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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorJ-B Vienney
   • CommentTimeAug 22nd 2024
   • (edited Aug 22nd 2024)
   • PermaLink
   Author: J-B Vienney
   Format: MarkdownItexNew entry. This is a bit experimental. I will finish later. I want to add examples for instance the category of groups from the category of sets etc... <a href="https://ncatlab.org/nlab/revision/category+with+extra+structure/1">v1</a>, <a href="https://ncatlab.org/nlab/show/category+with+extra+structure">current</a>

   New entry. This is a bit experimental. I will finish later. I want to add examples for instance the category of groups from the category of sets etc…

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorJ-B Vienney
   • CommentTimeAug 22nd 2024
   • (edited Aug 22nd 2024)
   • PermaLink
   Author: J-B Vienney
   Format: MarkdownItexdeleted

   deleted

3. • CommentRowNumber3.
   • CommentAuthorJ-B Vienney
   • CommentTimeAug 22nd 2024
   • PermaLink
   Author: J-B Vienney
   Format: MarkdownItexChanged the name to a better one. <a href="https://ncatlab.org/nlab/revision/category+with+extra+structure+on+objects/1">v1</a>, <a href="https://ncatlab.org/nlab/show/category+with+extra+structure+on+objects">current</a>

   Changed the name to a better one.

   v1, current

4. • CommentRowNumber4.
   • CommentAuthorHurkyl
   • CommentTimeAug 22nd 2024
   • (edited Aug 22nd 2024)
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexMechanically, what you're describing is, given a category $C$, a way to (simultaneously) construct a new category $D$ along with a faithful functor $D \to C$. Right? Albeit with the restriction that it's a literal injection on hom-sets. So, the idea is that "extra structure" can be conceived of as the objects of $D$ being the pair of an object of $C$ together with a choice of 'structure' on that object (and then we abstract further and just allow $Ob(D)$ to be an abstract set), right? Maybe it's because I'm sleepy, but I feel like the abstract is very confusing on this point. Maybe starting off by talking about the "structure of a category" using the same word in a different way is throwing me off, setting me up to misunderstand the next paragraph?

   Mechanically, what you’re describing is, given a category $C$, a way to (simultaneously) construct a new category $D$ along with a faithful functor $D \to C$. Right? Albeit with the restriction that it’s a literal injection on hom-sets.

   So, the idea is that “extra structure” can be conceived of as the objects of $D$ being the pair of an object of $C$ together with a choice of ’structure’ on that object (and then we abstract further and just allow $Ob(D)$ to be an abstract set), right? Maybe it’s because I’m sleepy, but I feel like the abstract is very confusing on this point. Maybe starting off by talking about the “structure of a category” using the same word in a different way is throwing me off, setting me up to misunderstand the next paragraph?

5. • CommentRowNumber5.
   • CommentAuthorJ-B Vienney
   • CommentTimeAug 22nd 2024
   • (edited Aug 22nd 2024)
   • PermaLink
   Author: J-B Vienney
   Format: MarkdownItexYou understand what I'm trying to do very well. It can probably be explained better. If you feel like you can make the abstract clearer, you're welcome to modify it! I'm going to try to replace the expression "structure of category" by something else.

   You understand what I’m trying to do very well. It can probably be explained better. If you feel like you can make the abstract clearer, you’re welcome to modify it! I’m going to try to replace the expression “structure of category” by something else.

6. • CommentRowNumber6.
   • CommentAuthorJ-B Vienney
   • CommentTimeAug 22nd 2024
   • PermaLink
   Author: J-B Vienney
   Format: MarkdownItexI made this change. I think it's a bit less confusing now.

   I made this change. I think it’s a bit less confusing now.