# nLab n-monomorphism

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Contents

## Definition

For $n \in \mathbb{N}$ a morphism $f \colon X \to Y$ in an (infinity,1)-category is an $n$-monomorphism equivalently if

Similarly a function $f \colon X \to Y$ in homotopy type theory is an $n$-monomorphism if its $n$-image factorization is via an equivalence in homotopy type theory.

The dual concept is that of n-epimorphism.

## Examples

• $0$-monomorphism are precisely the equivalences.

• Every morphism is an $\infty$-monomorphism.

• 1-monomorphisms are often just called monomorphisms in an (∞,1)-category. The 1-monomorphisms into a fixed object are called the subobjects of that object.

• A 1-monomorphism between 0-truncated objects is precisely an ordinary monomorphism in the underlying 1-topos.

Last revised on October 28, 2016 at 18:00:13. See the history of this page for a list of all contributions to it.