Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
An (∞,1)-functor is conservative if its reflects equivalences in an (∞,1)-category. This means that on homotopy categories it is a conservative functor.
In an (∞,1)-topos, (∞,1)-pullback along an effective epimorphism is a conservative (∞,1)-functor between slice (∞,1)-categories.
We write in the conjectural internal logic of an (∞,1)-topos expressed as homotopy type theory, with slice (∞,1)-categories represented using dependent types. The reader is free to translate this manually into an (∞,1)-category-theoretic proof.
Thus, suppose is a surjection (i.e. , where denotes the propositional truncation of ). Let be dependent types with a fiberwise map which becomes a fiberwise equivalence upon restricting to , i.e. such that . We must show that is itself a fiberwise equivalence.
Thus, let ; we must show . By assumption we have . But is an h-proposition, so we may strip the truncation and obtain an such that . Transporting along this equality, it suffices to show , but this follows from the other assumption.
See also Lemma 6.2.3.16 in Higher Topos Theory.
In the context of quasi-categories, def. 6.2.1 in
Last revised on February 13, 2017 at 22:38:53. See the history of this page for a list of all contributions to it.