Barr–Beck–Lurie in families

In this section we present a generalization of the result of [GHK22, Proposition 4.4.5] which is adapted to our setting.

Proposition 1.1. Given a diagram

𝒞 𝒟 Upr ℬ

in $Cat∞$ such that:

(i): $p$ and $r$ are coCartesian fibrations and $U$ preserves coCartesian edges;
(ii): $U$ has a left adjoint $F : 𝒟 → 𝒞$ such that $pF ≃ r$ ;
(iii): The adjunction $F ⊣ U$ restricts in each fiber to an adjunction $Fb ⊣ Ub$ . For all $b ∈ ℬ$ , the functor $Ub$ is conservative, and $𝒞b$ admits colimits of $Ub$ -split simplicial objects, which $Ub$ preserves.
(iv): For any edge $′ e : b → b$ in $ℬ$ , the coCartesian covariant transport $e! : 𝒞b → 𝒞b′$ preserves geometric realizations of $Ub$ -split simplicial objects.

Then, the adjunction $F ⊣ U$ is monadic.

Proof of Proposition 1.1. We verify the conditions of the Barr–Beck–Lurie theorem [Lur17, Theorem 4.7.3.5].

First we show that $U$ is conservative. We can argue in exactly the same way as [GHK22, Proposition 4.4.5]. Suppose that $′ f : c → c$ is a morphism in $𝒞$ such that $Uf$ is an equivalence in $𝒟$ . Then $e := qUf ≃ pf$ is an equivalence in $ℬ$ . One can factor $f$ as $φ f′ c −→ e!c−→ c′$ where $φ$ is a coCartesian lift of $e$ and $f′$ is a morphism in the fiber $𝒞b′$ above $b′ := p(c′)$ . Since $φ$ is coCartesian lift of an equivalence, it is an equivalence. Because of the fiberwise monadicity assumption (iii), $f′$ is an equivalence. Therefore $f$ is an equivalence and $U$ is conservative.

Now we will show that $𝒞$ admits and $U$ preserves colimits of $U$ -split simplicial objects. Let $q : Δop → 𝒞$ be a $U$ -split simplicial object, so that $U q$ extends to a diagram $U～q : Δop−∞ → 𝒟$ . Let $f : Δop−∞ → ℬ$ be the underlying diagram in $ℬ$ . There is a morphism

Δ1 × Δop → Δop −∞ −∞

(1)

which is the identity on ${0} × Δo−p∞$ and carries ${1} × Δo−p∞$ to $[− 1] ∈ Δo−p∞$ . It sends each horizontal morphism ${0}× [n] → {1} × [n]$ to the unique morphism $[n] → [− 1]$ . Consider the composite

P : Δ1 × Δop → Δop f−→ ℬ. −∞ −∞

(2)

Now we will take a coCartesian lifts, using the exponentiation for coCartesian fibrations [Lur18, Tag 01VG].

Let $Q$ be a coCartesian lift of $P |Δ1×Δop$ to $𝒞$ . Then $Q$ is a natural transformation between $q$ and a morphism $q′ : Δop → 𝒞b$ , where $b$ is the image under $f$ of $[− 1] ∈ Δo−p∞$ .
Let $～UQ$ be a coCartesian lift of $P$ to $𝒟$ . Then $～UQ$ is a natural transformation between $U～q$ and a morphism $～Uq′ : Δop−∞ → 𝒞b$ .

These natural transformations $Q$ and $～U Q$ are uniquely characterised by the property that their components are coCartesian edges [Lur18, Tag 01VG]. Because of the assumption (i) that $U$ preserves coCartesian edges, this unicity implies that $| U Q ≃ ～U Q|| Δ1×Δop$ . In particular $U q′ : Δop → 𝒞b$ extends to the split simplicial object $op U～q′ : Δ −∞ → 𝒞b$ . By the fiberwise monadicity assumption (iii), this implies that $q′$ extends to a colimit diagram $-′ q : (Δop)⊳ → 𝒞b$ such that $Uq′$ is also a colimit diagram. By assumption (iv) and [Lur09, Proposition 4.3.1.10] it then follows that $q′$ and $Uq′$ , when regarded as diagrams in $𝒞$ and $𝒟$ respectively, are $p$ -colimit diagrams. Now we can argue as in [Lur09, Corollary 4.3.1.11]. We have a commutative diagram

∐ (Δ1 × Δop) {1}×Δop({1} × (Δ𝒞op )⊳) (Qps(,f|q(Δ′)op)⊳)∘π (Δ1 × Δop)⊳ ℬ

in which $π : (Δ1 × Δop)⊳ → (Δop )⊳ = Δo+p⊆ Δop−∞$ denotes the morphism which is the identity on ${0}× Δop$ and which carries $({1}× Δop)⊳$ to the cone point. Because the left map is an inner fibration there exists a lift $s$ as indicated by the dashed arrow. Consider now the map $Δ1 × (Δop)⊳ → (Δ1 ×Δop )⊳$ which is the identity on $Δ1 × Δop$ and carries the other vertices of $Δ1 × (Δop)⊳$ to the cone point. Let $-- Q$ denote the composition

s Δ1 × (Δop )⊳ → (Δ1 × Δop)⊳−→ 𝒞

(3)

and define $- -| q := Q|{0}×(Δop)⊳$ . Then $-- Q$ is a natural transformation from $- q$ to $-′ q$ which is componentwise coCartesian. Then [Lur09, Proposition 4.3.1.9] implies that $- q$ is a $p$ -colimit diagram which fits into the diagram

Δop 𝒞 - op⊳ qpqf|(Δo(Δp)⊳ ) ℬ

By assumption (i), $U Q-$ is a natural transformation from $U q$ to $Uq′$ which is componentwise coCartesian. Hence [Lur09, Proposition 4.3.1.9] implies that $Uq$ is a $p$ -colimit diagram. The underlying diagram $f| op⊳ (Δ )$ of $q$ in $ℬ$ extends to the split simplicial diagram $f$ and hence admits a colimit in $ℬ$ . Hence [Lur09, Proposition 4.3.1.5(2)] implies that $q$ and $U q$ are colimit diagrams in $𝒞$ and $𝒟$ respectively. Hence $𝒞$ admits and $U$ preserves geometric realizations of $U$ -split simplicial objects. □

Barr–Beck–Lurie in families

1 Barr–Beck–Lurie in families

References