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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.
   • CommentAuthorUrs
   • CommentTimeNov 18th 2011
   • (edited Nov 18th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave created a "floating TOC" _[[notions of subcategory]]_ and have included it in the relevant entries

   have created a “floating TOC” notions of subcategory and have included it in the relevant entries

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeNov 19th 2011
   • (edited Nov 19th 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexMaybe a better and simpler title would be simply "subcategories" (this is a topic and it is potentially a bit wider than listing kinds of subcategories, for example it would include [[torsion theory]] which is a couple of subcategories satisfying some axioms). I think we also have entry [[thick subcategory]]. But here is somewhat related genuine question, not for subcategories but subfunctors. There is something I called in one paper "equivalence of subfunctors" and I did not know the standard term. Maybe somebody does ? So let $F_1,F_2:A\to B$ be subfunctors of a functor $G:A\to B$ in which $B$ is a category with images. We say that $F_1$ is equivalent to $F_2$ if there is a natural automorphism $\alpha:G\Rightarrow G$ such that for every $a\in Ob(A)$ the restriction $\alpha_a|_{Im F_1(a)}$ is isomorphism onto $Im F_2(a)$. Thus, roughly speaking there is an natural isomorphism of functors $F_1\cong F_2$ which extends to a natural automorphism of $G$.

   Maybe a better and simpler title would be simply “subcategories” (this is a topic and it is potentially a bit wider than listing kinds of subcategories, for example it would include torsion theory which is a couple of subcategories satisfying some axioms). I think we also have entry thick subcategory.

   But here is somewhat related genuine question, not for subcategories but subfunctors. There is something I called in one paper “equivalence of subfunctors” and I did not know the standard term. Maybe somebody does ? So let $F_1,F_2:A\to B$ be subfunctors of a functor $G:A\to B$ in which $B$ is a category with images. We say that $F_1$ is equivalent to $F_2$ if there is a natural automorphism $\alpha:G\Rightarrow G$ such that for every $a\in Ob(A)$ the restriction $\alpha_a|_{Im F_1(a)}$ is isomorphism onto $Im F_2(a)$. Thus, roughly speaking there is an natural isomorphism of functors $F_1\cong F_2$ which extends to a natural automorphism of $G$.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 19th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added [[thick subcategory]] and [[Serre subcategory]] and [[topologizing subcategory]] to the list.

   I have added thick subcategory and Serre subcategory and topologizing subcategory to the list.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 27th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded *[[bireflective subcategory]]* and touched the formatting of the first few lines <a href="https://ncatlab.org/nlab/revision/diff/notions+of+subcategory/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/notions+of+subcategory/10">v10</a>, <a href="https://ncatlab.org/nlab/show/notions+of+subcategory">current</a>

   added bireflective subcategory and touched the formatting of the first few lines

   diff, v10, current

5. • CommentRowNumber5.
   • CommentAuthornLab edit announcer
   • CommentTimeNov 2nd 2024
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexMissing comma L Spice <a href="https://ncatlab.org/nlab/revision/diff/notions+of+subcategory/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/notions+of+subcategory/11">v11</a>, <a href="https://ncatlab.org/nlab/show/notions+of+subcategory">current</a>

   Missing comma

   L Spice

   diff, v11, current