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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?
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?
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?
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