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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology 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 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.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 22nd 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: Specifically, a continuous functor $C\to Set$ is a right adjoint functor if and only if it is representable, in which case the left adjoint functor $Set\to C$ sends the [[singleton]] set to the representing object ## Related concepts * [[adjoint functor theorem]] <a href="https://ncatlab.org/nlab/revision/diff/representable+functor+theorem/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/representable+functor+theorem/3">v3</a>, <a href="https://ncatlab.org/nlab/show/representable+functor+theorem">current</a>

   Added:

   Specifically, a continuous functor $C\to Set$ is a right adjoint functor if and only if it is representable, in which case the left adjoint functor $Set\to C$ sends the singleton set to the representing object

   Related concepts

   • adjoint functor theorem

   diff, v3, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 22nd 2023
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   Author: Urs
   Format: MarkdownItexI see now that, originating with the creation of the entry back in 2011, it has a line starting with the words > As representable functors are ubiquitous,... This does not make sense to me, neither the claim itself nor the suggestion that it implies the statement that follows. I wonder what was really meant here. But it looks like just deleting these words would not take anything away from the paragraphs that follow. (?)

   I see now that, originating with the creation of the entry back in 2011, it has a line starting with the words

   As representable functors are ubiquitous,…

   This does not make sense to me, neither the claim itself nor the suggestion that it implies the statement that follows. I wonder what was really meant here. But it looks like just deleting these words would not take anything away from the paragraphs that follow. (?)

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 22nd 2023
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI changed the opening of the first sentence under "Related facts" a little so that it ties in better with the phrase that follows (the earlier version, about the ubiquity of representable functors, seemingly echoes similar statements about the ubiquity of (the concept of) adjoint functors -- see the quotations of Mac Lane given in [[Categories for the Working Mathematician]]). <a href="https://ncatlab.org/nlab/revision/diff/representable+functor+theorem/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/representable+functor+theorem/5">v5</a>, <a href="https://ncatlab.org/nlab/show/representable+functor+theorem">current</a>

   I changed the opening of the first sentence under “Related facts” a little so that it ties in better with the phrase that follows (the earlier version, about the ubiquity of representable functors, seemingly echoes similar statements about the ubiquity of (the concept of) adjoint functors – see the quotations of Mac Lane given in Categories for the Working Mathematician).

   diff, v5, current