nLab
monomorphism in an (infinity,1)-category

Contents

Idea

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.

Definition

For CC an (∞,1)-category, a morphism f:YZf : Y \to Z is a monomorphism if regarded as an object in the (∞,1)-overcategory X /ZX_{/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 5..5.6.13.

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.

Properties

Definition

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.

Proposition

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

This appears as HTT, prop. 6.2.1.4.

Proposition

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 5.5.6.12).

References

The definition appears after example 5.5.6.13 in

with further discussion in section 6.2.

For model categories, see

Revised on May 13, 2015 19:15:18 by Adeel Khan (93.131.11.231)