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 $[C,D]$. The terminology comes instead from the fact that an object $d\in D$ is initial (resp. terminal) just when the corresponding functor $d:1\to D$ is initial (resp. final).

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 $F^{op} : C^{op} \to D^{op}$ 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 $G : D \to Set$ the natural function between colimits

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

   is a bijection.

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

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

   is a isomorphism.

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

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

   is a bijection.

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

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

   is an isomorphism.

6. For all $d \in D$

   $${\lim_\to}_{c \in C} Hom_D(d,F(c)) \simeq * \,.$$

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 $(L \dashv R) : C \to D$ 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 $(L(d), d \to R L (d))$. Similarly, for

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

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

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

Example

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

See at idempotent completion in the section on Finality.

Example

The inclusion of the cospan diagram into its cocone

$$\left( \array{ a \\ \downarrow \\ c \\ \uparrow \\ b } \right) \hookrightarrow \left( \array{ a \\ \downarrow & \searrow \\ c &\longrightarrow & p \\ \uparrow & \nearrow \\ b } \right)$$

is initial.

Remark

By the characterization (here) of limits in a slice category, this implies that fiber products in a slice category are computed as fiber products in the underlying category, or in other words that dependent sum to the point preserves fiber products.

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.

Section IX.3 of

• Saunders Mac Lane, Categories for the Working Mathematician