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Forgive me if this question is not well thought out. I don't know how else to ask it.

The nlab page on completion gives some examples of completions which are left adjoints. These completions are "free" and describe the most effortless way of getting particular properties. Particularly, these left adjoint feel as though they don't really "look inside" the object to which they're applied.

On the other hand, consider the coskeleton functor $$\mathrm{coskel}_n:s\mathsf{Set}_{\leq n-1}\longrightarrow s\mathsf{Set}_{\leq n} $$ defined by "filling in" $n$-dimensional simplicial holes. This is also a process of "completion", but it is the opposite of effortless: it seems like the most laborious way to get an $n$-truncated simplicial set from an $(n-1)$-truncated one, involving a difficult "search for simplicial holes" throughout the $(n-1)^\text{th}$ level.

Thus the adjoint triple $\mathrm{skel}_n\dashv \mathrm{tr}_n\dashv\mathrm{coskel}_n$ describes a left adjoint to truncation which is truly effortless, and a right adjoint which is maximally laborious.

What are some more examples of right adjoints which deserve to be called "laborious completions"? Perhaps such right adjoint completions only arise as the rightmost adjoint in a triple, so I'd be especially interested in such examples.

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  • $\begingroup$ What is a laborious completion ? Something difficult to calculate ? $\endgroup$
    – Philippe Gaucher
    Commented Aug 30, 2017 at 8:24
  • $\begingroup$ @PhilippeGaucher that sounds like a good criterion, yes. $\endgroup$
    – Arrow
    Commented Aug 30, 2017 at 9:19
  • $\begingroup$ There are left adjoints which are also difficult to calculate. The colimit of a small diagram of globular $\omega$-categories is a left adjoint. It is difficult to calculate because the colimit contains all free compositions of all maps of all dimensions, divided by the relations they are supposed to satisfy in a globular $\omega$-category. $\endgroup$
    – Philippe Gaucher
    Commented Aug 30, 2017 at 9:28
  • $\begingroup$ @PhilippeGaucher in that sense, the criterion of being difficult to calculate is not what I mean. Almost all free constructions are difficult to calculate because they're often huge for the same reason - they contain many "terms" divided by possibly many relations. My coskeleton example is "geometrically laborious" in the sense you must search through the entire simplicial set and fill holes, rather than just superficially add trivial structure in higher dimensions. $\endgroup$
    – Arrow
    Commented Aug 30, 2017 at 9:30

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