Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 7th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added a little bit of discussion to the Idea-section at _[[anodyne morphism]]_, added references, added a mentioning of the dendroidal case, and made _[[anodyne extension]]_ redirect to it.

   I have added a little bit of discussion to the Idea-section at anodyne morphism, added references, added a mentioning of the dendroidal case, and made anodyne extension redirect to it.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 19th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added the original reference (ch. IV.2 of Grabriel-Zisman) to _[[anodyne extension]]_

   I have added the original reference (ch. IV.2 of Grabriel-Zisman) to anodyne extension

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 22nd 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexExpanded the Idea section: The original definition by Gabriel–Zisman (Definition IV.2.1.4 \cite{GabrielZisman67}) defined __anodyne extensions__ as the [[weak saturation]] of [[simplicial horn]] inclusions. More generally, the same definition can be used to talk about the [[weak saturation]] of any set $S$ of morphisms in any category. One also talks about _anodyne maps_ or _anodyne morphisms_. If the [[small object argument]] is applicable, anodyne maps are precisely maps with a [[left lifting property]] with respect to all [[fibrations]], where the latter is defined as morphisms with a [[right lifting property]] with respect to $S$. In particular, if $S$ is a set of generating [[acyclic cofibrations]] in a [[model category]] with applicable [[small object argument]], then anodyne maps are precisely [[acyclic cofibrations]]. <a href="https://ncatlab.org/nlab/revision/diff/anodyne+morphism/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/anodyne+morphism/14">v14</a>, <a href="https://ncatlab.org/nlab/show/anodyne+morphism">current</a>

   Expanded the Idea section:

   The original definition by Gabriel–Zisman (Definition IV.2.1.4 \cite{GabrielZisman67}) defined anodyne extensions as the weak saturation of simplicial horn inclusions.

   More generally, the same definition can be used to talk about the weak saturation of any set $S$ of morphisms in any category. One also talks about anodyne maps or anodyne morphisms.

   If the small object argument is applicable, anodyne maps are precisely maps with a left lifting property with respect to all fibrations, where the latter is defined as morphisms with a right lifting property with respect to $S$.

   In particular, if $S$ is a set of generating acyclic cofibrations in a model category with applicable small object argument, then anodyne maps are precisely acyclic cofibrations.

   diff, v14, current