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'}}\downarrow && \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


The definition appears after example in

with further discussion in section 6.2.

For model categories, see

Last revised on May 7, 2016 at 02:59:19. See the history of this page for a list of all contributions to it.