By Brown R., Higgins P., Sivera R.

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This definition gives another equivalence relation among the set of maps from X to Y. The quotient set is denoted [X, Y]A and the equivalence class of a map f is again denoted by [f]. Since all maps homotopic relative to, or rel to, A must agree with a map u : A → Y, this set for a fixed u is written [X, Y; u]. Thus [X, Y]A is the union of the disjoint sets [X, Y; u] for all u : A → X. A particular case of this definition is when we study maps sending a fixed subset A of X to a given point y ∈ Y.

3 Let X and Y be spaces, A a subset of X and f : A → Y a map. We consider subspaces X1 ⊆ X and Y1 ⊆ Y and define A1 = X1 ∪ A and f1 = f| : A1 → Y1 . If the inclusions A ⊆ X and A1 ⊆ X1 are closed cofibrations and the pairs (Y, Y1 ), (X, X1 ), (A, A1 ) are connected, then: (Con) The pair (X ∪f Y, X1 ∪f1 Y1 ) is connected. 2) is a pushout of crossed modules. 4 The term closed cofibration included in the hypothesis of the theorem is satisfied in a great number of useful cases. It can be intuitively interpreted as saying that the placing of A in X and of A1 in X1 are ‘locally not wild’.

This apparently ‘trivial’ groupoid will play a key role in the theory, since it determines homotopies, and it is also called the ‘unit interval groupoid’. A directed graph X is called connected if the free groupoid F(X) on X is connected, and is called a forest if every object group F(X)(a) of F(X), a ∈ Ob(X), is trivial. A connected forest is called a tree . If X is a tree, then the groupoid F(X) is indiscrete. Let G be a connected groupoid and let a0 be an object of G. For each a ∈ Ob(G) choose an arrow τa : a0 → a, with τa0 = 1a0 .