# On homotopically inverting maps in DG categories

In a DG category $\mathcal{C}$, call a (closed degree 0) morphism $f: X \to Y$ a homotopy equivalence, if there exists $g: Y \to X$ such that $gf-1_X$ and $fg-1_Y$ are exact.

For a full DG subcategory $\mathcal{D} \subset \mathcal{C}$, recall the explicit construction of Drinfeld quotient $\mathcal{C}/\mathcal{D}$: the objects are that of $\mathcal{C}$ and the morphisms are freely generated by the morphisms of $\mathcal{C}$ and, for every $X \in \mathcal{D}$, by $b_X: X \to X$ of degree $-1$ with $d(b_X)=1_X$.

Now consider the following procedure of "homotopically inverting" a chosen (closed degree 0) morphism $f$ in $\mathcal{C}$. Take the Joneda embedding $J: \mathcal{C} \to \mathcal{C}\mathrm{Mod} = \mathrm{DGFun}(\mathcal{C}^{\mathrm{Op}},Ch_k)$; in $\mathcal{C}\mathrm{Mod}$ there exists $\mathrm{Cone}(Jf)$, so pass to $\mathcal{C}\mathrm{Mod}/\mathrm{Cone}(Jf)$ and consider there a full subcategory on the objects of $J\mathcal{C}$. Explicitly, this means freely adding to $\mathcal{C}$ the maps $f':Y \to X$ of degree 0 with $d(f')=0$, $r_X: X \to X$ of degree $-1$ with $d(r_X)=f'f-1_X$, $r_Y: Y \to Y$ of degree $-1$ with $d(r_Y)=ff'-1_Y$ and $r_{XY}: X \to Y$ of degree $-2$ with $d(r_{XY})=fr_X-r_Yf$. Denote the result by $L_H(\mathcal{C},f)$. (For example, applying this to the single arrow in the category $\bullet \to \bullet$ gains a category quasiequivalent to a point which came up in Tabuada's construction of the model structure for DG categories.)

Now, for an arbitrary DG category $\mathcal{A}$, compare the following DG categories:

• a full subcategory in $\mathrm{DGFun}(\mathcal{C},\mathcal{A})$ consisting of DG functors that take $f$ into a homotopy equivalence in $\mathcal{A}$;
• $\mathrm{DGFun}(L_H(\mathcal{C},f),\mathcal{A})$.

These two categories are not the same but my expectation is that they are queasiequivalent.

My question is: are they?

• The answer is yes. Key words to look for such notions in the literature are: Drinfeld quotient. – Denis-Charles Cisinski Apr 30 '18 at 0:15
• @Denis-CharlesCisinski, wouldn't there be a problem because DGFun doesn't give RHom for DG categories? I've been discussing this question outside MO and people told me that I should replace DGFun with some model for RHom, e.g. with $A_\infty Fun$ (i.e., with $A_\infty$ natural transformations between them)... – Dasha Poliakova May 2 '18 at 16:59
• You are right. I overlooked that you were not considering RHom. However, the comparison map from DGFun(L_H(C,f),A) to DGFun(C,A) is always fully faithful in the sense that it is a DG functor which induces a quasi-isomorphism on Hom’s (embed A in the DG category A’ of cofibrant A-modules (with respect to the projective model structure and prove it replacing A by A’). The essential surjectivity on the DG functors quasi-inverting f does not seem reasonable without further assumptions on A or C (like C cofibrant, or A of the form B’ for some B). – Denis-Charles Cisinski May 2 '18 at 20:34