nLab monomorphism in an (infinity,1)-category

Idea

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

There are also a notions of (homotopy) monomorphism in model categories and derivators.

Definition

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

Equivalently this means that the projection

$$\mathcal{C}_{/f} \longrightarrow \mathcal{C}_{/Z}$$

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 $X \in C$ the induced morphism

$$\mathcal{C}(X,f) \;\colon\; \mathcal{C}(X,Y) \longrightarrow \mathcal{C}(X,Z)$$

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

So if

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

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

$$j \,\colon\, \pi_0 \mathcal{C}(X,Y) \longrightarrow \pi_0 \mathcal{C}(X,Z)$$

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

Properties

Definition

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

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

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

This is partially ordered under inclusion.

Proposition

If $\mathcal{C}$ is a presentable (∞,1)-category, then $Sub(Z)$ is a small category.

This appears as HTT, prop. 6.2.1.4.

Proposition

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

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

is a pullback diagram and $f$ is a monomorphism, then so is $f'$.

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

Proposition

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

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

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

(Rezk 19, p. 21)

Related pages

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

• The notion of monomorphism in an $(\infty,1)$-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.

Lecture notes:

• Charles Rezk, p. 10 of: Lectures on Higher Topos Theory, Leeds 2019 (pdf, pdf)

For model categories, see

• Fernando Muro, Homotopy units in A-infinity algebras, arXiv:1111.2723.