nLab n-monomorphism

Definition

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

• in its n-image factorization

  $$f \colon X \to im_n(f) \to Y$$

  the first morphism is an equivalence in an (infinity,1)-category

• it is an (n-2)-truncated morphism;

In homotopy type theory

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

More explicitly, given a natural number $n:\mathbb{N}$ and types $A$ and $B$, a function $f:A \to B$ is a $n$-monomorphism if for all terms $b:B$ the fiber of $f$ over $b$ has an homotopy level of $n$.

$$isMonic(n, f) \coloneqq \prod_{b:B} hasHLevel(n, fiber(f, b))$$

A equivalence is a $0$-monomorphism. $1$-monomorphisms are typically just called monomorphisms or embeddings.

The type of all $n$-monomorphisms with domain $A$ and codomain $B$ is defined as

$$Monos(n, A, B): \sum_{f:A \to B} isMonic(n, f)$$

Dual concept

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.

Related concepts

• Postnikov system

• (n-connected, n-truncated) factorization system

• subtype