9
votes
Accepted
Is there a notion of projective dg category?
If $X$ is a smooth projective threefold with a flopping curve $C$ then typically the variety $Y$ obtained from $X$ by a flop in $C$ is not projective, but smooth, proper, and derived equivalent to $X$....
- 39.3k
8
votes
Why Faonte called "small" and "big" dg-nerves?
He uses big and small because the one he calls small is smaller!
Trying things out with a dg-category (rather than the more complicated A-infinity structure), the big one goes via the simplicially ...
- 9,597
8
votes
Accepted
Derived categories of smooth proper varieties?
The study of derived categories is a special case of the more general study of semiorthogonal components of derived categories. By Chow lemma for any proper variety $X$ there is a blow up $\pi \colon ...
- 39.3k
6
votes
Is every dg-coalgebra the colimit of its finite dimensional dg-subcoalgebras?
For coassociative dg-coalgebras over any field $k$ the answer is positive, because:
Let $C$ be a $\mathbb Z$-graded coalgebra and $D\subset C$ a finite-dimensional ungraded subcoalgebra (of the ...
- 15.6k
6
votes
How to stop worrying about enriched categories?
Have a look around on my n-Lab 'home page':
https://ncatlab.org/timporter/show/HomePage
and go down to the `resources'. There are various quite old sets of notes that look at simplicially enriched ...
5
votes
Accepted
Is the tensor product of pretriangulated dg-categories a pretriangulated dg-category?
Assume that every dg-category is over a field $k$. My guess is that there is a natural (I believe fully faithful) dg-functor
\begin{equation}
\Phi \colon \mathrm{Perf}(A) \otimes \mathrm{Perf}(B) \to \...
- 2,079
5
votes
Is every dg-coalgebra the colimit of its finite dimensional dg-subcoalgebras?
Yes. The category of $\mathbb{Z}$-coalgebras is locally presentable, and objects are filtered colimits of finite dimensional subobjects. See the appendix to Coalgebraic models for combinatorial model ...
- 30.3k
4
votes
Accepted
Functorial cones
The cone construction can be written down very explicitly, just following the definition of mapping cone of chain complexes. Good sources are in my opinion:
https://arxiv.org/pdf/math/0401009.pdf ...
- 2,079
3
votes
Accepted
Perfect DG modules
Well by definition a DG module $X$ is perfect iff it is a direct summand of a module which is a finite iterated extension of free modules (it's then a nontrivial result that this property is the same ...
- 7,952
2
votes
Accepted
Deriving the functor $ \int_{\Gamma} F(-,-)$
We eventually found a solution which solved the original issue, though not in the way it is exactly stated here. Namely, we found what would appear to be the most general way that ends can be derived. ...
- 295
2
votes
Accepted
DG natural transformation Serre functors
It is possible to lift $\alpha \otimes \mathrm{id}$ to a dg-enhancement, one way is the following. Just to fix notation, this natural transformation is induced from
$$
\alpha \colon \mathcal{O}_X \to ...
2
votes
Functorial cones
A proof in the context of model categories can be found in Proposition 6.3.5 of Hovey's book Model Categories. You could easily rewrite the proof to work in the context of dg-categories, where it is ...
- 30.3k
2
votes
Accepted
Morphisms on fibre products
This is not true. For example take $F_1 = \bigoplus_{n \in \mathbf{N}} \mathcal{O}_X$, $F_2 = \mathcal{O}_X$, $G_1 = \bigoplus_{m \in \mathbf{N}} \mathcal{O}_Y$ and $G_2 = \mathcal{O}_Y$. Moreover, ...
- 1,144
2
votes
Accepted
Intuition for points of the moduli of objects for a dg-category
[It would be good to mention the original paper of Toën-Vaquié https://arxiv.org/abs/math/0503269 where these moduli spaces of objects are defined, and maybe that of Lowrey for the ``coherent" ...
- 24k
2
votes
homotopy limits of dg categories
I'm sorry this answer is not on time. Actually for cosimplicial diagrams of dg-categories, which is mostly common in algebraic geometry, the homotopy limit is given by the dg-category of twisted ...
- 9,009
1
vote
Colimits of DG-categories and functors between them
The answer to (2) is yes, by a nice result of Gaitsgory plus an easy categorical argument. To spell it out, the situation of the OP is the following, where functors going down are $ev_i$ and functors ...
- 30.3k
1
vote
Accepted
Are dg-modules over a cofibrant dg-category cofibrant?
My question is: suppose that C is a cofibrant dg-category. Then are either of Ĉ or dgMod_C^op cofibrant dg-categories?
A cofibrant object in a cofibrantly generated model category (such as dgCat)
is ...
- 37.8k
1
vote
Accepted
Definition of gluing of dg categories
This gluing can be viewed as a dg-category of "generalized morphisms". If you take $\mathcal D_1 = \mathcal D_2 = \mathcal D$ and $N$ to be the diagonal bimodule, then what you get is the dg-category ...
- 2,079
1
vote
$k$-linear $\infty$ stable categories and dg categories
For a dg-category $\mathcal C$, being flat means that all enriched Hom's are flat (as k-modules). In other words, for every two objects $a,b$, one requires that the functor $\mathcal C(x,y)\otimes-$ ...
- 8,385
1
vote
$k$-linear $\infty$ stable categories and dg categories
The derived tensor product of dg-categories was explored
by Toën, see his article
The homotopy theory of dg-categories and derived Morita theory,
in particular, Section 4, where Toën explains how to ...
- 37.8k
1
vote
How to show mapping cones are homotopy cofibers
A very short answer would be as follows. What you define to be $\mathrm{Cone}(f)$ lies in a triangle in $\mathcal C$:
\begin{equation}
C \xrightarrow{f} D \to \mathrm{Cone}(f),
\end{equation}
...
- 2,079
1
vote
Reference request: category of sheaves of O-modules with coherent cohomology
As mentioned in the comments, I think SGA 6 is one of the original references for this. More specifically:
[SGA 6, II, Corollaire 2.2.2.1] If $X$ is a Noetherian scheme, then the canonical fully ...
- 1,109
1
vote
Deriving the functor $ \int_{\Gamma} F(-,-)$
The answer is no in this generality, but I do not know what happens specifically for ${\bf dgCat}$ and ${\bf dgFun}$. For convenience, let me construct a counter-example to the dual of your question, ...
- 9,481
1
vote
Deriving the functor $ \int_{\Gamma} F(-,-)$
A long comment.
The way you state the question currently mixes the abstract notion of derived functor (has nothing to do with fibrant replacements), and the notion of a co/fibrant replacement. I am ...
- 660
1
vote
Could $RHom(A,-)$ distinguish non-quasi-equivalent dg-categories?
Actually the hom-set $[B,C]$ between two dg-categories in $\mathrm{Hqe}$ is described as \begin{equation} [B,C] \cong \mathrm{Iso}(H^0(RHom(B,C))),\end{equation} so if $RHom(A,B) \cong RHom(A,C)$ in $...
- 2,079
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