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Corollary 5.6.1.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

There exists a morphism of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{Set})$ and an isomorphism $\operatorname{\mathcal{E}}\simeq \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ which carries $U$ to the projection map $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$.

$(2)$

There exists a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \operatorname{N}_{\bullet }( \operatorname{\mathcal{D}}) \ar [d]^{ \operatorname{N}_{\bullet }(V) } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ), } \]

where $V: \operatorname{\mathcal{D}}\rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is a left covering functor (in the sense of Definition 4.2.3.1).

$(3)$

For every category $\operatorname{\mathcal{C}}'$ and every morphism of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}') \rightarrow \operatorname{\mathcal{C}}$, the fiber product $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}') \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is isomorphic to the nerve of a category $\operatorname{\mathcal{E}}'$ and the projection $\operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}'$ is left covering functor (in the sense of Definition 4.2.3.1).

$(4)$

For every $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the fiber product $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is isomorphic to the nerve of a category $\operatorname{\mathcal{E}}'$ and the projection $\operatorname{\mathcal{E}}' \rightarrow [n]$ is a left covering functor (in the sense of Definition 4.2.3.1).

$(5)$

The morphism $U$ is a left covering map of simplicial sets (in the sense of Definition 4.2.3.8).

Proof. Every morphism of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{Set})$ determines a functor of categories $\mathrm{h} \mathit{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$, and Example 5.5.4.8 supplies a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [r] \ar [d] & \operatorname{N}_{\bullet }( \int _{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} } \mathrm{h} \mathit{\mathscr {F}} ) \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ). } \]

The implication $(1) \Rightarrow (2)$ follows from Remark 5.5.1.9, the implication $(2) \Rightarrow (3)$ follows from Remark 4.2.3.6, the implication $(3) \Rightarrow (4)$ is trivial, and the implication $(4) \Rightarrow (5)$ follows by combining Remark 4.2.3.15 with Proposition 4.2.3.16. The implication $(5) \Rightarrow (1)$ follows from Proposition 5.6.1.5. $\square$