nLab
final functor

Contents

Idea

A functor F:CDF : C \to D is final (often called cofinal), if we can restrict diagrams on DD to diagrams on CC along FF without changing their colimit.

Dually, a functor is initial (sometimes called co-cofinal) if pulling back diagrams along it does not change the limits of these diagrams.

Beware that this property is pretty much unrelated to that of a functor being an initial object or terminal object in the functor category [C,D][C,D].

Definition

Definition

A functor F:CDF : C \to D is final if for every object dDd \in D the comma category (d/F)(d/F) is non-empty and connected.

A functor F:CDF : C \to D is initial if the opposite F op:C opD opF^{op} : C^{op} \to D^{op} is final, i.e. if for every object dDd \in D the comma category (F/d)(F/d) is non-empty and connected.

Properties

Proposition

Let F:CDF : C \to D be a functor

The following conditions are equivalent.

  1. FF is final.

  2. For all functors G:DSetG : D \to Set the natural function between colimits

    lim GFlim G \lim_\to G \circ F \to \lim_{\to} G

    is a bijection.

  3. For all categories EE and all functors G:DEG : D \to E the natural morphism between colimits

    lim GFlim G \lim_\to G \circ F \to \lim_{\to} G

    is a isomorphism.

  4. For all functors G:D opSetG : D^{op} \to Set the natural function between limits

    lim Glim GF op \lim_\leftarrow G \to \lim_\leftarrow G \circ F^{op}

    is a bijection.

  5. For all categories EE and all functors G:D opEG : D^{op} \to E the natural morphism

    lim Glim GF op \lim_\leftarrow G \to \lim_\leftarrow G \circ F^{op}

    is an isomorphism.

  6. For all dDd \in D

    lim cCHom D(d,F(c))*. {\lim_\to}_{c \in C} Hom_D(d,F(c)) \simeq * \,.
Proposition

If F:CDF : C \to D is final then CC is connected precisely if DD is.

Proposition

If F 1F_1 and F 2F_2 are final, then so is their composite F 1F 2F_1 \circ F_2.

If F 2F_2 and the composite F 1F 2F_1 \circ F_2 are final, then so is F 1F_1.

If F 1F_1 is a full and faithful functor and the composite is final, then both functors seperately are final.

Generalizations

The generalization of the notion of final functor from category theory to (∞,1)-higher category theory is described at

The characterization of final functors is also a special case of the characterization of exact squares.

Examples

Example

If DD has a terminal object then the functor F:*DF : {*} \to D that picks that terminal object is final: for every dDd \in D the comma category d/Fd/F is equivalent to **. The converse is also true: if a functor *D*\to D is final, then its image is a terminal object.

In this case the statement about preservation of colimits states that the colimit over a category with a terminal object is the value of the diagram at that object. Which is also readily checked directly.

Example

Every right adjoint functor is final.

Proof

Let (LR):CD(L \dashv R) : C \to D be a pair of adjoint functors.To see that RR is final, we may for instance check that for all dDd \in D the comma category d/Rd / R is non-empty and connected:

It is non-empty because it contains the adjunction unit (L(d),dRL(d))(L(d), d \to R L (d)). Similarly, for

d f g R(a) R(b) \array{ && d \\ & {}^{\mathllap{f}}\swarrow && \searrow^{\mathrlap{g}} \\ R(a) &&&& R(b) }

two objects, they are connected by a zig-zag going through the unit, by the universal factorization property of adjunctions

d R(a) Rf¯ RL(d) R(g¯) R(b). \array{ && d \\ & \swarrow &\downarrow& \searrow \\ R(a) &\stackrel{R \bar f}{\leftarrow}& R L (d)& \stackrel{R(\bar g)}{\to} & R(b) } \,.
Example

The inclusion 𝒞𝒞˜\mathcal{C} \to \tilde \mathcal{C} of any category into its idempotent completion is final.

See at idempotent completion in the section on Finality.

References

Section 2.5 of

Section 2.11 of

  • Francis Borceux, Handbook of categorical algebra 1, Basic category theory

Notice that this says “final functor” for the version under which limits are invariant.

Revised on May 29, 2013 15:21:16 by Urs Schreiber (89.204.153.151)