Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

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.
   • CommentAuthorzskoda
   • CommentTimeAug 22nd 2024
   • PermaLink
   Author: zskoda
   Format: MarkdownItexPrecursor of a left adjoint, Borceux I.3.1. <a href="https://ncatlab.org/nlab/revision/reflection+along+a+functor/1">v1</a>, <a href="https://ncatlab.org/nlab/show/reflection+along+a+functor">current</a>

   Precursor of a left adjoint, Borceux I.3.1.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorvarkor
   • CommentTimeAug 22nd 2024
   • PermaLink
   Author: varkor
   Format: MarkdownItexIsn't this simply a [[relative adjoint]], relative to the functor $1 \to D$ that picks out $d \in D$? Perhaps this content would be better there.

   Isn’t this simply a relative adjoint, relative to the functor $1 \to D$ that picks out $d \in D$? Perhaps this content would be better there.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeAug 22nd 2024
   • PermaLink
   Author: zskoda
   Format: MarkdownItexYes, but it may be slightly too difficult for a non-specialist to hide it there: the notion of a [[reflection along a functor]] is more elementary even than of the adjunction and then for relative adjoint we have already a non-self-dual notion so one has to think which version for coreflection etc. Rather, one can just make a link here -- rather have two treatments for two different levels of audience than to loose simple concepts in more complicated in my opinion.

   Yes, but it may be slightly too difficult for a non-specialist to hide it there: the notion of a reflection along a functor is more elementary even than of the adjunction and then for relative adjoint we have already a non-self-dual notion so one has to think which version for coreflection etc. Rather, one can just make a link here – rather have two treatments for two different levels of audience than to loose simple concepts in more complicated in my opinion.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeAug 22nd 2024
   • (edited Aug 22nd 2024)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexAdjoints to inclusions may also be viewed as ordinary adjoints of the usual [[corestriction]] of the original functor if you wish. So, using corestriction, one does not need a relative point of view. For example, a right adjoint $\bar{A}\to A$ to the corestriction of a fully faithful functor $F:A\to B$ to the subcategory $\bar{A}$ of all $b$ in $B$ such that $B(F(-),b): A^{op}\to Set$ is representable is in fact a coreflection in the sense used in the context of [[coreflective subcategory]] (and is sometimes usefully viewed as an example of a [[Q-category]], as noticed by Rosenberg in 1980-s).

   Adjoints to inclusions may also be viewed as ordinary adjoints of the usual corestriction of the original functor if you wish. So, using corestriction, one does not need a relative point of view.

   For example, a right adjoint $\bar{A}\to A$ to the corestriction of a fully faithful functor $F:A\to B$ to the subcategory $\bar{A}$ of all $b$ in $B$ such that $B(F(-),b): A^{op}\to Set$ is representable is in fact a coreflection in the sense used in the context of coreflective subcategory (and is sometimes usefully viewed as an example of a Q-category, as noticed by Rosenberg in 1980-s).

5. • CommentRowNumber5.
   • CommentAuthorvarkor
   • CommentTimeAug 22nd 2024
   • PermaLink
   Author: varkor
   Format: MarkdownItexMentioned the connection to [[relative adjunctions]]. <a href="https://ncatlab.org/nlab/revision/diff/reflection+along+a+functor/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/reflection+along+a+functor/3">v3</a>, <a href="https://ncatlab.org/nlab/show/reflection+along+a+functor">current</a>

   Mentioned the connection to relative adjunctions.

   diff, v3, current