nLab
final functor

Contents

Contents

Idea

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

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

Warning: In older references (and also some others like HTT), final functors are sometimes called cofinal, the terminology having been imported from order theory (e.g. cofinality). However, this is confusing in category theory because usually the prefix “co-” denotes dualization. In at least one place (Borceux) this non-dualization was treated as a dualization and the word “final” used for the dual concept, but in general it seems that the consensus is to use “final” for what used to be called “cofinal”, and “initial” for the dual concept (since “co-final” would be ambiguous). For example, Johnstone in Sketches of an Elephant before Proposition B2.5.12 says:
Traditionally, final functors were called ‘cofinal functors’; but this use of ‘co’ is potentially misleading as it has nothing to do with dualization — it is derived from the Latin ‘cum’ rather than ‘contra’ — and so it is now generally omitted.

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.

The first two statements of Proposition in fact follow from the stability properties of orthogonal factorization systems:

Proposition

Final functors and discrete fibrations form an orthogonal factorization system.

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.

Example

The inclusion of the cospan diagram into its cocone

(a c b)(a c p b) \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

Last revised on August 4, 2019 at 09:53:50. See the history of this page for a list of all contributions to it.