Homotopy pullback in the category of DG algebras I was wondering if somebody could tell me the definition of homotopy pullback. More precisely, is there a description of the homotopy pullback in the category of (graded-commutative) DG algebras over a commutative ring $R$? What properties does this homotopy pullback satisfy?
 A: 
I was wondering if somebody could tell me the definition of homotopy pullback.

Suppose $\def\C{{\cal C}}\C$ is a relative category, i.e., a category equipped with a subcategory,
morphisms in which are known as weak equivalences.
For example, take the category of dg algebras with quasi-isomorphisms.
We can take the Dwyer–Kan hammock localization $H(\C)$ of $\C$,
which is a simplicial category, i.e., a category enriched in simplicial sets.
We have a canonical functor $\C→H(\C)$ (of ordinary, nonenriched categories),
which allows us to convert diagrams in $\C$ into diagrams in $H(\C)$.
This functor is given by the identity map on objects.
The homotopy pullback of a diagram $A→B←C$ in $H(\C)$ is a pair of maps
$D→A$, $D→C$ that make the square with vertices $A$, $B$, $C$, $D$ commutative
and such that for any object $E$ of the category $\C$
the induced map $$\def\Hom{\mathop{\rm Hom}} \Hom(E,D)→\Hom(E,A)⨯^h_{\Hom(E,B)}\Hom(E,C)$$
is a weak equivalence of simplicial sets.
Here $\Hom$ denotes the simplicially enriched hom functor
and $⨯^h$ denotes the homotopy pullback of a span of simplicial sets,
which can be computed using an ad hoc construction:
to compute $P⨯_Q R$,
first make $Q$ into a Kan complex by composing the maps $P→Q$ and $R→Q$ with the fibrant replacement $Q→{\rm Ex}^∞ Q$, then compute (assuming $Q$ is a Kan complex now)
$$P⨯^h_Q R = P ⨯_Q Q^{Δ^1} ⨯_Q R.$$

More precisely, is there a description of the homotopy pullback in the category of (graded-commutative) DG algebras over a commutative ring R?

In the projective model structure on dg algebras over a commutative ring $R$,
all objects are fibrant.
Thus, the relevant theory of model categories implies that the homotopy pullback of a diagram $S→T←U$ of dg algebras can be computed
in two different ways:
either factor one of the maps (say, $S→T$) as an acyclic cofibration
$S→S'$ followed by a fibration $S'→T$, then take
the ordinary pullback $S' ⨯_T U$.
Alternatively, one can simply take $S ⨯^h_T U = S ⨯_T T^I ⨯_T U$,
where $T^I$ denotes the powering of $T$ over a (chain) interval $I$,
i.e., the simplicial chain complex of $Δ^1$, given by $R⊕R$ in degree 0
and $R$ in degree 1.

What properties does this homotopy pullback satisfy?

It satisfies the universal property given above, which is completely
analogous to the usual universal property of pullbacks.
