# Universal property of gluing [collage, cograph] of dg-categories

In some recent works, such as this one (3.2, page 15), a definition of "gluing of dg-categories along a dg-bimodule" is given. It is obviously the analogue of the notion of collage (or cograph) of a profunctor.

My question is: is there any "universal property" of this gluing? Something like "a dg-functor defined on the gluing is uniquely determined by something defined on the "pieces" (the dg-categories and the bimodule)". The article on ncat about the collage of a profunctor suggests the existence of a universal property, but I'm unable to write it down in a "honest" (hands-on) manner. Is there a way to do this in the world of dg-categories?

• A dg-category is the same as a category enriched in chain complexes. To do this in the world of dg-categories you can first switch to the world of enriched categories, where profunctors, collages and their universal properties generalize straightforwardly from the ordinary Set enriched case. May 13, 2014 at 12:16

I think it is more natural to ask for a universal property with respect to quasifunctors. There is one and you can find it in Appendix A of http://arxiv.org/abs/1212.6170.

• This is interesting (I knew the article but I overlooked the appendices), unfortunately this "gluing" is not the gluing I need. It's a different construction. May 13, 2014 at 14:20
• @FrancescoGenovese, they are the same for pretriangulated dg-categories, and otherwise the version of Orlov is the pretriangulated envelope of the version of Kuznetsov-Lunts.
– AAK
May 13, 2014 at 14:30
• Really? Is there any source for this claim? May 13, 2014 at 14:53
• See Remark 4.2 in the linked paper.
– AAK
May 13, 2014 at 15:11

$\newcommand{\nto}{\looparrowright}$ Collage along the terminal profunctor has, indeed, a universal property since it can be characterized via an adjunction.

But let me expand a little bit a piece of theory I never found written anywhere. This should turn into something extremely boring for skilled readers, so let me apologize in advance for my naivete.

I'll denote as $W\colon C\nto D$ a profunctor $W\colon C^{op}\times D\to \rm Set$. (generalizing to the enriched case is a matter of changing Set with a generic $\cal V$).

Fact. The collage operation $(C,D,W\colon C \nto D)\mapsto C\star_W D$ is functorial if we consider the category $\cal C$ having as objects triples $(C,D,W)$ where $C,D\in{\rm Cat}$ and $W\colon C\nto D$ is a profunctor. This category is (a particular presentation of?) the proarrow equipment $({\rm Cat},\rm Dist)$.

Proof. $\cal C$ becomes a category if I define arrows $(C,D,W)\to (C',D', W')$ to be triples $(f,g,\eta)$ where

1. $f\colon C\to C'$ and $g\colon D\to D'$ are functors;
2. $\eta\colon W\to W'\diamond \phi^f\diamond \phi_g$ is a natural transformation: $\diamond$ is the composition of profunctors defined as a coend, $\phi^f=\hom(f,1), \phi_g =\hom(1,g)$ are the representable'' profunctors.

(this is precisely what you need to define the equipment).

It's rather easy to see that this is a category: given two arrows $(f,g,\eta)$, $(f', g',\psi)$ composition is defined by $$W\xrightarrow{\eta} W'\phi^f\phi_g \xrightarrow{\psi \diamond \phi^f\diamond\phi_g} W''\phi^{f'}\phi_{g'}\phi^f\phi_g \cong W''\diamond\phi^{f'f}\diamond\phi_{g'g}$$ (since it is obvious that $\phi^f\diamond \phi_g\cong \phi_g\diamond \phi^f$) and the identity arrow is the obvious one since $\phi^\text{id}=\phi_\text{id}=\hom$. We now have to prove that this defines a functor $C\star_W D\to C'\star_{W'}D'$ for any morphism $(f,g,\eta)$; this definition obviously collapses to the classical one if we consider $W,W'$ to be the terminal profunctors.

To this end, let's consider that a pair of functors $F\colon C\to C', G\colon D\to D'$ in the terminal case induced a functor $F\star G\colon C\star D\to C'\star D'$ (this can be proved fixing a component and letting the second vary, as the interchange law obviously holds); $F\star G$ is defined to be $F$ (on objects and arrows) when restricted to $C$, $G$ when restricted to $D$, and the unique arrow $c\to d$ is sent to the unique arrow $Fc\to Gd$.

In the general case, there is no preferred choice for an arrow linking $\hom_{C\star_W D}(c,d)$ to $\hom_{C'\star_W D'}(Fc,Gd)$, so the assignment of $F\star_W G$ on arrows has to be made via a morphism between profunctors: this is the role of $\{ \eta_{c,d}\colon W(c,d)\to W'\diamond \phi^f\diamond \phi_g(c,d)\}$.

$F\star_W G$ is defined as $F$ on $C$, as $G$ on $D$, and the new-born'' arrows represented by $W(c,d)$ are sent to the arrows $W'(Fc,Gd)$ via $\eta$. Functoriality of $F,G$ and naturality of $\eta$ ensure that everything goes right. $\blacksquare$

Now for your real question: I think the popularization of this result is due to Joyal, or at least, I learned it in simplicial flavour through his notes (see chap. 3) about quategories, and reading his catlab.

$\def\Cat{\text{Cat}}$ Consider the inclusion of the boundary of the standard 1-simplex, $i\colon \{0,1\}\to [1]$ as a functor between the discrete category with two elements and the walking arrow'' $I=\{0 \leq 1\}$. It induces a functor $$i^*\colon \Cat / I \to \Cat \times \Cat$$ which admits a right adjoint. This right adjoint is precisely the bifunctor $\star\colon \Cat \times \Cat \to \Cat / I$, given by the join along the terminal profunctor.

Notice that this collapsing has precisely the effect of removing the obstruction for $(-)\star_{(-)}(-)$ to become "monoidally-flavoured".

The result is clear, once we noticed that the category $C\star C'$ comes naturally equipped with an arrow $C\star C'\to I=1\star 1$ induced by (bi)functoriality of $\star$, starting from the canonical arrows $C\to 1, C'\to 1$ to the terminal category: more precisely, it is clear that $i^*$ is defined by sending $C\to I$ to the pair of categories $i^\leftarrow(0)=C_0, C_1=i^\leftarrow(1)$. The bijection $$\Cat^{\mathbf{2}}\Big(i^*\big( C\to I\big), (A,B)\Big)\cong \Cat / I \Big( C, A\star B \Big)$$ is now rather obvious, since any functor $i^*\big( C\to I\big)\to (A,B)$ determines a functor $C\to A\star B$ and viceversa. $\blacksquare$

Remark. If I remember well, $(\Cat, \star)$ is a fairly blatant example of a non-closed, biclosed (see Joyal 3.1.1, 3.1.2) monoidal category.

• (My answer is not related to dg-categories, so it may appear as an Off Topic; but I'm rather sure that this is something Francesco needs since we chatted in real life about his problem. Please be kind if you want to downvote me...) May 13, 2014 at 22:02