Contents

Idea

The notion of monomorphism in an $(\mathrm{\infty },1)$-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.

Definition

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.

Properties

This is partially ordered under inclusion.

This appears as HTT, prop. 6.2.1.4.

This is a special case of the general statement that $k$-truncated morphisms are stable under pullback. (HTT, remark 5.5.6.12).

Related pages

• The equivalence class of a monomorphism is a subobject in an (∞,1)-category.

• The notion of monomorphism in an $(\mathrm{\infty },1)$-category can also be characterized in its underlying homotopy derivator; see monomorphism in a derivator.

References

The definition appears after example 5.5.6.13 in

• Jacob Lurie, Higher Topos Theory

with further discussion in section 6.2.