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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeJan 29th 2019

Added a stub of an Idea section.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJan 29th 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.

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