# A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category

I employ the vast majority of the post to develop the notion of quasi-functor between dg-categories: I think it is important to get the idea.

Let $k$ be a field, and let $\mathcal V =\mathbf C(k)$ be the category of cochain complexes of $k$-vector spaces and chain maps. $\mathcal V$ is a symmetric closed monoidal category, and also has a structure of model category (the weak equivalences are the quasi-isomorphisms, the fibrations are the surjective chain maps). Categories enriched over $\mathcal V$ are called dg-categories. $\mathcal V$ is also enriched over itself.

Let $\mathcal A, \mathcal B$ be categories enriched over $\mathcal V$. The category $\mathrm{Fun}_{\mathcal V} (\mathcal A, \mathcal B)$ of $\mathcal V$-functors and $\mathcal V$-natural transformations is also enriched over $\mathcal V$. The (enriched) Yoneda lemma holds, and we have the (fully faithful) Yoneda embedding: \begin{equation} \mathcal A \hookrightarrow \textrm{mod-}\mathcal A =: \mathrm{Fun}_{\mathcal V}(\mathcal A^{\mathrm{op}},\mathcal V). \end{equation}

The category $\textrm{mod-}\mathcal A$ of (right) $\mathcal A$-modules inherits a (levelwise) model structure from $\mathcal V$: for example, quasi-equivalences are given by levelwise quasi-equivalences.

Now, let me give the following definition: given $\mathcal A, \mathcal B$ two $\mathcal V$-categories, a $\mathcal V$-functor $F: \mathcal A \to \textrm{mod-}\mathcal B$ is called a quasi-functor if for any $A \in \mathcal A$, $F(A)$ is isomorphic to a representable right $\mathcal B$-module in the homotopy category $\mathrm{Ho}(\text{mod-}\mathcal B)$.

Now, observe that we can define a "tensor product" $\mathcal A \otimes \mathcal B$ of $\mathcal V$-categories. Moreover, $\mathcal V$-functors $\mathcal A \otimes \mathcal B^{\mathrm{op}} \to \mathcal V$ correspond exactly to $\mathcal V$-functors $\mathcal A \to \textrm{mod-}\mathcal B$. They are precisely the bimodules (or profunctors). Let me denote by $\mathcal A\text{-mod-}\mathcal B$ the $\mathcal V$-category of bimodules (covariant in $\mathcal A$, contravariant in $\mathcal B$). Since it is a category of modules, it is a model category, with the levelwise structure discussed above.

Now, I can define $\mathrm{rep}(\mathcal A,\mathcal B)$ as the full subcategory of $\mathrm{Ho}(\mathcal A\text{-mod-}\mathcal B)$ whose objects are the quasi-functors. Moreover, I can define $\mathrm{rep}_{\mathcal V}(\mathcal A,\mathcal B)$ as the full $\mathcal V$-subcategory of $\mathcal A\text{-mod-}\mathcal B$ whose objects are the quasi-functors which are also cofibrant as bimodules.

Where can we go from here? Well, the category $\mathcal V$-$\mathbf{Cat}$ of categories enriched over $\mathcal V$ has itself a model structure: it is a known result by G. Tabuada about dg-categories, more recently generalized. The homotopy category $\mathrm{Ho}(\mathcal V\text{-}\mathbf{Cat})$ is monoidal, and $\mathrm{rep}_{\mathcal V}(\mathcal A,\mathcal B)$ gives the internal hom. In the world of dg-categories, $\mathrm{rep}(\mathcal A,\mathcal B)$ is (equivalent to) $H^0(\mathrm{rep}_{\mathcal V}(\mathcal A,\mathcal B))$.

Finally, here is my question: generalize from this particular case $\mathcal V = \mathbf C(k)$ to a more general setting, possibly letting $\mathcal V$ be a monoidal model category (with some assumptions). Every definition given above should work without problems. Has someone developed a "general theory" of those quasi-functors? I've studied it in the case of dg-categories, but I guess it is just a particular case.

I think the right way to call this functors is "quasi-representable" functors (in the language of Toen) and "potentially distinguished" functors in the language of Dwyer-Hess. At least when $\mathbf{V}$ is one of these model categories 1) simplicial sets ($\mathbf{sSet}$), (compactly generated) topological space ($\mathbf{Top}$), symmetric spectra ($\mathbf{Sp}^{\Sigma}$) you can almost copy-paste Toen's argument to construct 1) the (derived) mapping space (using the moduli space of quasi representable functors) 2) the derived internal Hom of the model category $\mathbf{V}-Cat$.
• For all categories $\mathbf{V}$ (cited in my answer), the category $\mathbf{V}-Cat$ is symmetric closed monoidal category and a model category BUT the Monoidal closed structure and the model structure are NOT compatible in the sense of Hovey. What you can prove is that the homotopy category $Ho(\mathbf{V}-Cat)$ is a symmetric monodical closed category where the (derived) internal Hom is described in terms of $Rep_{\mathbf{V}}(A,B)$. It is a difficult theorem done by Toen. Aug 9 '14 at 10:05
• What I was saying in my answer is that in the case where $\mathbf{V}$ is the category of simplicial sets, topological spaces or Spectra you can repeat Toen's arguments i.e., the category $Ho(\mathbf{V}-Cat)$ is a symmetric monoidal closed category, where the internal Hom is given by $Rep_{\mathbf{V}}(-,-)$ and the mapping space is the moduli space of (right) quasi-representable functors. Aug 9 '14 at 10:08
• In the appendix of Higher Topos Theory there is a relatively easy construction of the internal hom (it follows from Lemma A.3.4.6 and Corollary A.3.4.14). On the other side, Toen does something much more difficult: he computes the mapping spaces. I am not aware of a generalization of this last result to $V \textrm{-} \mathbf{Cat}$. Aug 9 '14 at 10:56
In some sense, the "universal" version of this fact was proved by Blumberg-Gepner-Tabuada as Proposition 3.3 in this paper. That is, they proved the analogue for stable $\infty$-categories, which is the description of the $\infty$-category $Fun^{ex}(A, B)$, the internal hom in the $\infty$-category of idempotent complete stable $\infty$-categories and exact functors, via what they call "right-compact" $A$-$B$-bimodules, which are the analogue of right quasi-representable functors or quasi-functors. In fact, the proof is much nicer in this setting.
I claim that any "reasonable" version of Toën's theorem should follow from this version. By that I mean for any $\mathcal{V}$ such that $\mathcal{V}$-categories model some version of stable $\infty$-categories (so that my claim is basically tautological). For example, for spectral categories ($\mathcal{V}$-categories where $\mathcal{V}$ is the symmetric monoidal model category of symmetric spectra), this follows from the equivalence in Theorem 1.10 between idempotent complete stable $\infty$-categories, and spectral categories up to Morita equivalence. For dg-categories, it should follow similarly by using the equivalence between dg-categories and linear stable $\infty$-categories. (Note that the internal hom in the $\infty$-category of dg-categories is the same as the derived internal hom in the model category of dg-categories.)