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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 22nd 2020
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexCame across this categorification of [[pro-object]]. <a href="https://ncatlab.org/nlab/revision/2-pro-object/1">v1</a>, <a href="https://ncatlab.org/nlab/show/2-pro-object">current</a>

   Came across this categorification of pro-object.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 22nd 2020
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI've been sniffing around a categorified Stone/Isbell duality at the [nCafe](https://golem.ph.utexas.edu/category/2020/04/ultracategories_and_2monads.html), hence the interest in 2-pro-objects. I see it hasn't been taken up much. There's a use in * Yuliang Huang, Giulio Orecchia, Matthieu Romagny, _Unramified F-divided objects and the étale fundamental pro-groupoid in positive characteristic_, ([arXiv:1906.05072](https://arxiv.org/abs/1906.05072)) to construct the étale fundamental pro-groupoid of a flat finitely presented algebraic stack. I wonder what $2Pro(FP Cat)$ is for finitely-presented categories.

   I’ve been sniffing around a categorified Stone/Isbell duality at the nCafe, hence the interest in 2-pro-objects. I see it hasn’t been taken up much. There’s a use in

   • Yuliang Huang, Giulio Orecchia, Matthieu Romagny, Unramified F-divided objects and the étale fundamental pro-groupoid in positive characteristic, (arXiv:1906.05072)

   to construct the étale fundamental pro-groupoid of a flat finitely presented algebraic stack.

   I wonder what $2Pro(FP Cat)$ is for finitely-presented categories.

3. • CommentRowNumber3.
   • CommentAuthorTim_Porter
   • CommentTimeApr 22nd 2020
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexFor what its worth in the original work on shape theory, predating Borsuk by a long time, D. E. Christie (1944) used a homotopy version of 2-pro-object to define his invariants. The idea of homotopy coherent pro-objects was used by Batanin in his work on strong shape theory and of course, comes into Lurie's work.

   For what its worth in the original work on shape theory, predating Borsuk by a long time, D. E. Christie (1944) used a homotopy version of 2-pro-object to define his invariants. The idea of homotopy coherent pro-objects was used by Batanin in his work on strong shape theory and of course, comes into Lurie’s work.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 22nd 2020
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded * [[Maria Emilia Descotte]], _A theory of 2-pro-objects, a theory of 2-model 2-categories and the 2-model structure for 2-Pro(C)_, ([arXiv:2010.10636](https://arxiv.org/abs/2010.10636)) <a href="https://ncatlab.org/nlab/revision/diff/2-pro-object/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/2-pro-object/3">v3</a>, <a href="https://ncatlab.org/nlab/show/2-pro-object">current</a>

   Added

   • Maria Emilia Descotte, A theory of 2-pro-objects, a theory of 2-model 2-categories and the 2-model structure for 2-Pro(C), (arXiv:2010.10636)

   diff, v3, current