Definition 10.3.5.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ is regular if it satisfies the following conditions:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ admits finite limits.
- $(2)$
The $\infty $-category $\operatorname{\mathcal{C}}$ has images. That is, every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ can be extended to a $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & Y_0 \ar [dr]^{i} & \\ X \ar [ur]^{q} \ar [rr]^{f} & & Y, } \]where $q$ is a quotient morphism and $i$ is a monomorphism.
- $(3)$
The collection of quotient morphisms in $\operatorname{\mathcal{C}}$ is closed under pullback. That is, for every pullback diagram
\[ \xymatrix@C =50pt@R=50pt{ X' \ar [d]^{f'} \ar [r] & X \ar [d]^{f} \\ Y' \ar [r] & Y } \]of $\operatorname{\mathcal{C}}$, if $f$ is a quotient morphism, then $f'$ is also a quotient morphism.