# nLab sharp map

Contents

This entry is about the concept related to homotopy pullbacks. For a different concept of the same name see at sharp modality.

# Contents

## Idea

In a right proper model category a morphism is called sharp if its pullback along any other morphism is already a homotopy pullback. In a general model category a morphism is sharp if all its pullbacks preserve weak equivalences under (further) pullback.

Other terms used for sharp morphisms include right proper morphism, $h$-fibration, and $W$-fibration; see the references.

## Definition

In a model category $\mathcal{M}$, a sharp map is a morphism $p : X \to Y$ satisfying the following condition: for any commutative diagram in $\mathcal{M}$ of the form below,

$\array{ X'' & \stackrel{f}{\longrightarrow} & X' & \longrightarrow & X \\ \downarrow && \downarrow && \downarrow^{\mathrlap{p}} \\ Y'' & \stackrel{g}{\longrightarrow} & Y' & \longrightarrow & Y }$

if $g \colon Y'' \to Y'$ is a weak equivalence and both squares are pullback diagrams, then $f \colon X'' \to X'$ is also a weak equivalence.

## Properties

A model category is right proper if and only if every fibration is sharp. (Rezk 98, prop. 2.2)

In a right proper model category, the sharp maps in the full subcategory on sharp-fibrant objects form the fibrations of a category of fibrant objects. See there the section Examples – Right proper model categories.

## References

The concept was introduced in

The terminology arises by dualization of “flat morphism” which was used by Hopkins for the dual concept, which is presumably motivated by the fact that a ring homomorphism is flat if tensoring with it is exact, hence preserves weak equivalences of chain complexes.

The notion was rediscovered and renamed by various other authors. In

• Andrei Radulescu-Banu?, Cofibrations in Homotopy Theory, arxiv, 2006

it was called a “right proper morphism” (with focus on the dual notion of “left proper morphism”), presumably due to the connection with right proper model categories. In

sharp maps were renamed “weak fibrations”. The authors of

chose instead to rename them to “fibrillations”, because it sounds more like “fibration”. Whereas the authors of

chose to rename the dual notion to “$h$-cofibrations”, with reference to the use of that term for the related — but nevertheless distinct — notion of Hurewicz cofibration by Peter May and collaborators such as Johann Sigurdsson? and Kate Ponto. In

the dual notion was called a “$W$-cofibration”, where $W$ is the relevant class of weak equivalences (note that the definition only depends on the weak equivalences, not the whole model structure); apparently this terminology dates back to unpublished work of Grothendieck.