Introduction to Homotopy Theory in nLab https://ncatlab.org/nlab/print/Introduction+to+Homotopy+Theory
nLab Introduction to Homotopy Theory
This pages gives a detailed introduction to classical homotopy theory, starting with the concept of homotopy in topological spaces and motivating from this the “abstract homotopy theory” in general model categories.
For background on basic topology see at Introduction to Topology.
For application to homological algebra see at Introduction to Homological algebra.
For application to stable homotopy theory see at Introduction to Stable homotopy theory.
Contents Context
1. Topological homotopy theory Homotopy theory Universal constructions Homotopy Cell complexes Fibrations 2. Abstract homotopy theory Factorization systems Homotopy The homotopy category Derived functors Quillen adjunctions 3. The model structure on topological spaces The classical homotopy category Model structure on pointed spaces Model structure on compactly generated spaces Topological enrichment Model structure on topological functors 4. Homotopy fiber sequences Mapping cones Categories of fibrant objects Homotopy fibers Homotopy pullbacks Long sequences 5. The suspension/looping adjunction 6. References
While the field of algebraic topology clearly originates in topology, it is not actually interested in topological spaces regarded up to topological isomorphism, namely homeomorphism (“point-set topology”), but only in topological spaces regarded up to weak homotopy equivalence – hence it
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is interested only in the “weak homotopy types” of topological spaces. This is so notably because ordinary cohomology groups are invariants of the (weak) homotopy type of topological spaces but do not detect their homeomorphism class.
The category of topological spaces obtained by forcing weak homotopy equivalences to become isomorphisms is the “classical homotopy category” Ho(Top). This homotopy category however has forgotten a little too much information: homotopy theory really wants the weak homotopy equivalences not to become plain isomorphisms, but to become actual homotopy equivalences. The structure that reflects this is called a model category structure (short for “category of models for homotopy types”). For classical homotopy theory this is accordingly called the classical model structure on topological spaces. This we review here. 1. Topological homotopy theory
This section recalls relevant concepts from actual topology (“point-set topology”) and highlights facts that motivate the axiomatics of model categories below. We prove two technical lemmas (lemma 1.40 and lemma 1.52) that serve to establish the abstract homotopy theory of topological spaces further below.
Literature (Hirschhorn 15)
Throughout, let Top denote the category whose objects are topological spaces and whose morphisms are continuous functions between them. Its isomorphisms are the homeomorphisms.
(Further below we restrict attention to the full subcategory of compactly generated topological spaces.)
Universal constructions
To begin with, we recall some basics on universal constructions in Top: limits and colimits of diagrams of topological spaces; exponential objects.
Generally, recall:
Definition 1.1. A diagram in a category is a small category and a functor