By Emily Riehl

This e-book develops summary homotopy conception from the specific point of view with a specific specialize in examples. half I discusses competing views wherein one usually 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 supply the proper inspiration in definite examples. Riehl unifies those likely rival views and demonstrates that version constructions on diagram different types are inappropriate. Homotopy (co)limits are defined to be a different case of weighted (co)limits, a foundational subject in enriched class thought. partially II, Riehl additional examines this subject, keeping apart express arguments from homotopical ones. half III treats the main ubiquitous axiomatic framework for homotopy thought - Quillen's version different types. the following, Riehl simplifies general version express lemmas and definitions through concentrating on susceptible factorization structures. half IV introduces quasi-categories and homotopy coherence.

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In examples, this internal hom is the object of basepoint-preserving maps from v to w, and its basepoint is the constant map. 1 2 The smash product is not associative in the (inconvenient) category of all based spaces; (cf. 3). 14); your intuition is correct. 4 Underlying categories of enriched categories 39 There is a disjoint basepoint-forgetful adjunction (−)+ : V o ⊥ G V∗ : U whose left adjoint is defined to be the coproduct with the terminal object. 16 When V is a cartesian closed symmetric monoidal category, the functor (−)+ : V → V∗ is strong monoidal, that is, S0 ∼ = (∗)+ v+ ∧ w+ ∼ = (v × w)+ , and these natural isomorphisms are appropriately associative and unital.

Recall that a pointwise right Kan extension is one that is constructed as a limit in the target category. From this definition, one would not expect a total left derived functor to be a pointwise Kan extension. 13 The total left derived functor of a left deformable functor is a pointwise right Kan extension. Proof If F : M → N is left deformable, it has a total left derived functor (δF Q, δF q) constructed using a left deformation (Q, q) and the localization functor δ. Because Kan extensions are characterized by a universal property, any total derived functor LF is isomorphic to this one.

Write Ch≥0 (A) and Ch≥0 (B) for the categories of chain complexes in A and B concentrated in non-negative degrees. 83in 978 1 107 04845 4 February 4, 2014 Derived functors via deformations homotopical when we instead take the weak equivalences to be the quasiisomorphisms. For a counterexample, take A = B = Ab and consider the quasi-isomorphism 0 G Z/2 G 0 G Z/2 G Z/4 G Z/4 ··· A• f• B• ··· G 0 G Z/2 G Z/2 G 0 G 0 Applying the functor homZ (Z/2, −) pointwise yields homZ (Z/2, A)• ··· G Z/2 ··· G 0 homZ (Z/2,f )• homZ (Z/2, B)• ∼ = G Z/2 0 G Z/2 0 G Z/2 G 0 ∼ = G Z/2 G 0 0 G Z/2 G 0 which is not a quasi-isomorphism; indeed, these chain complexes are not quasiisomorphic.