equivalences in/of $(\infty,1)$-categories
The notion of monomorphism in an $(\infty,1)$-category is the generalization of the notion of monomorphism from category theory to (∞,1)-category theory. It is the special case of the notion of n-monomorphisms for $n = 1$. In an (∞,1)-topos every morphism factors by an effective epimorphism (1-epimorphism) followed by a monomorphism through its 1-image.
The dual concept is that of an epimorphism in an (∞,1)-category.
There is also the concept regular monomorphism in an (∞,1)-category, but beware that this need not be a special case of the definition given here.
There are also a notions of (homotopy) monomorphism in model categories and derivators.
For $C$ an (∞,1)-category, a morphism $f : Y \to Z$ is a monomorphism if regarded as an object in the (∞,1)-overcategory $X_{/Z}$ it is a (-1)-truncated object.
Equivalently this means that the projection
is a full and faithful (∞,1)-functor. This is in Higher Topos Theory after Example 5..5.6.13.
Equivalently this means that for every object $X \in C$ the induced morphism
of ∞-groupoids is such that its image in the homotopy category exhibits $C(X,Y)$ as a direct summand in a coproduct decomposition of $C(X,Z)$.
So if $C(X,Y) = \coprod_i C(X,Y)_{i \in \pi_0(C(X,Y))}$ and $C(X,Z) = \coprod_{j \in \pi_0((C(X,Z))} C(X,Z)_j$ is the decomposition into connected components, then there is an injective function
such that $C(X,f)$ is given by component maps $C(X,Y)_i \to C(X,Z)_{j(i)}$ which are each an equivalence.
For $Z$ an object of $C$, write $Sub(Z)$
for the category of subobjects of $C$.
This is partially ordered under inclusion.
If $C$ is a presentable (∞,1)-category, then $Sub(Z)$ is a small category.
This appears as HTT, prop. 6.2.1.4.
Monomorphisms are stable under (∞,1)-pullback: if
is a pullback diagram and $f$ is a monomorphism, then so is $f'$.
This is a special case of the general statement that $k$-truncated morphisms are stable under pullback. (HTT, remark 5.5.6.12).
The equivalence class of a monomorphism is a subobject in an (∞,1)-category.
The notion of monomorphism in an $(\infty,1)$-category can also be characterized in its underlying homotopy derivator; see monomorphism in a derivator.
The definition appears after example 5.5.6.13 in
with further discussion in section 6.2.
For model categories, see