# nLab final functor

### Context

#### Limits and colimits

limits and colimits

# Contents

## 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

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.

## 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.

Section IX.3 of

Revised on November 4, 2014 19:24:28 by Urs Schreiber (141.0.8.155)