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
   • CommentTimeMay 1st 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am feeling a bit dense that I have trouble giving a formal proof of the following. Maybe someone can give me a nudge in the right direction. A retract square of a pullback square is itself a pullback. Right? How to show this in the $\infty$-categorical context? So here is what I am talking about in more detail: let $S$ be the category with four objects and morphisms forming a square, so that functors $\mathbf{S} : S \to D$ are precisely commuting square diagrams in $D$. Consider a retract $Id : \mathbf{S}_1 \to \mathbf{S}_2 \to \mathbf{S}_1$ in the functor category $[S,D]$. If the square $\mathbf{S}_2$ is a pullback, then so should be $\mathbf{S}_1$. I want this statement for homotopy pullbacks / $\infty$-pullbacks.

   I am feeling a bit dense that I have trouble giving a formal proof of the following. Maybe someone can give me a nudge in the right direction.

   A retract square of a pullback square is itself a pullback. Right? How to show this in the $\infty$-categorical context?

   So here is what I am talking about in more detail: let $S$ be the category with four objects and morphisms forming a square, so that functors $\mathbf{S} : S \to D$ are precisely commuting square diagrams in $D$.

   Consider a retract $Id : \mathbf{S}_1 \to \mathbf{S}_2 \to \mathbf{S}_1$ in the functor category $[S,D]$. If the square $\mathbf{S}_2$ is a pullback, then so should be $\mathbf{S}_1$. I want this statement for homotopy pullbacks / $\infty$-pullbacks.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMay 2nd 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThis is easy with derivators. An object of $D(S)$ is a pullback square iff it is in the image of $r_* \colon D(C) \to D(S)$, where $C$ is the walking cospan and $r$ the inclusion. Since $r_*$ is a fully faithful right adjoint, this is equivalent to saying that the counit $X \to r_* r^* X$ is an isomorphism. But a retract of an isomorphism is also an isomorphism. I expect one can say an up-to-homotopy version of this for $\infty$-categories.

   This is easy with derivators. An object of $D(S)$ is a pullback square iff it is in the image of $r_* \colon D(C) \to D(S)$, where $C$ is the walking cospan and $r$ the inclusion. Since $r_*$ is a fully faithful right adjoint, this is equivalent to saying that the counit $X \to r_* r^* X$ is an isomorphism. But a retract of an isomorphism is also an isomorphism.

   I expect one can say an up-to-homotopy version of this for $\infty$-categories.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 2nd 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, Mike! That's very useful. I have now added a discussion along these lines to [[retract]]. I realize that I need to better include in my active repository of tools the fact that the (homotopy) limiting cone is equivalently the right Kan extension along the inclusion of the given diagram into its cone diagram.

   Thanks, Mike! That’s very useful.

   I have now added a discussion along these lines to retract.

   I realize that I need to better include in my active repository of tools the fact that the (homotopy) limiting cone is equivalently the right Kan extension along the inclusion of the given diagram into its cone diagram.