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Let us consider dgCat, the "collection" of all small dg-categories. In On differential graded categories and Lectures on dg categories the authors state that they form a category, i.e. dgCat has small dg categories as objects and the dg functors as morphisms. They then give a model structure on dgCat.

However, as pointed in What do DG-categories form?, it is more likely that dgCat is a 2-category: it does not make too much sense to say whether two dg-functors $F$ and $G$ are equal. Instead we can talk about morphisms between dg-functors.

How to integrate these two viewpoint? In particular, could we still have a model category structure on dgCat if we treat it as a 2-category?

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    $\begingroup$ It's an $(\infty, 2)$-category. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 29, 2015 at 4:58
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    $\begingroup$ Model structures present $(\infty, 1)$-categories, but there should be more structure here than that. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 29, 2015 at 5:14
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    $\begingroup$ You're aware that 2-categories are also ordinary categories, right? $\endgroup$
    – Fernando Muro
    Commented Sep 29, 2015 at 6:26
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    $\begingroup$ @Fernando: well, this is a bit tricky. It's true that a strict 2-category has an underlying category given by forgetting 2-morphisms, but I prefer to use a convention where n-category always means the weak thing by default, so with that convention, if you just forget 2-morphisms composition might fail to be associative (and in any case this operation doesn't respect equivalences of 2-categories). The model-independent thing to do is to take the homotopy category, but using the word "are" for this construction seems like sweeping something under the rug. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 29, 2015 at 16:18
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    $\begingroup$ @QiaochuYuan the fact is that dgCat is what you call a strict 2-category, and hence has an underlying category. $\endgroup$
    – Fernando Muro
    Commented Sep 29, 2015 at 20:11

2 Answers 2

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The model structure on the category of dg-categories presents an $(\infty,1)$-category DGCat. This structure is essentially provided by the existence of mapping spaces (or mapping $\infty$-groupoids) between dg-categories.

To see why DGCat admits the further structure of an $(\infty,2)$-category, it is sufficient to see why these mapping $\infty$-groupoids can be refined to mapping $(\infty,1)$-categories, which give back our mapping $\infty$-groupoids when we pass to the sub-$\infty$-groupoids of invertible morphisms.

Such structure is provided by functor dg-categories, which are the internal hom objects in DGCat (in the "derived" sense -- these are denoted $R\underline{Hom}$ by Toen). These functor dg-categories give $(\infty,1)$-categories (by using for example the Dold-Kan correspondence), which provide the mapping $(\infty,1)$-categories in the $(\infty,2)$-category DGCat.

(Toen showed that these mapping $(\infty,1)$-categories can also be described as the $(\infty,1)$-categories of right quasi-representable bimodules.)

Edit: I just noticed that the preprint [Giovanni Faonte, $A_\infty$-functors and homotopy theory of dg-categories, arXiv:1412.1255] appears to contain a precise construction of the $(\infty,2)$-category of dg-categories.

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    $\begingroup$ Can we construct an "actual" $(\infty, 2)$-category, say a $\Theta_2$-space or a quasicategory-enriched category? After all, there are such things as sesquicategories which have hom-categories but otherwise fail to completely satisfy the 2-category axioms. $\endgroup$
    – Zhen Lin
    Commented Sep 29, 2015 at 7:47
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    $\begingroup$ @ZhenLin: Gepner-Haugseng show that the $\infty$-category of $V$-enriched $\infty$-categories has an enrichment over itself, for $V$ a presentably $E_2$-monoidal $\infty$-category. In particular this gives the enrichment of DGCat over itself, and hence via Dold-Kan an enrichment over $\infty$-categories. (I'm using implicitly the rectification result of Haugseng, which implies that DGCat is equivalent to the $\infty$-category of $\infty$-categories enriched over the $\infty$-category of chain complexes.) $\endgroup$
    – AAK
    Commented Sep 29, 2015 at 8:08
  • $\begingroup$ @Adeel Could you explain a little bit what does it mean by The model structure on the category of dg-categories "presents" an $(\infty,1)$=category DGCAT? $\endgroup$
    – Zhaoting Wei
    Commented Oct 14, 2015 at 2:00
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    $\begingroup$ Any model category (in fact, even just a category with weak equivalences) gives rise to an $(\infty,1)$-category. For example, if one takes simplicially enriched categories as a model of $(\infty,1)$-categories, this is given by the Dwyer-Kan simplicial localization. $\endgroup$
    – AAK
    Commented Oct 14, 2015 at 8:36
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Adeel answer is perfect, I will be more basic. There is many notions of "2-category" structure here around. 1) the category of small dg-categories $\mathbf{dgCat}$ is symmetric monoidal closed category. Closed means that there is an internal $HOM$ in the sense that for any two small dg-categories $A$ and $B$ there is $HOM(A,B)\in \mathbf{dgCat} $ such that there is a natural isomorphism of sets $\mathbf{dgCat}(X,HOM(A,B))\cong \mathbf{dgCat}(X\otimes A,B)$. That means $\mathbf{dgCat}$ is enriched over it self.

There is a functor from $H^{0}:\mathbf{dgCat}\rightarrow \mathbf{Cat}$ which gives you an enrichment of the category $\mathbf{dgCat}$ over $\mathbf{Cat}$, hence you can see $\mathbf{dgCat}$ as a 2-category.

On an other hand, the internal $HOM(-,-)$ described before has the wrong homotopy type, you can not derive it since it does not take Dwyer-Kan equivalences (between fibrant-cofibrant objects) to Dwyer-Kan equivalences (the $\mathbf{dgCat}$ is not symmetric monoidal model category in the sense of Hovey). Bertand Toen constructed, for the model category $\mathbf{dgCat}$, the right notion of the derived internal hom denoted by $RHOM(A,B)\in \mathbf{dgCat}$ (using bimodules, I will not write the details). Moreover this new derived $RHOM(A,B)$ induces the derived Mapping space $Map _{\mathbf{dgCat}}(A,B)$ via the nerve functor of some well choosen subcategory of $RHOM(A,B)$). This new derived internal allows you to see the category $\mathbf{dgCat}$ as $(2,\infty)$-category and in the same time as symmetric monoidal $(1,\infty)$-category.

An important consequence is the following isomorphism in $Ho(\mathbf{sSet})$: $$Map_{\mathbf{dgCat}}(A\otimes^{L}B,C)\cong Map_{\mathbf{dgCat}}(A,RHOM(B,C))$$

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