A definition of ‘exact’ for an augmented (semi)simplicial object can be found in Tierney and Vogel (1969), Simplicial resolutions and derived functors (MR, DOI 10.1007/BF01110914):
Definition. For a projective class $\mathcal{P}$, an object $A$ and a (semi)simplicial object $X_\bullet$, we say $\partial^0 : X_0 \to A$ is $\mathcal{P}$-exact if $\partial^0$ is a $\mathcal{P}$-epimorphism and, for every $n$, the comparison morphism $X_{n+1} \to K_{n+1}$ is a $\mathcal{P}$-epimorphism, where $K_{n+1}$ is the simplicial kernel of $X_n \mathrel{\mbox{$\begin{matrix} \smash{\to} \newline \smash{\scriptstyle\vdots} \newline \smash{\to} \end{matrix}$}} X_{n-1}$ i.e. we have morphisms $k^{n+1}_{0}, \ldots, k^{n+1}_{n+1} : K_{n+1} \to X_n$ which are universal with respect to the property that $\partial^n_i \circ k^{n+1}_i = \partial^n_i \circ k^{n+1}_{i+1}$ (Recall that, by the simplicial identities, $\partial^n_i \circ \partial^{n+1}_i = \partial^n_i \circ \partial^{n+1}_{i+1}$, so there is a comparison morphism $X_{n+1} \to K_{n+1}$ by universality.)
In particular, in a regular category, if $\mathcal{P}$ is the class of regular projectives, and $X_0 \to A$ is $\mathcal{P}$-exact, we have a kernel pair diagram $K_1 \rightrightarrows X_0 \to A$ which is also a coequaliser diagram (by unique image factorisation), so we have an exact fork.