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.
   • CommentAuthorHarry Gindi
   • CommentTimeMar 2nd 2011
   • (edited Mar 2nd 2011)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexSuppose $T$ is a (grothendieck) topos, and suppose $C\subseteq T$ is a full subcategory. If the inclusion $C\hookrightarrow T$ has a lex left adjoint, is $C$ necessarily a topos? What if the embedding is not necessarily full?

   Suppose $T$ is a (grothendieck) topos, and suppose $C\subseteq T$ is a full subcategory. If the inclusion $C\hookrightarrow T$ has a lex left adjoint, is $C$ necessarily a topos?

   What if the embedding is not necessarily full?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 2nd 2011
   • (edited Mar 2nd 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAs you know, Grothendieck toposes are the exact reflective subcategories of presheaf toposes. So if $T$ is a Grothendieck topos, there exists a small category $B$ and a left exact reflective inclusion $T \hookrightarrow PSh(B)$. Since left exact reflective inclusions are closed under composition, it follows that also $$ C \hookrightarrow T \hookrightarrow PSh(B) $$ is a left exact reflective inclusion into a presheaf topos.

   As you know, Grothendieck toposes are the exact reflective subcategories of presheaf toposes. So if $T$ is a Grothendieck topos, there exists a small category $B$ and a left exact reflective inclusion $T \hookrightarrow PSh(B)$. Since left exact reflective inclusions are closed under composition, it follows that also

   $$C \hookrightarrow T \hookrightarrow PSh(B)$$

   is a left exact reflective inclusion into a presheaf topos.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeMar 4th 2011
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexThanks!

   Thanks!