nLab sharp map

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

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.

Related concepts

• The dual notion is (most commonly) known as “h-cofibration”.

References

The concept was introduced in

• Charles Rezk, Fibrations and homotopy colimits of simplicial sheaves (arXiv:9811038), 1998

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

• Denis-Charles Cisinski, Invariance de la K-théorie par équivalences dérivées, 2010

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

• Clark Barwick and Daniel Kan, Quillen Theorems Bn for homotopy pullbacks of (infinity, k)-categories, arxiv, 2012

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

• Michael Batanin and Clemens Berger, Homotopy theory for algebras over polynomial monads, arxiv, 2013

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

• Dimitri Ara and Georges Maltsiniotis, Towards a Thomason model structure on the category of strict $n$-categories, arxiv, 2013

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.

Some discussion about these various names was had at

• Mike Shulman, Pullbacks That Preserve Weak Equivalences, blog post