Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Search

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

91 results

Page 1 of 5

  • A 2-categorical proof of Frobenius for fibrations defined from a generic point

  • Sina Hazratpour, Emily Riehl
  • Journal:
    Mathematical Structures in Computer Science / Volume 34 / Issue 4 / April 2024
    Published online by Cambridge University Press:
    15 April 2024, pp. 258-280
    • Article
      • You have access
      • Open access
    • PDF
    • HTML
    • Export citation

  • Consider a locally cartesian closed category with an object $\mathbb{I}$ and a class of trivial fibrations, which admit sections and are stable under pushforward and retract as arrows. Define the fibrations to be those maps whose Leibniz exponential with the generic point of $\mathbb{I}$ defines a trivial fibration. Then the fibrations are also closed under pushforward.

  • 2 - Homotopical categories: from model categories to (∞, 1)-categories

  • Edited by Andrew J. Blumberg, Columbia University, New York, Teena Gerhardt, Michigan State University, Michael A. Hill, University of California, Los Angeles
  • Mathematical Sciences Research Institute
  • Book:
    Stable Categories and Structured Ring Spectra
    Published online:
    29 May 2025
    Print publication:
    21 July 2022, pp 5-74

  • Summary

    A homotopical category is a category equipped with some collection of morphisms traditionally called “weak equivalences” that somewhat resemble isomorphisms but fail to be invertible in any reasonable sense, and might in fact not even be reversible: that is, the presence of a weak equivalence X → Y need not imply the presence of a weak equivalence Y → X. Frequently, the weak equivalences are defined as the class of morphisms in a category K that are “inverted by a functor” F : K → L, in the sense of being precisely those morphisms in K that are sent to isomorphisms in L. For instance:

    • – Weak homotopy equivalences of spaces or spectra are those maps inverted by the homotopy group functors π∗ : Top → GrSet or π∗ : Spectra → GrAb.

    • – Quasi-isomorphisms of chain complexes are those maps inverted by the homology functor H∗ : Ch → GrAb.

    • – Equivariant weak homotopy equivalences of G-spaces are those maps inverted by the homotopy functors on the fixed point subspaces for each compact subgroup of G.

    The term used to describe the equivalence class represented by a topological space up to weak homotopy equivalence is a homotopy type. Since the weak homotopy equivalence relation is created by the functor π∗ , a homotopy type can loosely be thought of as a collection of algebraic invariants of the space X, as encoded by the homotopy groups π∗X. Homotopy types live in a category called the homotopy category of spaces, which is related to the classical category of spaces as follows: a genuine continuous function X → Y certainly represents a map (graded homomorphism) between homotopy types. But a weak homotopy equivalence of spaces, defining an isomorphism of homotopy types, should now be regarded as formally invertible.

    In their 1967 manuscript Calculus of fractions and homotopy theory, Gabriel and Zisman [100] formalized the construction of what they call the category of fractions associated to any class of morphisms in any category together with an associated localization functor π : K → K[𝑊−1] that is universal among functors with domain K that invert the class 𝑊 of weak equivalences. This construction and its universal property are presented in §2.2.

Page 1 of 5