Initial and final functors allow limits and colimits to be computed over a smaller or more convenient indexing category. This section records their basic criterion and consequences, taking the following form of Quillen’s Theorem A as a black box:
Theorem 21.5.1 (Joyal’s version of Quillen’s Theorem A, [Lurie (2009), Theorem 4.1.3.1]). For a functor \(\alpha \colon I \to J\), the following two conditions are equivalent:
- (1)
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A functor \(F\colon J \to C\) has a colimit if and only if the functor \(F \circ \alpha \colon I \to C\) has a colimit, and in this case the canonical map \[ \colim _I (F \circ \alpha ) \to \colim _J F \] is an isomorphism in \(C\);
- (2)
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For every object \(j\) of \(J\), the relative slice category \(I_{j/} = I \times _J J_{j/}\) is weakly contractible, meaning that its geometric realization \(\geom {I_{j/}}\) is contractible.
The dual result is also true, where in (1) we use limits and in (2) we use the over categories \(I_{/j}\).
Definition 21.5.2. A functor \(\alpha \) satisfying the conditions of the theorem is called final. We say that \(\alpha \) is initial if \(\alpha \catop \colon I\catop \to J\catop \) is final (i.e. each relative slice \(I_{/j}\) is weakly contractible, or equivalently limits of functors \(F\colon J \to C\) may be computed as limits of \(F \circ \alpha \colon I \to C\)).
Proof. Let \(F\colon C[W^{-1}]\to D\) be a functor. By the universal property of the localization, precomposition with \(q\) is fully faithful. Hence for every object \(X\in D\) it induces an equivalence \[ \Nat (F,\const _X)\iso \Nat (F\circ q,\const _X). \] Thus the cocone functors of \(F\) and \(F\circ q\) are naturally equivalent, so either is representable if and only if the other is, by the same object. Condition (1) of Theorem 21.5.1 now shows that \(q\) is final. □
Warning 21.5.4. The terminology regarding final functors is extremey inconsistent in the literature. Because of historic reasons, final functors are also called cofinal functors, where the ‘co’ roughly means ‘jointly final’ and has nothing to do with the usual meaning of ‘co’ in category theory. Nevertheless, some authors use the words ‘final’ and ‘cofinal’ for the two notions from Definition 21.5.2, and there is no consensus as to which of the two words should refer to which of the two notions.
We hope that the convention in Definition 21.5.2 is easy to remember in light of the following lemma:
Lemma 21.5.5. Consider an object \(j\) of an \(\infty \)-category \(J\). Then \(j\colon * \to C\) is an initial functor if and only if \(j\) is an initial object, and it is a final functor if and only if \(j\) is a final (i.e. terminal) object of \(J\).
Proof. Given an object \(x\) of \(C\), unwinding definitions reveals that the relative slice category of the functor \(j\colon * \to C\) is equivalent to the hom anima \(\Hom _C(j,x)\). Since this is already an anima, it is equivalent to its geometric realization, and hence it is weakly contractible if and only if it is contractible. So we see that \(j\) is an initial functor if and only if the hom anima \(\Hom _C(j,x)\) is contractible for every object \(x\) of \(C\), which by is the case by definition precisely if \(j\) is an initial object. □
A common way of showing that the relative slice categories \(I_{j/}\) or \(I_{/j}\) are weakly contractible is by showing they admit an initial or terminal object:
Lemma 21.5.6. Let \(C\) be an \(\infty \)-category that admits an initial (resp. terminal) object \(x\colon * \to C\). Then \(C\) is weakly contractible: \(\geom {C} \simeq *\).
Proof. By Remark 21.2.3 we have an adjunction \(x\colon * \rightleftarrows C \noloc p_C\), which by Lemma 21.1.7 induces an equivalence \(* \simeq \geom {*} \simeq \geom {C}\). □
Corollary 21.5.7. If a functor \(\alpha \colon I \to J\) admits a left adjoint, then \(\alpha \) is a final functor. Dually, if it admits a right adjoint it is an initial functor.
Proof. We need to show that for every object \(j\) the relative slice category \(I_{j/}\) is weakly contractible. Let \(\beta \colon J \to I\) be a left adjoint to \(\alpha \). Then the relative slice category \(I_{j/}\) is equivalent to the slice category \(I_{\beta (j)/}\) of \(I\). But this admits an initial object given by \((\beta (j), \id _{\beta (j)})\), hence is weakly contractible by Lemma 21.5.6. □
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