monomorphism in an (infinity,1)-category




The notion of monomorphism in an (,1)(\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=1n = 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 CC an (∞,1)-category, a morphism f:YZf : Y \to Z is a monomorphism if regarded as an object in the (∞,1)-overcategory C /ZC_{/Z} it is a (-1)-truncated object.

Equivalently this means that the projection

C /fC /Z C_{/f} \to C_{/Z}

is a full and faithful (∞,1)-functor. This is in Higher Topos Theory after Example

Equivalently this means that for every object XCX \in C the induced morphism

C(X,f):C(X,Y)C(X,Z) C(X,f) : C(X,Y) \to C(X,Z)

of ∞-groupoids is such that its image in the homotopy category exhibits C(X,Y)C(X,Y) as a direct summand in a coproduct decomposition of C(X,Z)C(X,Z).

So if C(X,Y)= iC(X,Y) iπ 0(C(X,Y))C(X,Y) = \coprod_i C(X,Y)_{i \in \pi_0(C(X,Y))} and C(X,Z)= jπ 0((C(X,Z))C(X,Z) jC(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

j:π 0(C(X,Y))π 0(C(X,Z)) j : \pi_0(C(X,Y)) \to \pi_0(C(X,Z))

such that C(X,f)C(X,f) is given by component maps C(X,Y) iC(X,Z) j(i)C(X,Y)_i \to C(X,Z)_{j(i)} which are each an equivalence.



For ZZ an object of CC, write Sub(Z)Sub(Z)

Sub(Z)τ 1C /Z. Sub(Z) \simeq \tau_{\leq -1} C_{/Z} \,.

for the category of subobjects of CC.

This is partially ordered under inclusion.


If CC is a presentable (∞,1)-category, then Sub(Z)Sub(Z) is a small category.

This appears as HTT, prop.


Monomorphisms are stable under (∞,1)-pullback: if

A B f f C D \array{ A &\to& B \\ {}^{\mathllap{f'}}\big\downarrow && \big\downarrow^{\mathrlap{f}} \\ C &\to& D }

is a pullback diagram and ff is a monomorphism, then so is ff'.

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


In an (∞,1)-topos, monomorphisms are stable under (∞,1)-pushout: if

A B f f C D \array{ A &\to& B \\ {}^{\mathllap{f}}\big\downarrow && \big\downarrow^{\mathrlap{f'}} \\ C &\to& D }

is a homotopy pushout diagram and ff is a monomorphism, then so is ff'.

(Rezk 19, p. 21)


The definition appears after example 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.