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Results tagged with model-categories
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user 12166
A model category is a category equipped with notions of weak equivalences, fibrations and cofibrations allowing to run arguments similar to those of classical homotopy theory.
3
votes
Accepted
Moerdijk Model Structure on Bisimplicial sets
That's not the Moerdijk model structure. In the Moerdijk model structure, weak equivalences and fibrations are created by the diagonal simplicial set construction. Your model structure resembles the B …
- 15.2k
8
votes
A model category which is an additive category
Yes, because the projection functor from the model category to the homotopy category preserves coproducts of cofibrant objects. That is actually the way of showing that the homotopy category has copro …
- 15.2k
7
votes
Accepted
Does the right adjoint of a Quillen equivalence preserve homotopy colimits?
The homotopy colimit functor $Ho(D^I)\rightarrow Ho(D)$ is the left adjoint of the constant diagram functor $Ho(D)\rightarrow Ho(D^I)$. Quillen equivalences induce Quillen equivalences between diagram …
- 15.2k
2
votes
Accepted
Is the projective model structure simplicial?
The answer is positive when the target is the category of simplicial sets. You can find a proof in Chaper VIII of Goerss-Jardine's book. Then it follows for presented combinatorial model categories in …
- 15.2k
5
votes
Accepted
Homotopy excision and homotopy pushout
1) I assume your three model structures have the same weak equivalences, correct me if I'm wrong. Let $\mathcal C$ be a model category and $I$ a small category, e.g. $I=\bullet\leftarrow \bullet\right …
- 15.2k
5
votes
Accepted
When does the projective model structure on functors exist?
I found it in Hirschhorn's book. It's in 11.6 Diagrams in a cofibrantly generated model category. The answer is always, provided K
is cofibrantly generated (Theorem 11.6.1). For him (as well as for Ho …
- 15.2k
3
votes
Accepted
dg-flat complexes and their characters
I take 'dg-injective' as fibrant in the injective model structure on complexes of right dg-$R$-modules, whose weak equivalences are the quasi-isomorphisms and whose cofibrations are the levelwise inje …
- 15.2k
4
votes
Homotopy limit-colimit diagrams in stable model categories
Yes, because both statements are equivalent to the existence of an exact triangle of the form
$$W\longrightarrow X\oplus Y\oplus Z\longrightarrow V\longrightarrow\Sigma W.$$
The arrows are the same …
- 15.2k
6
votes
Accepted
Analogues of 'cone' distinguished triangles for pointed model categories?
1) Mikhail, mapping cones etc, are defined for arbitrary maps. The problem is that they are not homotopy invariant unless your model category is left proper. Therefore, in general you must take cofibr …
- 15.2k
5
votes
If $A$ is a cofibrant commutative dg-algebra over a commutative ring of characteristic $0$, ...
Cofibrant CDGAs are retracts of cellular ones. A cellular cofibrant CDGA is a free commutative graded algebra on a (possibly transfinite) sequence of generators $x_1,x_2,\dots$ such that $d(x_i)$ only …
- 15.2k
10
votes
Accepted
Fubini theorem for hocolim
This property holds actually for right derivable categories in the sense of:
MR2729017 Reviewed Cisinski, Denis-Charles Catégories dérivables. (French) [Derivable categories] Bull. Soc. Math. France 1 …
- 15.2k
3
votes
Characterizing the rationalization of spaces.
I think the following is a trivial counterexample, which may lead you to reflect about your question:
\begin{align*}
F\colon Spaces & \longrightarrow Spaces\\\\
X&\;\mapsto\;\bigvee_{H_1(X,\mathbb{F …
- 15.2k
3
votes
Reference for homotopy (co)limits of (co)chain complexes via totalization of double complexes
Rodríguez González, Beatriz(E-CSIC-IM)
Simplicial descent categories. (English summary)
J. Pure Appl. Algebra 216 (2012), no. 4, 775–788
- 15.2k
5
votes
Accepted
Bousfield localization before and after taking homotopy
Let $\tilde C$ be the left Bousfield localization of $C$. As categories $C=\tilde C$, but $\tilde C$ has more weak equivalences. In particular, the identity functor induces a functor $\varphi\colon \o …
- 15.2k
8
votes
An example of two cofibrant dg categories whose tensor product is not cofibrant
You can consider DGAs, which are DG-categories with only one object. The polynomial ring $k[x]$ concentrated in degree $0$ (with trivial differential) is cofibrant, since it is free as a (graded) $k$- …
- 15.2k