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

a b c d \array{a & \to & b \\ \downarrow & & \downarrow \\ c & \to & d}

that is the “free-living commutative square”, let II be the interval category (01)(0\to 1), and let p:Ip\colon \square \to I denote the functor collapsing a,b,ca,b,c to 00 and sending dd to 11.

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

Examples

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

    A id A id f A f B \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 (,1)(\infty,1)-category, one can check that this reduces to the usual notion of monomorphism in an (∞,1)-category.

Created on June 12, 2010 04:44:22 by Mike Shulman (75.3.130.212)