Kerodon

Warning 3.4.0.4. For any diagram of simplicial sets $X_0 \rightarrow X \leftarrow X_1$, the simplicial set $X_0 \times _{ \operatorname{Fun}( \{ 0\} , X) } \operatorname{Fun}( \Delta ^1, X ) \times _{ \operatorname{Fun}( \{ 1\} , X) } X_1$ is well-defined. However, we will refer to it as a homotopy fiber product (and denote it by $X_0 \times ^{\mathrm{h}}_{X} X_1$) only in the case where $X$ is a Kan complex. In more general situations, we will refer to this simplicial set as the oriented fiber product of $X_0$ with $X_1$ over $X$, and denote it by $X_0 \operatorname{\vec{\times }}_{X} X_1$ (Definition 4.6.4.1). In the setting of $\infty $-categories, we will adopt a slightly different definition for the homotopy fiber product $X_0 \times _{X}^{\mathrm{h}} X_1$: see Construction 4.5.2.1.