# Why do the model structures on dg-algebras and on dg-categories are not compatible?

First we talk about dg-algebras. According to this n-lab page, we write $dgAlg$ for the category of cochain dg-algebras in non-negative degree over a field $k$ of characteristic $0$. Write $CdgAlg\subset dgAlg$ for the subcategory of (graded-)commutative dg-algebras. Then The projective model category structure on $CdgAlg$ and on $dgAlg$ is given by setting:

• weak equivalences are the quasi-isomorphisms
• fibrations are the degreewise surjections.

Then we look at dg-categories. According to this n-lab page, we write $dgCat$ for the category of small dg-categories over $k$. Then the Dwyer-Kan model structure on $dgCat$ is given by setting:

• a dg-functor $F:A\to B$ is a weak equivalence if $(1)$ for all objects $x,y\in A$ the component $F_{x,y}:A(x,y)\to B(F(x),F(y))$ is a quasi-isomorphism of chain complexes $(2)$ the induced functor on homotopy categories $H^0(F)$ (obtained by taking degree 0 chain homology in each hom-object) is an equivalence of categories.
• a dg-functor $F:A\to B$ is a fibration if $(1)$ for all objects $x,y\in A$ the component $F_{x,y}$ is a degreewise surjection of chain complexes; $(2)$ for each isomorphism $F(x)\to Z$ in $H^0(B)$ there is a lift to an isomorphism in $H^0(A)$.

Now we compare this two definitions. A dg-algebra can be considered as a dg-category which consists of one object hence we have a inclusion $dgAlg\subset dgCat$. However, although very similar at the first glance, the two model structures are not compatible. In particular there are dg-maps $F: A\to B$ between dg-algebras which is a fibration in the projective model structure on $dgAlg$ but not a fibration in the Dwyer-Kan model structure on $dgCat$.

For example let $A$ be the de Rham algebra of the closed interval $[0,1]$ and $B$ be the de Rham algebra of the two end points ${0,1}$ (hence $B$ is concentrated in degree $0$). Let $F: A\to B$ be the restriction map. It is clear that $F$ is degreewise surjective hence $F$ is a fibration in $dgAlg$. However if we take $f\in B$ to be $f(0)=1$, $f(1)=-1$ then $f$ is invertible in $H^0(B)$ but $f$ cannot be a restriction of a closed element in $A^0$ hence $F: A\to B$ is not a fibration when considered as a morphism in $dgCat$.

$\textbf{My question}$ is: what is the reason and consequence of this incompatibility? Does it mean that we can construct a new model structure on $dgAlg$ by restricting the Dwyer-Kan model structure to $dgAlg$?

• At least they have the same weak equivalences. So in addition to your question --- does dgAlg have a model structure with the standard weak equivalences but the Dwyer--Kan fibrations --- one can ask whether dgCat has a model structure with its standard weak equivalences but the projective fibrations. Note that if the answers are "yes", then the identity functor will be a Quillen equivalence, which is often good enough. Jul 31, 2015 at 17:18
• A minor comment is that to obtain a model structure on dg-algebras you should most likely either restrict to connected algebras, or change the definition of a fibration to a surjection in positive degrees. Jul 31, 2015 at 17:20

I will try to give a example in topology and then in dg-world. Suppose that $G$ and $H$ are topological groups. There is a fiber sequence $$Map_{\ast}(BG,BH)\rightarrow Map(BG,BH)\rightarrow BH$$ where $B$ is the classifying space. Here is a homotopy categorical interpretation of the previous fiber sequence. $$Map^{\otimes}(G,H)\rightarrow Map_{top-cat}(\mathbf{G},\mathbf{H})\rightarrow Map_{Top-cat}(\ast, \mathbf{H})$$ where $Map^{\otimes}$ is the derived mapping space in the model category of topological mono ids. $Map_{Top-cat}$ is the derived mapping space in the model category of small topological categories. The notation $\mathbf{H}$ is for the topological category with one object such that the endomorphism monoid is $H$.
So what happens in the dg-world. Well, there is a fiber sequence $$Map_{dg-algebras}(A,B)\rightarrow Map_{dg-cat}(\mathbf{A},\mathbf{B})\rightarrow Map_{dg-cat}(\mathbf{k}, \mathbf{B})$$
The notation $\mathbf{A}$ is for the $dg_{k}$-category with one object such that the endomorphism monoid is $A$. $k$ is the ground commutative ring. Now we see why $Map_{dg-algebras}(A,B)$ is different from
$Map_{dg-cat}(\mathbf{A},\mathbf{B})$.
• It's also worth to say that $Map_{dg-cat}(\mathbf{k},\mathbf{B})$ is the classifying space of the topological/simplicial group of derived units in $\mathbf{B}$, i.e. $Map_{dg-algebras}(\mathbf{k}[t^{\pm1}],\mathbf{B})$. Aug 28, 2015 at 15:47
• A small remark: the functor of $\infty$-categories from (dg-algebras) to (dg-categories) factors through a functor to the $\infty$-category of pointed dg-categories which is fully faithful. As you point out, the mapping spaces in (pointed dg-categories) are obtained from the mapping spaces in (dg-categories) by quotienting out by the action of the simplicial group of units.