Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2016, Director: Carles Casacuberta
Whitehead’s Theorem is a classical result in algebraic topology which states that any continuous map between CW complexes which is both inducing a bijection of path connected components and isomorphisms in homotopy groups for any choice of base point is an homotopy equivalence.
CW complexes are topological spaces built through an interative process of cell attachment. In the 1990s a more general notion of cellular object in the framework of model categories was given and it started a really productive work on cellular objects in many other areas like commutative algebra, group theory or algebraic geometry. The first aim of this work is to write down the proof of Whitehead’s Theorem in pointed model categories which states that an $A$-equivalence between $A$-cellular fibrant objects is an homotopy equivalence for any cofibrant object $A$.