By Emily Riehl
This booklet develops summary homotopy thought from the explicit viewpoint with a specific specialize in examples. half I discusses competing views during which one often first encounters homotopy (co)limits: both as derived functors definable while the proper diagram different types admit a suitable version constitution, or via specific formulae that provide the correct concept in definite examples. Riehl unifies those probably rival views and demonstrates that version buildings on diagram different types are inappropriate. Homotopy (co)limits are defined to be a unique case of weighted (co)limits, a foundational subject in enriched classification conception. partially II, Riehl additional examines this subject, setting apart express arguments from homotopical ones. half III treats the main ubiquitous axiomatic framework for homotopy thought - Quillen's version different types. right here, Riehl simplifies commonplace version specific lemmas and definitions via targeting susceptible factorization structures. half IV introduces quasi-categories and homotopy coherence.
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Extra info for Categorical Homotopy Theory
The preceding axioms ensure that multiplication distributes over addition, and so on. 3 Topological spaces are naturally enriched over groupoids. Given spaces X and Y , define the hom-object from X to Y to be the groupoid whose objects are continuous maps X → Y and whose morphisms are homotopy classes of homotopies between these maps. 6 – any unenriched category C has an associated free V-category. Its objects are those of C. The hom-object from a to b is the copower C(a,b) ∗. The morphisms identifying the identities are given by including ∗ at the component of the identity arrow in C(a,a) ∗.
4. A major obstacle to understanding the category HoM is that it is not necessarily locally small: one could easily imagine non-equivalent zigzags from x to y in M snaking through each object of M. Quillen proves that homotopy categories associated to model categories are locally small, but even so, it is often preferable to try to avoid working in the homotopy category at all and to seek instead to understand which point-set level constructions are homotopically meaningful, that is, which descend to functors between the appropriate homotopy categories.
Hence, your favorite adjoint functor theorem implies that L has a right adjoint R : E → sSet. From the desired adjoint correspondence and the Yoneda lemma, for any e ∈ E, (Re)n ∼ = sSet( n , Re) ∼ = E(L n , e) ∼ = E( n , e). Thus we define the n-simplices of Re to be the maps in E from n to e. The face and degeneracy maps for this simplicial set are given by precomposition by the appropriate maps in the cosimplicial object • . Levelwise postcomposition defines a map of simplicial sets for each e → e ∈ E and makes R a functor.
Categorical Homotopy Theory by Emily Riehl