Categorical Homotopy Theory

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ERRATA

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[page 49, Definitions 3.7.2-3] The isomorphism in 3.7.2 should be V-natural in n, while the isomorphism in 3.7.3 should be V-natural in m. See here for a discussion of this point. Alternatively, Remark 3.7.4 can be taken as a definition of what it means for a V-category to be tensored and/or cotensored.

[page 65, proof of Lemma 4.4.2] There is nothing technically incorrect but the notation is very misleading. Because Σ : el(ND) op → Δ op , the objects in the comma category Σ/[n] are pairs given by a composable sequence of arrows d 0 →...→d m together with a morphism [m] → [n] in Δ op . More directly, if we instead consider Σ as a functor Σ: el(ND) → Δ then the comma category appearing in the text is the opposite of the comma category [n]/Σ. To see that [n]/Σ decomposes as a disjoint union of categories with an initial object, consider a general object [n] → Σ(d 0 →...→d m ). By the Eilenberg-Zilber lemma, the map [n] → [m] factors uniquely as an epimorphism [n] → [k] followed by a monomorphism [k] → [m]. The initial object in this component is formed by taking the subsequence of k composable arrows identified by the monomorphism [k] → [m], then inserting identities as specified by the epimorphism [n] → [k], and then using the resulting sequence of n composable arrows to define an object in [n]/Σ whose morphism cmponent is the identity at [n]. This then proves the claim made in the text, that the opposite of this category decomposes as a disjoint union of categories with a terminal object.

→ Δ , the objects in the comma category Σ/[n] are pairs given by a composable sequence of arrows d →...→d together with a morphism [m] → [n] in Δ . More directly, if we instead consider Σ as a functor Σ: el(ND) → Δ then the comma category appearing in the text is the opposite of the comma category [n]/Σ. [page 107, Remark 7.2.10] The category C § introduced by Mac Lane is for computing ends as ordinary limits; to compute coends as colimits, you need to use the opposite category. Taking Mac Lane's definition, the obvious functor C § → twC is initial (not final).

introduced by Mac Lane is for computing ends as ordinary limits; to compute coends as colimits, you need to use the opposite category. Taking Mac Lane's definition, the obvious functor C → twC is initial (not final). [page 111, Definition 9.1.5] The enriched bar and cobar constructions are defined relative to a cosimplicial object in V. Thus, the domain of the functor Δ • should not have an “op”, a typo that appears in the first sentence of Definition 9.1.5 and also in the sentence preceding the definition.

should not have an “op”, a typo that appears in the first sentence of Definition 9.1.5 and also in the sentence preceding the definition. [page 129, Example 8.3.9] As remarked above, the functor C § → twC is initial (not final). A final functor, relevant for computing coends as ordinary colimits, is given by replacing both categories by their opposites.

→ twC is initial (not final). A final functor, relevant for computing coends as ordinary colimits, is given by replacing both categories by their opposites. [page 195, proof of Theorem 12.2.2] In the first displayed diagram, the index of the interior coproduct should be Sq(j,R n f), not Sq(j,f).

f), not Sq(j,f). [page 299, after Theorem 17.1.1] The category of finite ordinals and order-preserving maps is strictly monoidal, with the monoidal product given by ordinal sum, but it is not symmetric monoidal. On objects, we have [n] + [m] = [n+m+1] = [m] + [n], but this (identity) isomorphism is not natural.

[page 302, end of section 17.1] The proof of Theorem 17.1.1 reveals that you can solve lifting problems between outer horn inclusions and inner fibrations between quasi-categories, provided the image of the initial or terminal edge of the horn, respectively, is an isomorphism. More discussion on a closely related question may be found here.

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