# 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 $\left[C,D\right]$.

## Definition

###### Definition

A functor $F:C\to D$ is final if for every object $d\in D$ the comma category $\left(d/F\right)$ 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 $\left(F/d\right)$ 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

$\underset{\to }{\mathrm{lim}}G\circ F\to \underset{\to }{\mathrm{lim}}G$\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

$\underset{\to }{\mathrm{lim}}G\circ F\to \underset{\to }{\mathrm{lim}}G$\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

$\underset{←}{\mathrm{lim}}G\to \underset{←}{\mathrm{lim}}G\circ {F}^{\mathrm{op}}$\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

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

is an isomorphism.

6. For all $d\in D$

${\underset{\to }{\mathrm{lim}}}_{c\in C}{\mathrm{Hom}}_{D}\left(d,F\left(c\right)\right)\simeq *\phantom{\rule{thinmathspace}{0ex}}.${\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 $\left(L⊣R\right):C\to D$ 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 $\left(L\left(d\right),d\to RL\left(d\right)\right)$. Similarly, for

$\begin{array}{ccc}& & d\\ & {}^{f}↙& & {↘}^{g}\\ R\left(a\right)& & & & R\left(b\right)\end{array}$\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

$\begin{array}{ccc}& & d\\ & ↙& ↓& ↘\\ R\left(a\right)& \stackrel{R\overline{f}}{←}& RL\left(d\right)& \stackrel{R\left(\overline{g}\right)}{\to }& R\left(b\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $𝒞\to \stackrel{˜}{𝒞}$ 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)