In broad terms, the main goal of homotopy theory is to study the objects of a category up to a "weak equivalence". These weak equivalences satisfy certain properties, that are also satisfied by isomorphisms, but with the downside that they are not always invertible. Thus, over the years, homotopy theorists have tried to produce various axiomatizations that guarantee that certain "point-level-constructions" respect weak equivalences.

The goal of this lecture is to give a brief introduction to abstract homotopy theory, through the model categories of Daniel Quillen. We will consider a model structure on the category of topological spaces and one on the category of simplicial sets, establishing a nice equivalence between the homotopy theories stemming from the respective model categories.

References

[Car07] Jacobien Carstens. Homotopy Theory of Topological Spaces and Simplicial Sets. 2007. URL: https://web.math.ku.dk/~jg/students/carstens.bs.2007.pdf.

[Fried11] Greg Friedman. An elementary illustrated introduction to simplicial sets. 2011. URL: https://faculty.tcu.edu/gfriedman/papers/simp.pdf.

[GZ67] P. Gabriel and M. Zisman. Calculus of Fractions and Homotopy Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3Island Press, 1967. ISBN: 9783642858451. URL: https://books.google.pt/books?id=rtDWwQEACAAJ.

[GJ09] P.G. Goerss and J.F. Jardine. Simplicial Homotopy Theory. Modern Birkh ̈auser Classics. Birkh ̈auser Basel, 2009. ISBN: 9783034601894. URL: https://books.google.pt/books?id=ED1bVh5K-5YC.

[HM22] G. Heuts and I. Moerdijk. Simplicial and Dendroidal Homotopy Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. Springer International Publishing, 2022. ISBN: 9783031104473. URL: https://books.google.pt/books?id=aPd-EAAAQBAJ.