Eilenberg and Steenrod proved that ordinary homology is characterized by five axioms. Later, Atiyah, Hirzebruch and Whitehead observed that there are other families of functors that satisfy the four "most important" axioms. They defined the so called "generalized homology theories", now just called homology theories, which are examples of stable phenomena in homotopy theory.
The concept of a prespectrum was first introduced by Elon Lages Lima in his PhD thesis to study some kinds of stable phenomena, such as Spanier-Whitehead duality and Stable Postnikov invariants. Later, Adams and Boardman proposed the first homotopy category of prespectrums. This was the starting point of the research field called stable homotopy theory. Nowadays stable homotopy categories are fundamental for studying all kind of stable phenomena in homotopy theory, including generalized homology theories and cohomology theories.
The goal of the talk is to introduce the concept of prespectrum, proving the first basic property: each prespectrum represents a (reduced) homology theory.
To reach this goal, we present some results of basic algebraic topology. Firstly, we introduce some model categorical language and results, mostly related to the so called homotopy pushout. Secondly, we assume two classical theorems: homotopy excision and long exact sequences of homotopy groups. And then we derive some consequences, such as Freudenthal suspension theorem and basic results concerning homotopy groups. At the end, we prove that every prespectrum represents a homology theory.