homotopy double groupoid of a Hausdorff space

0.1 Homotopy double groupoid of a Hausdorff space

Let X be a Hausdorff space. Also consider the HDA concept of a double groupoid, and how it can be completely specified for a Hausdorff space, X. Thus, in ref. [2] Brown et al. associated to X a double groupoid, ρ₂^□(X) , called the homotopy double groupoid of X which is completely defined by the data specified in Definitions 0.1 to 0.3 in this entry and related objects.

Generally, the geometry of squares and their compositions leads to a common representation of a double groupoid in the following form:

---s1--- S|--------H|| ||| t1 ||| D = s2|||t2 s|||t , ---s--- V -------M t

(0.1)

where M is a set of ‘points’, H,V are ‘horizontal’ and ‘vertical’ groupoids, and S is a set of ‘squares’ with two compositions.

The laws for a double groupoid are also defined, more generally, for any topological space 𝕋, and make it also describable as a groupoid internal to the category of groupoids. Further details of this general definition are provided next.

Given two groupoids H,V over a set M, there is a double groupoid □(H,V ) with H,V as horizontal and vertical edge groupoids, and squares given by quadruples

( ) h ′ v ′ v h

(0.2)

for which we assume always that h,h′∈ H, v,v′∈ V and that the initial and final points of these edges match in M as suggested by the notation, that is for example sh = sv,th = sv′,…, etc. The compositions are to be inherited from those of H,V , that is:

( h ) ( h′ ) ( h ) ( h ) ( k ) ( hk ) v v ′ ∘1 w w′ = vw v′w ′ , v v′ ∘2 v′ v′′ = v v′′ . h′ h ′′ h′′ h′ k′ h′k′

(0.3)

Alternatively, the data for the above double groupoid D can be specified as a triple of groupoid structures:

(D2, D1,∂1−,∂+1 ,+1,𝜀1),(D2, D1, ∂−2 ,∂+2 ,+2,𝜀2),(D1, D0, ∂−1 ,∂+1 ,+, 𝜀),

where:

D0 = M , D1 = V = H , D2 = S,

s1 = ∂− , t1 = ∂+ , s2 = s = ∂− 2 2 1

and

t2 = t = ∂+ . 1

Then, as a first step, consider this data for the homotopy double groupoid specified in the following definition; in order to specify completely such data one also needs to define the related concepts of thin equivalence and the relation of cubically thin homotopy, as provided in the two definitions following the homotopy double groupoid data specified above and in the (main) Definition 0.1.

Definition 0.1. The data for the homotopy double groupoid, ρ^□(X), will be denoted by :

(ρ□2 (X ),ρ □1(X ),∂−1 ,∂+1 ,+1, 𝜀1),ρ□2(X ),ρ□1 (X ),∂ −2 ,∂+2 ,+2,𝜀2) (ρ□1(X ),X, ∂− ,∂+,+, 𝜀).

Here ρ₁(X) denotes the path groupoid of X from ref. [1] where it was defined as follows. The objects of ρ₁(X) are the points of X. The morphisms of ρ₁^□(X) are the equivalence classes of paths in X with respect to the following (thin) equivalence relation ∼_T, defined as follows. The data for ρ₂^□(X) is defined last; furthermore, the symbols specified after the thin square symbol specify both the sides (or the groupoid ‘dimensions’) of the square which are involved (i.e., 1 and 2, respectively), and also the order in which the shown operations (∂₁⁻, 𝜀₂... , etc) are to be performed relative to the thin square specified for each groupoid, ρ₁ or ρ₂; moreover, all such symbols are explicitly and precisely defined in the related entries of the concepts involved in this definition. These two groupoids can also be pictorially represented as the (H,V ) pair depicted in the large diagram (0.1), or D, shown at the top of this page.

Definition 0.2. Thin Equivalence

Let a,a′ : x ≃ y be paths in X. Then a is thinly equivalent to a′, denoted a ∼_Ta′, if there is a thin relative homotopy between a and a′.

We note that ∼ _T is an equivalence relation, see [2]. We use ⟨a⟩ : x ≃ y to denote the ∼ _T class of a path a : x ≃ y and call ⟨a⟩ the semitrack of a. The groupoid structure of ρ₁^□(X) is induced by concatenation, +, of paths. Here one makes use of the fact that if a : x ≃ x′, a′ : x′ ≃ x′′, a′′ : x′′ ≃ x′′′ are paths then there are canonical thin relative homotopies

(a + a′) + a′′ ≃ a + (a′ + a′′) : x ≃ x′′′ (rescale) a + ex′ ≃ a : x ≃ x′; ex + a ≃ a : x ≃ x′ (dilation) a + (− a) ≃ ex : x ≃ x (cancellation).

The source and target maps of ρ₁^□(X) are given by

− + ∂ 1 ⟨a⟩ = x, ∂ 1 ⟨a⟩ = y,

if ⟨a⟩ : x ≃ y is a semitrack. Identities and inverses are given by

𝜀(x) = ⟨ex⟩ resp.− ⟨a⟩ = ⟨− a⟩.

At the next step, in order to construct the groupoid ρ₂^□(X) data in Definition 0.1, R. Brown et al. defined as follows a relation of cubically thin homotopy on the set R₂^□(X) of squares.

Definition 0.3. Cubically Thin Homotopy

Let u,u′ be squares in X with common vertices.

1.

A cubically thin homotopy U : u ≡_T^□u′ between u and u′ is a cube U ∈ R₃^□(X) such that

(i) U is a homotopy between u and u′,

i.e. ∂₁⁻(U) = u, ∂₁⁺(U) = u′,

(ii) U is rel. vertices of I²,

i.e. ∂₂⁻∂₂⁻(U), ∂₂⁻∂₂⁺(U), ∂₂⁺∂₂⁻(U), ∂₂⁺∂₂⁺(U) are constant,

(iii) the faces ∂_i^α(U) are thin for α = ±1, i = 1, 2.

2.

The square u is cubically T-equivalent to u′, denoted u ≡_T^□u′ if there is a cubically thin homotopy between u and u′.

Remark 0.1. By removing from the above double groupoid construction the condition that all morphisms must be invertible one obtains the prototype of a double category.

References

[1] K.A. Hardie, K.H. Kamps and R.W. Kieboom., A homotopy 2-groupoid of a Hausdorff Applied Categorical Structures, 8 (2000): 209-234.

[2] R. Brown, K.A. Hardie, K.H. Kamps and T. Porter., A homotopy double groupoid of a Hausdorff space , Theory and Applications of Categories 10,(2002): 71-93.

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