# Connection between quasifibrations and homotopy cartesian squares

Let me first fix the definitions.

A map $p\colon E\rightarrow B$ is called a quasi-fibration, iff the canonical inclusion $p^{-1}(b)\rightarrow hofib_b(p)$ is a weak equivalence for all for all $b\in B$.

Consider a commutative square $$\begin{array}{ccc} Y & \to & E \cr\downarrow&&\downarrow \cr X& \to &B \end{array}.$$ Replace $E\rightarrow B$ by a fibration $E'\rightarrow B$ via the usual construction and pull back the fibration to a fibration over $X$. We then obtain a canonical map from $Y$ to the pullback. I call the diagram homotopy cartesian iff this map is a weak equivalence.

Obviously a map $p\colon E\rightarrow B$ is a quasi-fibration, iff the square $$\begin{array}{ccc} p^{-1}(\{b\}) & \to & E \cr\downarrow&&\downarrow \cr \{b\}& \to &B \end{array}$$ is homotopy cartesian for all $b\in B$.

Can one characterize homotopy cartesian squares in terms of quasi-fibrations to obtain a statement of the shape "The square is cartesian iff a map constructed out of the square is a quasifibration"?

• Are you looking for something like this fibre-wise characterization: ncatlab.org/nlab/show/…? – AAK Jan 23 '15 at 23:59
• No, I'm aware of the fiberwise characterization. I'd rather would like to have a statement of the shape I proposed. – Jennifer Jan 24 '15 at 16:56