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 finite 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.
   • CommentAuthorMike Shulman
   • CommentTimeJan 29th 2019
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI have a silly question. Suppose $E$ is a simplicially enriched category (or maybe a category enriched over any cartesian monoidal category, or maybe even any monoidal category, though that seems more dubious). Then its slice categories $E/x$ are also simplicially enriched, where for $f:y\to x$ and $g:z\to x$ the hom-object $(E/x)(f,g)$ is the fiber of $(g\circ -) : E(y,z) \to E(y,x)$ over $f$. Simplicial copowers in these slice categories are, like ordinary colimits, created in $E$, which means in particular that for $f:x \to y$ the induced functor $f_! : E/x \to E/y$ preserves copowers. Equivalently, the adjunction $f_! \dashv f^*$ is simplicially enriched, and the pullback functor $f^*$ preserves simplicial powers. Now what if $E$ is also locally cartesian closed, so that $f^*$ has a right adjoint $f_*$? Is the adjunction $f^* \dashv f_*$ also necessarily simplicially enriched? Does pullback $f^*$ preserve copowers, and dependent product $f_*$ preserve powers? I feel like I should know the answer, and I know that in the past I've believed that this was true; but I also can't see how it to prove it right now.

   I have a silly question. Suppose is a simplicially enriched category (or maybe a category enriched over any cartesian monoidal category, or maybe even any monoidal category, though that seems more dubious). Then its slice categories are also simplicially enriched, where for and the hom-object is the fiber of over . Simplicial copowers in these slice categories are, like ordinary colimits, created in , which means in particular that for the induced functor preserves copowers. Equivalently, the adjunction is simplicially enriched, and the pullback functor preserves simplicial powers.

   Now what if is also locally cartesian closed, so that has a right adjoint ? Is the adjunction also necessarily simplicially enriched? Does pullback preserve copowers, and dependent product preserve powers? I feel like I should know the answer, and I know that in the past I’ve believed that this was true; but I also can’t see how it to prove it right now.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJan 30th 2019
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexActually I think the question can be reduced to the simpler case of ordinary cartesian closure. That is, if $E$ is an enriched category with products (in the enriched sense) whose underlying category is cartesian closed, is the cartesian-closure adjunction necessarily an enriched adjunction?

   Actually I think the question can be reduced to the simpler case of ordinary cartesian closure. That is, if is an enriched category with products (in the enriched sense) whose underlying category is cartesian closed, is the cartesian-closure adjunction necessarily an enriched adjunction?