Homotopy pullback in the category of DG algebras

Question

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?

Answer 1

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.