Higher Categories and Homotopical Algebra

Cisinski, Denis-Charles

doi:10.1017/9781108588737

Cited by 109 publications

(127 citation statements)

References 12 publications

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“…For

n = 0

, the argument is similar to [, Proposition 2.1.1], and for

n > 0

, the proof of [, Lemma 1.6.6] is easily adapted to this purpose. Then, following the proof of the Approximation Theorem in , or applying [, Lemma 7.6.7], we conclude that the functor

w {Cob}^{sym} {({scriptC}^{f} false(X false))}_{n} \to w {Cob}^{sym} {({scriptC}^{h f} false(X false))}_{n}

induces a homotopy equivalence after passing to the classifying spaces — as shown in the proof of [, Lemma 7.6.7], only the existence of an initial object, rather than a zero object, is required. The result then follows.

□

…”

Section: Example: Cobordism Categories and A‐theorysupporting

confidence: 55%

A cobordism model for Waldhausen K‐theory

Raptis

Steimle²

2018

J. London Math. Soc.

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We study a categorical construction called the cobordism category, which associates to each Waldhausen category a simplicial category of cospans. We prove that this construction is homotopy equivalent to Waldhausen's S•‐construction and therefore it defines a model for Waldhausen K‐theory. As an example, we discuss this model for A‐theory and show that the cobordism category of homotopy finite spaces has the homotopy type of Waldhausen's A(∗). We also review the canonical map from the cobordism category of manifolds to A‐theory from this viewpoint.

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“…For

n = 0

, the argument is similar to [, Proposition 2.1.1], and for

n > 0

, the proof of [, Lemma 1.6.6] is easily adapted to this purpose. Then, following the proof of the Approximation Theorem in , or applying [, Lemma 7.6.7], we conclude that the functor

w {Cob}^{sym} {({scriptC}^{f} false(X false))}_{n} \to w {Cob}^{sym} {({scriptC}^{h f} false(X false))}_{n}

□

…”

Section: Example: Cobordism Categories and A‐theorysupporting

confidence: 55%

A cobordism model for Waldhausen K‐theory

Raptis

Steimle²

2018

J. London Math. Soc.

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“…A comparison result ensures that if the relative category happens to be a model category then this recovers the usual homotopy limits and homotopy colimits. This is explained in Remark 7.9.10 of [7] or Remark 2.5.8 in [2]. In particular it follows from this that any weak equivalence of relative categories preserves homotopy limits, which we will need below.…”

Section: Relative Categoriesmentioning

confidence: 88%

“…We will use Joyal's theory of ∞-categories as quasi-categories, see [17,18] for further background. Given any simplicial set K with a subsimplicial set W we may consider it as an object of qCat and define its localization L W K , see [7,Proposition 7.1.3]. It has the universal property that for any quasi-category C the functor category Fun(L W K , C) is equivalent to the subcategory of Fun(K , C) consisting of functors sending any map in W to an invertible map in C. See also the section on homotopy localization in [17].…”

Section: Localization Of ∞-Categoriesmentioning

confidence: 99%

Homotopy theory of monoids and derived localization

Chuang

Holstein

Lazarev

2021

J. Homotopy Relat. Struct.

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We use derived localization of the bar and nerve constructions to provide simple proofs of a number of results in algebraic topology, both known and new. This includes a recent generalization of Adams’s cobar-construction to the non-simply connected case, and a new algebraic model for the homotopy theory of connected topological spaces as an $$\infty $$ ∞ -category of discrete monoids.

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“…The definition presented naturally extends the notion of ∞colimits appearing in [3,10], as demonstrated by the following result.…”

Section: Introductionmentioning

confidence: 85%

Marked colimits and higher cofinality

García

2021

J. Homotopy Relat. Struct.

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Given a marked $$\infty $$ ∞ -category $$\mathcal {D}^{\dagger }$$ D † (i.e. an $$\infty $$ ∞ -category equipped with a specified collection of morphisms) and a functor $$F: \mathcal {D}\rightarrow {\mathbb {B}}$$ F : D → B with values in an $$\infty $$ ∞ -bicategory, we define "Equation missing", the marked colimit of F. We provide a definition of weighted colimits in $$\infty $$ ∞ -bicategories when the indexing diagram is an $$\infty $$ ∞ -category and show that they can be computed in terms of marked colimits. In the maximally marked case $$\mathcal {D}^{\sharp }$$ D ♯ , our construction retrieves the $$\infty $$ ∞ -categorical colimit of F in the underlying $$\infty $$ ∞ -category $$\mathcal {B}\subseteq {\mathbb {B}}$$ B ⊆ B . In the specific case when "Equation missing", the $$\infty $$ ∞ -bicategory of $$\infty $$ ∞ -categories and $$\mathcal {D}^{\flat }$$ D ♭ is minimally marked, we recover the definition of lax colimit of Gepner–Haugseng–Nikolaus. We show that a suitable $$\infty $$ ∞ -localization of the associated coCartesian fibration $${\text {Un}}_{\mathcal {D}}(F)$$ Un D ( F ) computes "Equation missing". Our main theorem is a characterization of those functors of marked $$\infty $$ ∞ -categories $${f:\mathcal {C}^{\dagger } \rightarrow \mathcal {D}^{\dagger }}$$ f : C † → D † which are marked cofinal. More precisely, we provide sufficient and necessary criteria for the restriction of diagrams along f to preserve marked colimits

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Higher Categories and Homotopical Algebra

Cited by 109 publications

References 12 publications

A cobordism model for Waldhausen K‐theory

A cobordism model for Waldhausen K‐theory

Homotopy theory of monoids and derived localization

Marked colimits and higher cofinality

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