nLab reflection along a functor

Redirected from "universal arrow".

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Idea

The reflection of an object dd in a category DD along a functor F:CDF \colon C\to D is a morphism η d\eta_d in DD, playing the role of a would-be unit of an adjunction LFL\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 FRF\dashv R.

Definition

Let F:CDF \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 dCL_d\in C

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

which is universal in the sense that for any object cCc \in C and morphism f:dF(c)f \colon d\to F(c) there is a unique α:L dc\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 1D1 \to D which picks out dDd \in D.

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

  • an object R cCR_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 cCc\in C and a morphism g:F(c)dg \colon F(c)\to d there is a unique β:cR d\beta \colon c\to R_d such that ϵ dF(β)=g\epsilon_d\circ F(\beta) = g.

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

If F:CDF: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:LF(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 fF(f)uf\mapsto F(f)\circ u, γF(γ)1 u\gamma\mapsto F(\gamma)\circ 1_u (where f,f:dcf,f':d\to c and γ:ff\gamma:f\to f' is a 2-cell) is an equivalence of categories.

Literature

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

Last revised on September 22, 2024 at 14:25:06. See the history of this page for a list of all contributions to it.