equivalences in/of -categories
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 5.5.6.13.
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.
This is partially ordered under inclusion.
If is a presentable (∞,1)-category, then 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 is a monomorphism, then so is .
This is a special case of the general statement that -truncated morphisms are stable under pullback. (HTT, remark 5.5.6.12).
In an (∞,1)-topos, monomorphisms are stable under (∞,1)-pushout: if
is a homotopy pushout diagram and is a monomorphism, then so is .
The equivalence class of a monomorphism is a subobject in an (∞,1)-category.
The notion of monomorphism in an -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.
Lecture notes:
For model categories, see
Last revised on June 27, 2020 at 07:09:21. See the history of this page for a list of all contributions to it.