sharp map

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

sharp modality.

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.

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.

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*.

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

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

Last revised on January 26, 2019 at 03:38:03. See the history of this page for a list of all contributions to it.