nLab final functor

Idea

A functor $F \colon C \to D$ is final, if restricting diagrams along $F$ does not change 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 $[C,D]$. The terminology comes instead from the fact that an object $d\in D$ is initial (resp. terminal) just when the corresponding functor $d \colon 1\to D$ is initial (resp. final).

Remark

(warning on terminology)
In older references (and also some others like HTT), final functors are sometimes called cofinal, the terminology having been imported from order theory (cf. 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 says (before Proposition B2.5.12 ):

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.

See also the warning at final $(\infty,1)$-functor (here).

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 (the non-emptiness condition is redundant since connected categories are non-empty by convention).

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 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 an 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 $d \in D$

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

7. for all functors $G: D \to E$, the canonical map

   $$Nat(G,\Delta-)\to Nat(GF,\Delta-) : E \to Set$$

   between the functors of cocones on $G$ and $GF$ is an isomorphism. (Here, the $\Delta$‘s denote the functors sending objects to constant functors, and $Nat$ stands for the set of natural transformations.)

(Kashiwara & Schapira 2006, Prop. 2.5.2)

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 $F_1$.

Final functors are stable under pushout.

The coproduct of final functors in the arrow category $\mathbf{Cat}^{\mathbf{2}}$ is a final functor.

If $F_1$ 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 called the comprehensive factorization system.

Generalizations

Exact squares

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

Final enriched functors

Finality for enriched functors with respect to weighted colimits (for enrichment in a cartesian closed category) is discussed in Kelly 1982 §4.5: see Theorem 4.67 in particular.

See also discussion of finality specifically for coends (initiality for weighted limits and ends) by:

• Tim Campion: Cofinality for Coends (2020) [MO:q/353876, MO:a/354097]

Final $\infty$-functors

The generalization of the notion of final functor from category theory to (∞,1)-higher category theory is described at

• final (∞,1)-functor.

Examples

Example

(inclusion of a terminal object is final functor)
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.

More generally (see Lemma 3.7 of Clarke 2024):

Lemma

A functor whose domain is a discrete category is final if and only if it is a right adjoint, in which case the functor is fully faithful.

Proof

That every right adjoint functor is final is proven below. For the converse, suppose that we have a final functor $F : C \to D$ where $C$ is discrete. Finality says that, for each $d \in D$, $d \downarrow F$ is connected: - Nonemptyness of $d \downarrow F$ says that there exists some $c_d \in C$ and morphism $\eta_d : d \to F(c_d)$ in $D$. - The existence of a path between any pair of objects says that, given any $c, c' \in C$ and span $F(c) \leftarrow d \rightarrow F(c')$, we have $c = c'$ and that the two legs are equal.

Consequently, we have an assignment $|D| \to |C|$ sending $d \mapsto c_d$. For this to be an adjunction, we must have $C(c_d, c') \cong D(d, R(c'))$. Since $C$ is discrete, this says that there exists at most one morphism $d \to R(c')$ in $D$, and there exists a morphism iff $c_d = c'$. If $c_d = c'$, then we take $\eta_d$. Conversely, if there exists $d \to R(c')$, we necessarily have $c' = c_d$ by the path existence property.

Full faithfulness now follows, since $D(Fc, Fc') \cong C(c_{Fc}, c') = C(c, c')$.

Example

A functor between groupoids is final iff it is essentially surjective and full.

(eg. Cigoli 2018, Prop. 1.1)

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.

Example

For $\Delta^{op}$ the opposite of the simplex category, the non-full subcategory inclusion of the lowest two face maps

$$\big( [1] \underoverset {d_0} {d_1} {\rightrightarrows} [0] \big) \;\xrightarrow{\;\;\;\;}\; \Delta^{op}$$

is a final functor.

It follows that the colimit over a simplicial diagram is equivalently the coequalizer of the lowest two face maps.

(e.g. Riehl 14, Exp. 8.3.8)

Related concepts

• cofinal diagram

• homotopy final functor

• final (∞,1)-functor

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • embedding of categories

  • essentially surjective functor

    • surjective-on-objects functor

    • bijective-on-objects functor

    • dominant functor

  • split essentially surjective functor

  • essentially surjective and full functor

  • equivalence of categories

  • conservative functor

  • localization functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor, dense functor

• 2-functors

  • full and faithful 2-functor

  • locally fully faithful 2-functor

  • equivalence of 2-categories

• (∞,1)-functors:

  • fully faithful (∞,1)-functor

  • essentially surjective (∞,1)-functor

  • equivalence of (∞,1)-categories

  • homotopy final functor, final (∞,1)-functor

References

• Masaki Kashiwara, Pierre Schapira, Section 2.5 of: Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006) [doi:10.1007/3-540-27950-4, pdf]

• Saunders Mac Lane, Section IX.3 of: Categories for the Working Mathematician

• Francis Borceux, Section 2.11 of: Handbook of Categorical Algebra 1, Basic category theory, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994)

  (this says “final functor” for the version under which limits are invariant)

• Emily Riehl, Section 8.3 of: Categorical Homotopy Theory, Cambridge University Press, 2014 (pdf, doi:10.1017/CBO9781107261457)

• Paolo Perrone, Walter Tholen, Kan extensions are partial colimits, Applied Categorical Structures, 2022. (arXiv:2101.04531)

  (this says “confinal functor” for the version under which colimits are invariant)

In enriched category theory:

• Max Kelly, §4.5 in: Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories 10 (2005) 1-136 [ISBN:9780521287029, tac:tr10, pdf]

In internal category theory for internal functors between internal groupoids in exact categories:

• Alan S. Cigoli, A characterization of final functors between internal groupoids in exact categories, Theory and Applications of Categories 33 11 (2018) 265-275. [arXiv:1711.10747, tac:33-11]

• Bryce Clarke, Lifting twisted coreflections against delta lenses, Theory and Applications of Categories 41.26 (2024): 838-877.