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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 3rd 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexStarting something... prodded by discussion with user varkor, [here](https://nforum.ncatlab.org/discussion/14797/joyal-locus/?Focus=109925#Comment_109925). This entry gets its name from the notion in * [[Hongde Hu]], [[Walter Tholen]], *Quasi-coproducts and accessible categories with wide pullbacks*, Appl Categor Struct **4** (1996) 387–402 &lbrack;[doi:10.1007/BF00122686](https://doi.org/10.1007/BF00122686)&rbrack; which is currently (just) briefly recalled in the first Definition ([here](https://ncatlab.org/nlab/show/quasi-coproduct#QuasiCoproduct)). But, for the time being the entry has nothing more to say on Hu & Tholen's discussion (though feel invited to change this!) but turns to a slight variant (effectively a special case) of their definition, which I am proposing and proposing to call "homotopical quasi-coproducts" ([here](https://ncatlab.org/nlab/show/quasi-coproduct#CategoryWithHomotopicalQuasiCoproducts)). The main motivation is to have a transparent proof that for any cocomplete $\mathcal{C}$ its Grothendieck construction $$ Loc_{\mathcal{C}} \;\;\;\coloneqq\;\;\; \underset{\mathcal{X}\in Grpd}{\textstyle{\int}} \mathcal{C}^{\mathcal{X}} $$ is its "free homotopy quasi-coproduct completion" in direct analogy to how the category of familes $$ Fam_{\mathcal{C}} \;\;\;\coloneqq\;\;\; \underset{X\in Set}{\textstyle{\int}} \mathcal{C}^{X} $$ is the free coproduct completion. I think I have proven this in the final theorem ([here](https://ncatlab.org/nlab/show/quasi-coproduct#LocalSystemsAreFreeHomotopyQuasiCoproductCompletion)) but this is fresh from the press, so please handle with care for the moment. <a href="https://ncatlab.org/nlab/revision/quasi-coproduct/1">v1</a>, <a href="https://ncatlab.org/nlab/show/quasi-coproduct">current</a>

   Starting something… prodded by discussion with user varkor, here.

   This entry gets its name from the notion in

   • Hongde Hu, Walter Tholen, Quasi-coproducts and accessible categories with wide pullbacks, Appl Categor Struct 4 (1996) 387–402 [doi:10.1007/BF00122686]

   which is currently (just) briefly recalled in the first Definition (here).

   But, for the time being the entry has nothing more to say on Hu & Tholen’s discussion (though feel invited to change this!) but turns to a slight variant (effectively a special case) of their definition, which I am proposing and proposing to call “homotopical quasi-coproducts” (here).

   The main motivation is to have a transparent proof that for any cocomplete $\mathcal{C}$ its Grothendieck construction

   $$Loc_{\mathcal{C}} \;\;\;\coloneqq\;\;\; \underset{\mathcal{X}\in Grpd}{\textstyle{\int}} \mathcal{C}^{\mathcal{X}}$$

   is its “free homotopy quasi-coproduct completion” in direct analogy to how the category of familes

   $$Fam_{\mathcal{C}} \;\;\;\coloneqq\;\;\; \underset{X\in Set}{\textstyle{\int}} \mathcal{C}^{X}$$

   is the free coproduct completion.

   I think I have proven this in the final theorem (here) but this is fresh from the press, so please handle with care for the moment.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 6th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexFor what it's worth, a bit more of a writeup of this point is now in the notes [here](https://ncatlab.org/schreiber/show/Entanglement+of+Sections#Notes).

   For what it’s worth, a bit more of a writeup of this point is now in the notes here.