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1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeFeb 7th 2014
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexIn [[completion ]] the following occurs: >Non-idempotent completions These completions add a property-like structure, are often lax-idempotent or colax-idempotent. There follows a list of supposedly non-idempotent completion processes, but the pro-completion of a category seems to be an idempotent process since a pro-object $F:\mathsf{I}\to Pro-C$ has a limit in Pro-C which is just itself! (If you see what I mean!) You form an indexing category from $\mathsf{I}$ and all the indexing categories that are used in the various $F(i)$. (That looks very like some sort of Grothendieck construction, but I cannot quite see my way to state it correctly!) That gives a new pro-object which is the limit of $F$, but allows the reconstruction of $F$ from that data as well. (I should know if this gives an equivalence of categories, but cannot find it in the literature.) The entry is also suffering from a bad attack of query boxes.

   In completion the following occurs:

   Non-idempotent completions These completions add a property-like structure, are often lax-idempotent or colax-idempotent.

   There follows a list of supposedly non-idempotent completion processes, but the pro-completion of a category seems to be an idempotent process since a pro-object $F:\mathsf{I}\to Pro-C$ has a limit in Pro-C which is just itself! (If you see what I mean!) You form an indexing category from $\mathsf{I}$ and all the indexing categories that are used in the various $F(i)$. (That looks very like some sort of Grothendieck construction, but I cannot quite see my way to state it correctly!) That gives a new pro-object which is the limit of $F$, but allows the reconstruction of $F$ from that data as well. (I should know if this gives an equivalence of categories, but cannot find it in the literature.)

   The entry is also suffering from a bad attack of query boxes.

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeFeb 7th 2014
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexNo, it is not idempotent. The canonical embedding $\mathbf{Pro}(\mathcal{C}) \to \mathbf{Pro}(\mathbf{Pro}(\mathcal{C}))$ is rarely ever an equivalence.

   No, it is not idempotent. The canonical embedding $\mathbf{Pro}(\mathcal{C}) \to \mathbf{Pro}(\mathbf{Pro}(\mathcal{C}))$ is rarely ever an equivalence.