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?



Homotopy pullback in the category of DG algebras

> 
> I was wondering if somebody could tell me the definition of homotopy pullback.
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> 


Suppose $\def\C{{\cal C}}\C$ is a [relative category](https://ncatlab.org/nlab/show/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](https://ncatlab.org/nlab/show/hammock+localization) $H(\C)$ of $\C$,
which is a [simplicial category](https://ncatlab.org/nlab/show/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?
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It satisfies the universal property given above, which is completely
analogous to the usual universal property of pullbacks.


