nLab
monomorphism in a derivator

Idea

A monomorphism in a derivator is the generalization to the context of a derivator of the notion of monomorphism in ordinary category theory. Viewing a derivator as the “shadow” of an (∞,1)-category, the notion of monomorphism therein coincides with the notion of monomorphism in an (∞,1)-category.

Definition

Let $\square$ denote the category

$$\begin{array}{ccc}a& \to & b\\ \downarrow & & \downarrow \\ c& \to & d\end{array}$$\array{a & \to & b \\ \downarrow & & \downarrow \\ c & \to & d}

that is the “free-living commutative square”, let $I$ be the interval category $(0\to 1)$, and let $p:\square \to I$ denote the functor collapsing $a,b,c$ to $0$ and sending $d$ to $1$.

Let $D$ be a prederivator and $f:X\to Y$ a morphism in $D(1)$. By one of the axioms of a derivator, there exists an object $F\in D(I)$ representing $f$, which is unique up to non-unique isomorphism. We say that $f$ is a monomorphism in $D$ if $p^{*}(F)\in D(\square )$ is a pullback square.

Examples

• It is well-known and easy to verify that a morphism $f:A\to B$ in a 1-category is a monomorphism if and only if the square

  $$\begin{array}{ccc}A& \stackrel{\mathrm{id}}{\to }& A\\ {}^{\mathrm{id}}\downarrow & & {\downarrow }^f\\ A& \underset{f}{\to }& B\end{array}$$\array{ A & \overset{id}{\to} & A \\ ^{id}\downarrow & & \downarrow^f \\ A &\underset{f}{\to} & B }

  is a pullback. Therefore, in representable prederivators this definition reduces to the usual notion of monomorphism.

• In the homotopy derivator of an $(\mathrm{\infty },1)$-category, one can check that this reduces to the usual notion of monomorphism in an (∞,1)-category.