nLab
Day's reflection theorem

Day’s reflection theorem gives conditions under which a reflective subcategory of a symmetric monoidal closed category is closed under internal homs (or, more strongly, is an “exponential ideal”). The formulation and efficient proof we give are modeled on some notes by Ross Street.

Theorem (Day)

Let R:CDR: C \to D be a fully faithful functor with left adjoint L:DCL \colon D \to C, and suppose given a symmetric monoidal closed structure on DD with tensor \otimes and internal hom [,][-, -]. Then for any object cc of CC and dd of DD, if any one of the following natural transformations is invertible, then all are:

  1. u[d,Rc]:[d,Rc]RL[d,Rc]u[d, R c] \colon [d, R c] \to R L[d, R c];

  2. [u,1]:[RLd,Rc][d,Rc][u, 1] \colon [R L d, R c] \to [d, R c];

  3. L(u1):L(dd)L(RLdd)L(u \otimes 1) \colon L(d \otimes d') \to L(R L d \otimes d');

  4. L(uu):L(dd)L(RLdRLd)L(u \otimes u) \colon L(d \otimes d') \to L(R L d \otimes R L d').

In particular, if DD is cartesian closed and LL preserves products, then R:CDR \colon C \to D realizes CC as an exponential ideal of DD.

Proof

I will put this in later.

Created on October 9, 2011 03:44:07 by Todd Trimble (69.118.58.208)