nLab reflection along a functor

Idea

The reflection of an object $d$ in a category $D$ along a functor $F \colon C\to D$ is a morphism $\eta_d$ in $D$, playing the role of a would-be unit of an adjunction $L\dashv F$ (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 $D$ along $F$ exists and is chosen then the left adjoint $L$ to $F$ does exists.

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

Definition

Let $F \colon C\to D$ be a functor and $d$ an object of the category $D$.

A reflection of $d$ along $F$ (synonym: universal arrow from $d$ to $F$) is a pair $(L_d,\eta_d)$ of

• an object $L_d\in C$

• a morphism $\eta_d \colon d\to F(L_d)$ in $D$

which is universal in the sense that for any object $c \in C$ and morphism $f \colon d\to F(c)$ there is a unique $\alpha \colon L_d\to c$ such that $F(\alpha)\circ\eta_d = f$. In other words, it is an initial object in the comma category $d/F$.

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

Dually, a coreflection of $d$ along $F$ is a pair $(R_d,\epsilon_d)$ of

• an object $R_c\in C$

• a morphism $\epsilon_d \colon F(R_d)\to d$ in $D$

which is universal in the sense that for any $c\in C$ and a morphism $g \colon F(c)\to d$ there is a unique $\beta \colon c\to R_d$ such that $\epsilon_d\circ F(\beta) = g$.

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

If $F:C\to D$ is a pseudofunctor among bicategories (homomorphism of bicategories) and $d$ an object of $D$, then a biuniversal arrow from $d$ to $F$ is an object $L$ in $C$ and a morphism $u:L\to F(c)$ such that for every object $c$ in $C$ the functor $Hom_C(L,c)\to Hom_D(d,F(c))$ given by $f\mapsto F(f)\circ u$, $\gamma\mapsto F(\gamma)\circ 1_u$ (where $f,f':d\to c$ and $\gamma:f\to f'$ is a 2-cell) is an equivalence of categories.

Related concepts

• 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