Not signed in (Sign In)

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

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 nforum 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

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 29th 2019

    Added a stub of an Idea section.

    diff, v5, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 30th 2019

    Re codense subspace: how standard is this terminology? At least one reference in a google search defines it as a subspace whose complement is dense, which is different from how the nLab article defines it. Do you have a counter-reference?

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 30th 2019
    • (edited Jan 30th 2019)

    Here’s one paper that defines it that way (sec. 4). Also this page on Google books.

    Is it perhaps the case that the line of development went: dense subspace, dense subcategory, dense functor, then to its dual, codense functor, and from there to condensity monad?

    Then it need not be the case that codense functor has a relation to some dual to dense subspace?

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 30th 2019

    We have an occurrence of codense subcategory at convex space:

    The subcategory consisting of the single object, the unit interval, is dense and codense (adequate and coadequat) in the category, and consequently every convex space is a canonical colimit. Equivalently, the restricted Yoneda embedding is still full and faithful. This follows from Isbell’s theorem on left adequate subcategories for algebraic theories. A more detailed description is given by Sturtz (2017), where the existence of the codense subcategory is exploited to relate the category of convex spaces to the Giry algebras.

    So is a subcategory dense (codense) when objects of the category can be expressed as colimits (limits) of the subcategory?

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 30th 2019

    So I removed mention of codense subspace and added an example of a codense subcategory from Ulmer.

    diff, v6, current

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 30th 2019

    Added as a reference Lawvere on Isbell. He talks there about ’coadequate’. Presumably that’s a synonym of ’codense’.

    diff, v7, current