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1. • CommentRowNumber1.
   • CommentAuthorjademaster
   • CommentTimeJan 15th 2023
   • PermaLink
   Author: jademaster
   Format: MarkdownItexJust some notes about generalizing the Grothendieck construction to models of a theory. <a href="https://ncatlab.org/nlab/revision/T-Grothendieck+construction/1">v1</a>, <a href="https://ncatlab.org/nlab/show/T-Grothendieck+construction">current</a>

   Just some notes about generalizing the Grothendieck construction to models of a theory.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorvarkor
   • CommentTimeJan 16th 2023
   • (edited Jan 16th 2023)
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   Author: varkor
   Format: MarkdownItexA few minor issues that are easily addressed: - $T$ is said to be an algebraic theory, but the theory of categories is not algebraic. - The categories of "models of a theory" are here given as functor categories, but the definition of model of a theory requires some limit preservation. - The information about 2-functoriality of functor categories is useful to have, but would probably make more sense on [[functor category]]. - In the first example "we recover the displayed category construction" seems a little bit of a stretch, given that the equivalence you reference is doing a lot of work. - It would be good to explain how the main equivalence is an instance of the Grothendieck construction. The concern I have with the page is that the main theorem is really nothing about (algebraic) theories: it's an immediate consequence of the (discrete) Grothendieck construction, as you explain in the proof. It's not clear that this deserves its own name. Perhaps it would be better to explain that this is a useful consequence of the Grothendieck construction on the [[Grothendieck construction]] page?

   A few minor issues that are easily addressed:

   • $T$ is said to be an algebraic theory, but the theory of categories is not algebraic.
   • The categories of “models of a theory” are here given as functor categories, but the definition of model of a theory requires some limit preservation.
   • The information about 2-functoriality of functor categories is useful to have, but would probably make more sense on functor category.
   • In the first example “we recover the displayed category construction” seems a little bit of a stretch, given that the equivalence you reference is doing a lot of work.
   • It would be good to explain how the main equivalence is an instance of the Grothendieck construction.

   The concern I have with the page is that the main theorem is really nothing about (algebraic) theories: it’s an immediate consequence of the (discrete) Grothendieck construction, as you explain in the proof. It’s not clear that this deserves its own name. Perhaps it would be better to explain that this is a useful consequence of the Grothendieck construction on the Grothendieck construction page?