nLab
final functor

Idea

A functor $F:C\to D$ is final (often called cofinal), if we can restrict diagrams on $D$ to diagrams on $C$ along $F$ 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 $[C,D]$.

Definition

Definition

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

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

Properties

Proposition

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

The following conditions are equivalent.

1. $F$ is final.

2. For all functors $G:D\to \mathrm{Set}$ the natural function between colimits

   $$\underrightarrow{\lim }G\circ F\to \underrightarrow{\lim }G$$\lim_\to G \circ F \to \lim_{\to} G

   is a bijection.

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

   $$\underrightarrow{\lim }G\circ F\to \underrightarrow{\lim }G$$\lim_\to G \circ F \to \lim_{\to} G

   is a isomorphism.

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

   $$\underleftarrow{\lim }G\to \underleftarrow{\lim }G\circ F^{\mathrm{op}}$$\lim_\leftarrow G \to \lim_\leftarrow G \circ F^{op}

   is a bijection.

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

   $$\underleftarrow{\lim }G\to \underleftarrow{\lim }G\circ F^{\mathrm{op}}$$\lim_\leftarrow G \to \lim_\leftarrow G \circ F^{op}

   is an isomorphism.

6. For all $d\in D$

   $${\underrightarrow{\lim }}_{c\in C}{\mathrm{Hom}}_D(d,F(c))\simeq *.$${\lim_\to}_{c \in C} Hom_D(d,F(c)) \simeq * \,.

Proposition

If $F:C\to D$ is final then $C$ is connected precisely if $D$ is.

Proposition

If $F_1$ and $F_2$ are final, then so is their composite $F_1\circ F_2$.

If $F_2$ and the composite $F_1\circ F_2$ are final, then so is $F_1$.

If $F_1$ 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

• final (∞,1)-functor.

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

Examples

Example

If $D$ has a terminal object then the functor $F:*\to D$ that picks that terminal object is final: for every $d\in D$ the comma category $d/F$ is equivalent to $*$. The converse is also true: if a functor $*\to D$ 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 $(L⊣R):C\to D$ be a pair of adjoint functors.To see that $R$ is final, we may for instance check that for all $d\in D$ the comma category $d/R$ is non-empty and connected:

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

$$\begin{array}{ccc}& & d\\ & {}^f\swarrow & & {\searrow }^g\\ R(a)& & & & R(b)\end{array}$$\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

$$\begin{array}{ccc}& & d\\ & \swarrow & \downarrow & \searrow \\ R(a)& \stackrel{R\overline{f}}{\leftarrow }& RL(d)& \stackrel{R(\overline{g})}{\to }& R(b)\end{array}.$$\array{ && d \\ & \swarrow &\downarrow& \searrow \\ R(a) &\stackrel{R \bar f}{\leftarrow}& R L (d)& \stackrel{R(\bar g)}{\to} & R(b) } \,.

Related concepts

• final functor, cofinal diagram

• homotopy final functor

• final (∞,1)-functor

References

Section 2.5 of

• Kashiwara, Shapira, Categories and Sheaves

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.