reflection along a functor in nLab

Contents

Idea

The reflection of an object dd in a category DD along a functor F:C→DF \colon C\to D is a morphism η d\eta_d in DD, playing the role of a would-be unit of an adjunction L⊣FL\dashv F (which may not exist in general), satisfying that part of the universal property of the unit for that component. If a reflection of each object in DD along FF exists and is chosen then the left adjoint LL to FF does exists.

Similarly, one can define a coreflection along a functor by a morphism ϵ d\epsilon_d in DD playing the role of the would-be counit of an adjunction F⊣RF\dashv R.

Definition

Let F:C→DF \colon C\to D be a functor and dd an object of the category DD.

A reflection of dd along FF (synonym: universal arrow from dd to FF) is a pair (L d,η d)(L_d,\eta_d) of

• an object L d∈CL_d\in C

• a morphism η d:d→F(L d)\eta_d \colon d\to F(L_d) in DD

which is universal in the sense that for any object c∈Cc \in C and morphism f:d→F(c)f \colon d\to F(c) there is a unique α:L d→c\alpha \colon L_d\to c such that F(α)∘η d=fF(\alpha)\circ\eta_d = f. In other words, it is an initial object in the comma category d/Fd/F.

In other words this means that a reflection is an adjoint relative to the functor 1→D1 \to D which picks out d∈Dd \in D.

Dually, a coreflection of dd along FF is a pair (R d,ϵ d)(R_d,\epsilon_d) of

• an object R c∈CR_c\in C

• a morphism ϵ d:F(R d)→d\epsilon_d \colon F(R_d)\to d in DD

which is universal in the sense that for any c∈Cc\in C and a morphism g:F(c)→dg \colon F(c)\to d there is a unique β:c→R d\beta \colon c\to R_d such that ϵ d∘F(β)=g\epsilon_d\circ F(\beta) = g.

In other words this means that a coreflection is a coadjoint relative to the functor 1→D1 \to D which picks out d∈Dd \in D.

If F:C→DF:C\to D is a pseudofunctor among bicategories (homomorphism of bicategories) and dd an object of DD, then a biuniversal arrow from dd to FF is an object LL in CC and a morphism u:L→F(c)u:L\to F(c) such that for every object cc in CC the functor Hom C(L,c)→Hom D(d,F(c))Hom_C(L,c)\to Hom_D(d,F(c)) given by f↦F(f)∘uf\mapsto F(f)\circ u, γ↦F(γ)∘1 u\gamma\mapsto F(\gamma)\circ 1_u (where f,f′:d→cf,f':d\to c and γ:f→f′\gamma:f\to f' is a 2-cell) is an equivalence of categories.

• adjunction
• relative adjunction

Literature

• Francis Borceux, Handbook of Categorical Algebra, I.3.1

For the bicategorical case see for example around def. 9.4 in

• Thomas M. Fiore, Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory, Memoirs of the Amer. Math. Soc. 182 (2006), no. 860. 171 pages. arXiv:math.CT/0408298