# Exact sequences with parallel arrows

In Milne's Étale Cohomology (both the book and the online notes), he sometimes says a diagram of the form

$$0 \rightarrow A \rightarrow B \rightrightarrows C$$

is an exact sequence if $A \to B$ is the equaliser of $B \rightrightarrows C$. My lecturer also said the same thing, so I suspect this standard usage at least in some circles. I'm wondering if this is ad-hoc, or if this is a special case of a more general definition of exact sequence I'm unaware of; for instance, what might an exact sequence of the form

$$A \rightarrow B \rightrightarrows C \mathrel{\hbox{\begin{matrix} \smash{\to} \newline \smash{\to} \newline \smash{\to} \end{matrix}}} D$$

actually be?

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I've seen the dual situation in Tim van der Linden's thesis and his subsequent work, where the dual diagram of $A \rightrightarrows B \to C$ is called a fork (see page 29). Exactness notions are considered in non-additive settings for developing "nonabelian" homology theories. – t.b. Dec 2 '11 at 20:31

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.