Barr-Beck-Lurie in Families

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 $f:c\to c^{\prime }$ is a morphism in $𝒞$ such that $\mathit{𝑈𝑓}$ is an equivalence in $𝒟$. Then $e:=\mathit{𝑞𝑈𝑓}\simeq \mathit{𝑝𝑓}$ is an equivalence in $B$. One can factor $f$ as $c\underset{}{\overset{\phi }{\to }}{e}_{!}c\underset{}{\overset{f^{\prime }}{\to }}{c}^{\prime }$ where $\phi$ is a coCartesian lift of $e$ and $f^{\prime }$ is a morphism in the fiber ${𝒞}_{b^{\prime }}$ above $b^{\prime }:=p(c^{\prime })$. Since $\phi$ is coCartesian lift of an equivalence, it is an equivalence. Because of the fiberwise monadicity assumption (iii), $f^{\prime }$ 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 $q:{\mathrm{\Delta }}^{\mathrm{𝗈𝗉}}\to 𝒞$ be a $U$-split simplicial object, so that $\mathit{𝑈𝑞}$ extends to a diagram $\widetilde{\mathit{𝑈𝑞}}:{\mathrm{\Delta }}_{-\infty }^{\mathrm{𝗈𝗉}}\to 𝒟$. Let $f:{\mathrm{\Delta }}_{-\infty }^{\mathrm{𝗈𝗉}}\to B$ be the underlying diagram in $B$. There is a morphism

which is the identity on $\{0\}\times {\mathrm{\Delta }}_{-\infty }^{\mathrm{𝗈𝗉}}$ and carries $\{1\}\times {\mathrm{\Delta }}_{-\infty }^{\mathrm{𝗈𝗉}}$ to $[-1]\in {\mathrm{\Delta }}_{-\infty }^{\mathrm{𝗈𝗉}}$. It sends each horizontal morphism $\{0\}\times [n]\to \{1\}\times [n]$ to the unique morphism $[n]\to [-1]$. Consider the composite

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

• Let $Q$ be a coCartesian lift of ${P|}_{{\mathrm{\Delta }}^1\times {\mathrm{\Delta }}^{\mathrm{𝗈𝗉}}}$ to $𝒞$. Then $Q$ is a natural transformation between $q$ and a morphism $q^{\prime }:{\mathrm{\Delta }}^{\mathrm{𝗈𝗉}}\to {𝒞}_b$, where $b$ is the image under $f$ of $[-1]\in {\mathrm{\Delta }}_{-\infty }^{\mathrm{𝗈𝗉}}$.
• Let $\widetilde{\mathit{𝑈𝑄}}$ be a coCartesian lift of $P$ to $𝒟$. Then $\widetilde{\mathit{𝑈𝑄}}$ is a natural transformation between $\widetilde{\mathit{𝑈𝑞}}$ and a morphism $\widetilde{U{q}^{\prime }}:{\mathrm{\Delta }}_{-\infty }^{\mathrm{𝗈𝗉}}\to {𝒞}_b$.

These natural transformations $Q$ and $\widetilde{\mathit{𝑈𝑄}}$ 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 $\mathit{𝑈𝑄}\simeq {\widetilde{\mathit{𝑈𝑄}}|}_{{\mathrm{\Delta }}^1\times {\mathrm{\Delta }}^{\mathrm{𝗈𝗉}}}$. In particular $U{q}^{\prime }:{\mathrm{\Delta }}^{\mathrm{𝗈𝗉}}\to {𝒞}_b$ extends to the split simplicial object $\widetilde{U{q}^{\prime }}:{\mathrm{\Delta }}_{-\infty }^{\mathrm{𝗈𝗉}}\to {𝒞}_b$. By the fiberwise monadicity assumption (iii), this implies that $q^{\prime }$ extends to a colimit diagram ${\overline{q}}^{\prime }:{({\mathrm{\Delta }}^{\mathrm{𝗈𝗉}})}^{⊳}\to {𝒞}_b$ such that $U{\overline{q}}^{\prime }$ is also a colimit diagram. By assumption (iv) and [Lur09, Proposition 4.3.1.10] it then follows that ${\overline{q}}^{\prime }$ and $U{\overline{q}}^{\prime }$, 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

$$$$

(4)

in which $\pi :{({\mathrm{\Delta }}^1\times {\mathrm{\Delta }}^{\mathrm{𝗈𝗉}})}^{⊳}\to {({\mathrm{\Delta }}^{\mathrm{𝗈𝗉}})}^{⊳}={\mathrm{\Delta }}_{+}^{\mathrm{𝗈𝗉}}\subseteq {\mathrm{\Delta }}_{-\infty }^{\mathrm{𝗈𝗉}}$ denotes the morphism which is the identity on $\{0\}\times {\mathrm{\Delta }}^{\mathrm{𝗈𝗉}}$ and which carries ${(\{1\}\times {\mathrm{\Delta }}^{\mathrm{𝗈𝗉}})}^{⊳}$ 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 ${\mathrm{\Delta }}^1\times {({\mathrm{\Delta }}^{\mathrm{𝗈𝗉}})}^{⊳}\to {({\mathrm{\Delta }}^1\times {\mathrm{\Delta }}^{\mathrm{𝗈𝗉}})}^{⊳}$ which is the identity on ${\mathrm{\Delta }}^1\times {\mathrm{\Delta }}^{\mathrm{𝗈𝗉}}$ and carries the other vertices of ${\mathrm{\Delta }}^1\times {({\mathrm{\Delta }}^{\mathrm{𝗈𝗉}})}^{⊳}$ to the cone point. Let $\overline{Q}$ denote the composition

and define $\overline{q}:={\overline{Q}|}_{\{0\}\times {({\mathrm{\Delta }}^{\mathrm{𝗈𝗉}})}^{⊳}}$. Then $\overline{Q}$ is a natural transformation from $\overline{q}$ to ${\overline{q}}^{\prime }$ which is componentwise coCartesian. Then [Lur09, Proposition 4.3.1.9] implies that $\overline{q}$ is a $p$-colimit diagram which fits into the diagram

$$$$

(6)

By assumption (i), $U\overline{Q}$ is a natural transformation from $U\overline{q}$ to $U{\overline{q}}^{\prime }$ which is componentwise coCartesian. Hence [Lur09, Proposition 4.3.1.9] implies that $U\overline{q}$ is a $p$-colimit diagram. The underlying diagram ${f|}_{{({\mathrm{\Delta }}^{\mathrm{𝗈𝗉}})}^{⊳}}$ of $\overline{q}$ 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 $\overline{q}$ and $U\overline{q}$ are colimit diagrams in $𝒞$ and $𝒟$ respectively. Hence $𝒞$ admits and $U$ preserves geometric realizations of $U$-split simplicial objects. □