PlanetPhysics/Hamiltonian Algebroid

Homotopy addition lemma

Let Failed to parse (syntax error): {\displaystyle f: \boldsymbol{\rho ^\square(X) \to \mathsf D} be a [[../TrivialGroupoid/|morphism]] of double groupoids with connection. If $\alpha \in {\mathit{\rho }}_2^{◻}(X)$ is thin, then $f(\alpha )$ is thin.}

Remarks

The [[../GroupoidHomomorphism2/|groupoid]] ${\mathit{\rho }}_2^{◻}(X)$ employed here is as defined by the [[../CubicallyThinHomotopy2/|cubically thin homotopy]] on the set $R_2^{◻}(X)$ of squares. Additional explanations of the data, including [[../PreciseIdea/|concepts]] such as path groupoid and [[../ThinEquivalence/|homotopy]] [[../WeakHomotopy/|double groupoid]] are provided in an attachment.

Corollary

\emph{Let $u:I^3\to X$ be a singular cube in a Hausdorff space $X$. Then by restricting $u$ to the faces of $I^3$ and taking the corresponding elements in ${\mathit{\rho }}_2^{◻}(X)$, we obtain a cube in ${\mathit{\rho }}^{◻}(X)$ which is commutative by the Homotopy addition lemma for ${\mathit{\rho }}^{◻}(X)$ (, [[../Predicate/|proposition]] 5.5). Consequently, if $f:{\mathit{\rho }}^{◻}(X)\to 𝖣$ is a morphism of double groupoids with connections, any singular cube in $X$ determines a [3-shell commutative]{http://www.math.purdue.edu/research/atopology/BrownR-Kamps-Porter/vkt7.txt} in $𝖣$.}