The notion of monomorphism in an -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 . 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 an (∞,1)-category, a morphism is a monomorphism if regarded as an object in the (∞,1)-overcategory 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 18.104.22.168.
Equivalently this means that for every object the induced morphism
of ∞-groupoids is such that its image in the homotopy category exhibits as a direct summand in a coproduct decomposition of .
So if and is the decomposition into connected components, then there is an injective function
such that is given by component maps which are each an equivalence.
For an object of , write
for the category of subobjects of .
This is partially ordered under inclusion.
This appears as HTT, prop. 22.214.171.124.
Monomorphisms are stable under (∞,1)-pullback: if
is a pullback diagram and is a monomorphism, then so is .
This is a special case of the general statement that -truncated morphisms are stable under pullback. (HTT, remark 126.96.36.199).
The definition appears after example 188.8.131.52 in
with further discussion in section 6.2.
For model categories, see