Background
Basic concepts
equivalences in/of $(\infty,1)$-categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
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
the first morphism is an equivalence in an (infinity,1)-category
it is an (n-2)-truncated morphism;
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 $n$.
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
The dual concept is that of n-epimorphism.
$0$-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 November 18, 2023 at 05:10:13. See the history of this page for a list of all contributions to it.