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 definitions 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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics 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 object 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.
   • CommentAuthorMike Shulman
   • CommentTimeNov 18th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI have created [[final lift]], and added to [[adjoint triple]] a proof that in a fully faithful adjoint triple between cocomplete categories, the middle functor admits final lifts of *small* structured [[sinks]] (and dually). This means that it is kind of like a [[topological concrete category]], except that the forgetful functor need not be faithful. I find this interesting because it means that in the situation of axiomatic cohesion, where the forgetful functor from "spaces" is not necessarily faithful, we can still construct such "spaces" in "initial" and "final" ways, as long as we restrict to small sources and sinks.

   I have created final lift, and added to adjoint triple a proof that in a fully faithful adjoint triple between cocomplete categories, the middle functor admits final lifts of small structured sinks (and dually). This means that it is kind of like a topological concrete category, except that the forgetful functor need not be faithful.

   I find this interesting because it means that in the situation of axiomatic cohesion, where the forgetful functor from “spaces” is not necessarily faithful, we can still construct such “spaces” in “initial” and “final” ways, as long as we restrict to small sources and sinks.

2. • CommentRowNumber2.
   • CommentAuthorFinnLawler
   • CommentTimeNov 18th 2011
   • (edited Nov 18th 2011)
   • PermaLink
   Author: FinnLawler
   Format: MarkdownItexI've replaced the $U$s in the proof with $G$s, and swapped $C$ and $D$, to match the notation used on the rest of the page, and also added a few words that I hope make the whole thing clearer (but do check that I've understood it right).

   I’ve replaced the $U$s in the proof with $G$s, and swapped $C$ and $D$, to match the notation used on the rest of the page, and also added a few words that I hope make the whole thing clearer (but do check that I’ve understood it right).

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeNov 18th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThanks! I caught a couple of $U$s that you missed. That notation mismatch was because I couldn't decide for a while what page to put that proof on, and $U$ is used on some other pages.

   Thanks! I caught a couple of $U$s that you missed. That notation mismatch was because I couldn’t decide for a while what page to put that proof on, and $U$ is used on some other pages.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeDec 15th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI added to [[final lift]] the example that (semi-)final lifts along the map to the terminal category are just colimits, and the observation that in [[HTT]] (strictly) final lifts for $(\infty,1)$-categories are called "$U$-colimits".

   I added to final lift the example that (semi-)final lifts along the map to the terminal category are just colimits, and the observation that in HTT (strictly) final lifts for $(\infty,1)$-categories are called “$U$-colimits”.