# 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?

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.