Quasicategories of Frames of Cofibration Categories

Kapulkin, Krzysztof; Szumiło, Karol

doi:10.1007/s10485-015-9422-y

Cited by 15 publications

(18 citation statements)

References 16 publications

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“…Theorem 2.11 [14,Corollary 4.6]. For any fibration category C, the quasi-categories Ho ∞ C and N f C are weakly equivalent in Joyal's model structure.…”

Section: Fibration Categories and The Quasi-category Of Framesmentioning

confidence: 99%

“…Barwick and Kan constructed a model structure on

sans-serifweCat

and showed that it is Quillen equivalent to the Joyal's model structure on

sans-serifsSet

, in which fibrant objects are exactly quasi‐categories [, Theorem 6.12]. While there are many constructions assigning to a category with weak equivalences a quasi‐category that are equivalences of homotopy theories, by result of Toën [, Theorem‐ 6.2], all of them are equivalent either to the functor constructed by Barwick and Kan or its opposite (see also [, Remark 4.7]).…”

Section: Background and Statement Of The Main Theoremmentioning

confidence: 99%

“…Thus the inclusion

D_{a} false[n false] ↪ D false[n + 1 false]

satisfies the assumptions of [, Lemma 2.16] and precomposing with it gives a well‐defined map

. Explicitly, given an

n

‐simplex

X : D {[n + 1]}^{op} \to C

{normalN}_{normalf} scriptC ↓ A

, the

n

‐simplex

{scriptS}_{A} X : D_{a} {[n]}^{op} \to C

is given by:

{({scriptS}_{A} X)}_{a_{1} a_{2} \dots a_{i}} : = X_{a_{1} a_{2} \dots a_{i} false(n + 1 false)} .

Theorem Let

scriptC

be a fibration category and let

A : D {[0]}^{op} \to C

be a 0‐simplex in

{normalN}_{normalf} C

.…”

Section: Slices Of the Quasi‐category Of Framesmentioning

confidence: 99%

“…We start by defining

{\tilde{Y}}_{false[1 + n false]}

as a limit:

{\overset{Y}{true}}_{[1 + n]} = \underset{0 \in S ⊊ [1 + n]}{trueprefixlim} Y_{S} .

Moreover, we set

{\overset{B}{true}}_{(1 + n]} : = F {\overset{Y}{true}}_{[1 + n]}

By construction, the canonical map

{\overset{Y}{true}}_{[1 + n]} \to Y_{S}

is a fibration for any

S ⊊ [1 + n]

such that

0 \in S

. By [, Lemma 2.17], it is moreover an acyclic fibration. Given

S \subseteq [n]

such that

0 \notin S

, there is a map

Y_{S \cup {0}} \to Y_{S} = G B_{S}

yielding:

\underset{0 \notin S}{trueprefixlim} Y_{S \cup {0}} ⟶ \underset{0 \notin S}{trueprefixlim} G B_{S} ≅ G false(lim_{0 \notin S} B_{S} false)

since

G

is a right adjoint.…”

Section: Adjoints Between Quasicategories Of Framesmentioning

confidence: 99%

See 3 more Smart Citations

Locally cartesian closed quasi‐categories from type theory

Kapulkin

2017

Journal of Topology

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“…Theorem 2.11 [14,Corollary 4.6]. For any fibration category C, the quasi-categories Ho ∞ C and N f C are weakly equivalent in Joyal's model structure.…”

Section: Fibration Categories and The Quasi-category Of Framesmentioning

confidence: 99%

“…Barwick and Kan constructed a model structure on

sans-serifweCat

and showed that it is Quillen equivalent to the Joyal's model structure on

sans-serifsSet

Section: Background and Statement Of The Main Theoremmentioning

confidence: 99%

“…Thus the inclusion

D_{a} false[n false] ↪ D false[n + 1 false]

satisfies the assumptions of [, Lemma 2.16] and precomposing with it gives a well‐defined map

. Explicitly, given an

n

‐simplex

X : D {[n + 1]}^{op} \to C

{normalN}_{normalf} scriptC ↓ A

, the

n

‐simplex

{scriptS}_{A} X : D_{a} {[n]}^{op} \to C

is given by:

{({scriptS}_{A} X)}_{a_{1} a_{2} \dots a_{i}} : = X_{a_{1} a_{2} \dots a_{i} false(n + 1 false)} .

Theorem Let

scriptC

be a fibration category and let

A : D {[0]}^{op} \to C

be a 0‐simplex in

{normalN}_{normalf} C

.…”

Section: Slices Of the Quasi‐category Of Framesmentioning

confidence: 99%

“…We start by defining

{\tilde{Y}}_{false[1 + n false]}

as a limit:

{\overset{Y}{true}}_{[1 + n]} = \underset{0 \in S ⊊ [1 + n]}{trueprefixlim} Y_{S} .

Moreover, we set

{\overset{B}{true}}_{(1 + n]} : = F {\overset{Y}{true}}_{[1 + n]}

By construction, the canonical map

{\overset{Y}{true}}_{[1 + n]} \to Y_{S}

is a fibration for any

S ⊊ [1 + n]

such that

0 \in S

. By [, Lemma 2.17], it is moreover an acyclic fibration. Given

S \subseteq [n]

such that

0 \notin S

, there is a map

Y_{S \cup {0}} \to Y_{S} = G B_{S}

yielding:

\underset{0 \notin S}{trueprefixlim} Y_{S \cup {0}} ⟶ \underset{0 \notin S}{trueprefixlim} G B_{S} ≅ G false(lim_{0 \notin S} B_{S} false)

since

G

is a right adjoint.…”

Section: Adjoints Between Quasicategories Of Framesmentioning

confidence: 99%

See 2 more Smart Citations

Locally cartesian closed quasi‐categories from type theory

Kapulkin

2017

Journal of Topology

Self Cite

View full text Add to dashboard Cite

show abstract

“…The third one [22] shows that it is possible to reconstruct a cofibration category from a cocomplete quasicategory in a way that establishes an equivalence between the homotopy theory of cofibration categories and the homotopy theory of cocomplete quasicategories. Moreover, in a paper joint with Chris Kapulkin [13] we prove that the quasicategory of frames models the simplicial localization of a cofibration category.…”

Section: Introductionmentioning

confidence: 99%