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 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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure 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.
   • CommentAuthorUrs
   • CommentTimeApr 28th 2023
   • (edited Apr 28th 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItextouched wording and hyperlinking of this old entry, for streamlining also added a couple of sentences to the beginning highlighting the ordinary category of enriched functors with the set of enriched natural transformations between them. <a href="https://ncatlab.org/nlab/revision/diff/enriched+functor+category/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/enriched+functor+category/20">v20</a>, <a href="https://ncatlab.org/nlab/show/enriched+functor+category">current</a>

   touched wording and hyperlinking of this old entry, for streamlining

   also added a couple of sentences to the beginning highlighting the ordinary category of enriched functors with the set of enriched natural transformations between them.

   diff, v20, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 10th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItex**Question:** Can the following basic fact be conveniently cited from the literature? For * $V$ a (co)complete symmetric closed monoidal category * $\mathbf{C}$ a $V$-enriched and -(co)tensored closed monoidal category * $\mathbf{X}$ a small $V$-enriched category then the $V$-enriched functor category $Func(\mathbf{X},\mathbf{C})$ is $Func(\mathbf{X},\mathbf{V})$-enriched and -(co)-tensored: The tensoring is objectwise over $\mathbf{X}$ the tensoring of $\mathbf{C}$ over $\mathbf{V}$ the enrichement (powering) is objectwise an end over hom- (power-) objects into the codomain object out of the tensoring of the domain with the corresponding representable. This follows via standard end-yoga, I may spell it out in the nLab entry later. But what I'd like to know is if there is an existing textbook or other publication that makes this explicit?

   Question: Can the following basic fact be conveniently cited from the literature?

   For

   • $V$ a (co)complete symmetric closed monoidal category

   • $\mathbf{C}$ a $V$-enriched and -(co)tensored closed monoidal category

   • $\mathbf{X}$ a small $V$-enriched category

   then the $V$-enriched functor category $Func(\mathbf{X},\mathbf{C})$ is $Func(\mathbf{X},\mathbf{V})$-enriched and -(co)-tensored:

   The tensoring is objectwise over $\mathbf{X}$ the tensoring of $\mathbf{C}$ over $\mathbf{V}$

   the enrichement (powering) is objectwise an end over hom- (power-) objects into the codomain object out of the tensoring of the domain with the corresponding representable.

   This follows via standard end-yoga, I may spell it out in the nLab entry later. But what I’d like to know is if there is an existing textbook or other publication that makes this explicit?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 11th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have typed it out: [here](https://ncatlab.org/nlab/show/enriched+functor+category#EnhancedEnichmentFormulas). <a href="https://ncatlab.org/nlab/revision/diff/enriched+functor+category/24">diff</a>, <a href="https://ncatlab.org/nlab/revision/enriched+functor+category/24">v24</a>, <a href="https://ncatlab.org/nlab/show/enriched+functor+category">current</a>

   I have typed it out: here.

   diff, v24, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 31st 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded a brief paragraph ([here](https://ncatlab.org/nlab/show/enriched+functor+category#AsInternalHomInVCat)) mentioning adjointness to the construction of enriched product categories <a href="https://ncatlab.org/nlab/revision/diff/enriched+functor+category/27">diff</a>, <a href="https://ncatlab.org/nlab/revision/enriched+functor+category/27">v27</a>, <a href="https://ncatlab.org/nlab/show/enriched+functor+category">current</a>

   added a brief paragraph (here) mentioning adjointness to the construction of enriched product categories

   diff, v27, current