nLab 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 𝒞\mathcal{C} an (∞,1)-category, a morphism f:YZf \colon Y \to Z is a monomorphism if regarded as an object in the (∞,1)-overcategory 𝒞 /Z\mathcal{C}_{/Z} it is a (-1)-truncated object.

Equivalently this means that the projection

𝒞 /f𝒞 /Z \mathcal{C}_{/f} \longrightarrow \mathcal{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) \mathcal{C}(X,f) \;\colon\; \mathcal{C}(X,Y) \longrightarrow \mathcal{C}(X,Z)

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

So if

𝒞(X,Y)=iπ 0𝒞(X,Y)𝒞(X,Y) iand𝒞(X,Z)=jπ 0(𝒞(X,Z)𝒞(X,Z) j \mathcal{C}(X,Y) \;=\; \underset{i \in \pi_0\mathcal{C}(X,Y)}{\coprod} \mathcal{C}(X,Y)_{i } \;\;\;\;\;\; \text{and} \;\;\;\;\;\; \mathcal{C}(X,Z) \;=\; \underset{j \in \pi_0(\mathcal{C}(X,Z)}{\coprod} \mathcal{C}(X,Z)_j

is the decomposition into connected components, then there is an injective function

j:π 0𝒞(X,Y)π 0𝒞(X,Z) j \,\colon\, \pi_0 \mathcal{C}(X,Y) \longrightarrow \pi_0 \mathcal{C}(X,Z)

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



For ZZ an object of 𝒞\mathcal{C}, write Sub(Z)Sub(Z)

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

for the category of subobjects of 𝒞\mathcal{C}.

This is partially ordered under inclusion.


If 𝒞\mathcal{C} 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

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