X_{0})$ for $n\geq 2$ and the groupoid $\pi _{1}(X_{1},X_{0})$. Then the fundamental crossed complex $\Pi X_{*}$ of the filtered space $X_{*}$ is the sequence

$...\to (\Pi X_{*})_{n}{\xrightarrow[{}]{\delta _{n}}}(\Pi X_{*})_{n-1}\to ...{\xrightarrow[{}]{\delta _{3}}}(\Pi X_{*})_{2}{\xrightarrow[{}]{\delta _{2}}}(\Pi X_{*})_{1}$

where $(\Pi X_{*})_{n}=\pi _{n}(X_{n},X_{n-1},X_{0})$ for $n\geq 2$ and $(\Pi X_{*})_{1}=\pi _{1}(X_{1},X_{0})$. Boundary maps $\delta _{n}$ for $n\geq 3$ are defined via the composition of maps from the long exact sequence of relative homotopy groups:

$\pi _{n}(X_{n},X_{n-1},x){\xrightarrow[{}]{\partial _{n}}}\pi _{n-1}(X_{n-1},x){\xrightarrow[{}]{i_{n-1}}}\pi _{n-1}(X_{n-1},X_{n-2},x)$

where $x\in X_{0}$. Here $\delta _{2}$ is the standart boundary map.

Fundamental crossed complex construction defines a functor $\Pi \colon {\text{FTop}}\to {\text{Crs}}$ from the category of filtered topological spaces to category of crossed complexes. This functor plays a central role in the formulation of a Higher Homotopy Seifert-van Kampen Theorem.

External links

• Brown, R.; Higgins, P.J.; Sivera, R. (2011). Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids. EMS Tracts in Mathematics. Vol. 15. arXiv:math/0407275. doi:10.4171/083. ISBN 978-3-03719-583-3.
• Brown, R. (1999). "Groupoids and crossed objects in algebraic topology" (PDF). Homology, Homotopy and Applications. 1 (1): 1–78. doi:10.4310/HHA.1999.v1.n1.a1.
• crossed complex at the nLab
• filtered+topological+space at the nLab

References

• Brown, R.; Higgins, P.J.; Sivera, R. (2011). Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids. EMS Tracts in Mathematics. Vol. 15. arXiv:math/0407275. doi:10.4171/083. ISBN 978-3-03719-583-3.
• Brown, R. (1999). "Groupoids and crossed objects in algebraic topology" (PDF). Homology, Homotopy and Applications. 1 (1): 1–78. doi:10.4310/HHA.1999.v1.n1.a1.

Category:Group actions Category:Algebraic topology