Closed Model Category Structure Research Articles

Koszul duality for compactly generated derived categories of second kind

For any dg algebra $A$ we construct a closed model category structure on dg $A$-modules such that the corresponding homotopy category is compactly generated by dg $A$-modules that are finitely generated and free over $A$ (disregarding the differential). We prove that this closed model category is Quillen equivalent to the category of comodules over a certain, possibly nonconilpotent dg coalgebra, a so-called extended bar construction of $A$. This generalises and complements certain aspects of dg Koszul duality for associative algebras.

Koszul duality and homotopy theory of curved Lie algebras

This paper introduces the category of marked curved Lie algebras with curved morphisms, equipping it with a closed model category structure. This model structure is---when working over an algebraically closed field of characteristic zero---Quillen equivalent to a model category of pseudo-compact unital commutative differential graded algebras; this extends known results regarding the Koszul duality of unital commutative differential graded algebras and differential graded Lie algebras. As an application of the theory developed within this paper, algebraic deformation theory is extended to functors on pseudo-compact, not necessarily local, commutative differential graded algebras.

Cocommutative coalgebras: homotopy theory and Koszul duality

We extend a construction of Hinich to obtain a closed model category structure on all differential graded cocommutative coalgebras over an algebraically closed field of characteristic zero. We further show that the Koszul duality between commutative and Lie algebras extends to a Quillen equivalence between cocommutative coalgebras and formal coproducts of curved Lie algebras.

Maurer–Cartan moduli and models for function spaces

We set up a formalism of Maurer–Cartan moduli sets for L∞ algebras and associated twistings based on the closed model category structure on formal differential graded algebras (a.k.a. differential graded coalgebras). Among other things this formalism allows us to give a compact and manifestly homotopy invariant treatment of Chevalley–Eilenberg and Harrison cohomology. We apply the developed technology to construct rational homotopy models for function spaces.

Closed model categories for uniquely S-divisible spaces

For each integer n>1 and a multiplicative system S of non-zero integers, we give a distinct closed model category structure to the category of pointed spaces Top ★ and we prove that the corresponding localized category Ho(Top ★ ( S, n) ), obtained by inverting the weak equivalences, is equivalent to the standard homotopy category of uniquely ( S, n)-divisible, ( n−1)-connected spaces. A space X is said to be uniquely ( S, n)-divisible if for k⩾ n the homotopy group π k X is uniquely S-divisible. This equivalence of categories is given by an ( S, n)-colocalization functor that carries a pointed space X to a space X ( S, n) . There is also a natural map X ( S, n) → X which is (finally) universal among all the maps Z→ X with Z a uniquely ( S, n)-divisible, ( n−1)-connected space. The structure of closed model category given by Quillen to Top ★ is based on maps which induce isomorphisms on all homotopy group functors π k and for any choice of base point. For each pair ( S, n), the closed model category structure given here take as weak equivalences those maps that for the given base point induce isomorphisms on the homotopy groups functors π k( Z[S −1];−) with coefficients in Z[S −1] for k⩾ n. We note that the category Ho( Top ★ ( Z−{0},2) ) is the homotopy category of rational 1-connected spaces.

Cooperads and coalgebras as closed model categories

This paper proves that (linear) quasi-isomorphisms and monomorphisms define a closed model category (CMC) structure on the category of positively graded cooperads. Z-coalgebras over a quasi-cofree cooperad Z supports also an analogous CMC structure.

Théories homotopiques dans les topos

The purpose of these notes is to give an ad hoc construction of a closed model category structure on a topos inverting an arbitrary small set of arrows. Moreover, a necessary and sufficient condition for those structures to be proper is given. As an example, the Joyal closed model category structure on the category of simplicial objects of a topos is constructed without the use of (boolean) points.

Virtual operad algebras and realization of homotopy types

We prove that the category of algebras over a cofibrant operad admits a closed model category structure. This leads to the notion of “virtual operad algebra” – the algebra over a cofibrant resolution of the given operad. In particular, virtual commutative algebras can serve to an algebraic description of homotopy p-types as in the recent preprint of Mandell (M. Mandell, E ∞-algebras and p-adic homotopy theory, Hopf preprint server, October, 1998). Our main result allows one to simplify the proof of Mandell's theorem.

Models for torsion homotopy types

Given an integern>1 and any setP of positive integers, one can assign to each topological spaceX a homotopy universal mapX(P,n)→X whereX(P,n) is an (n−1)-connected CW-complex whose homotopy groups areP-torsion. We analyze this construction and its properties by means of a suitable closed model category structure on the pointed category of topological spaces.

A closed model category for (𝑛-1)-connected spaces

For each integer n > 0 n > 0 , we give a distinct closed model category structure to the category of pointed spaces Top ⋆ \operatorname {Top}_\star such that the corresponding localized category Ho ⁡ ( Top ⋆ n ) \operatorname {Ho}(\operatorname {Top}_\star ^n) is equivalent to the standard homotopy category of ( n − 1 ) (n-1) -connected CW-complexes. The structure of closed model category given by Quillen to Top ⋆ \operatorname {Top}_\star is based on maps which induce isomorphisms on all homotopy group functors π q \pi _q and for any choice of base point. For each n > 0 n>0 , the closed model category structure given here takes as weak equivalences those maps that for the given base point induce isomorphisms on π q \pi _q for q ≥ n q\ge n .

Closed model categories for the n-type of spaces and simplicial sets

AbstractFor each integer n ≥ 0, we give a distinct closed model category structure to the categories of spaces and of simplicial sets. Recall that a non-empty map is said to be a weak equivalence if it induces isomorphisms on the homotopy groups for any choice of base point. Putting the condition on dimensions ≥ n, we have the notion of a weak n-equivalence which is at the base of the nth closed model category structure given here.

Homotopy Theory of Diagrams and CW-Complexes Over a Category

The purpose of this paper is to introduce the notion of a CW complex over a topological category. The main theorem of this paper gives an equivalence between the homotopy theory of diagrams of spaces based on a topological category and the homotopy theory of CW complexes over the same base category.A brief description of the paper goes as follows: in Section 1 we introduce the homotopy category of diagrams of spaces based on a fixed topological category. In Section 2 homotopy groups for diagrams are defined. These are used to define the concept of weak equivalence and J-n equivalence that generalize the classical definition. In Section 3 we adapt the classical theory of CW complexes to develop a cellular theory for diagrams. In Section 4 we use sheaf theory to define a reasonable cohomology theory of diagrams and compare it to previously defined theories. In Section 5 we define a closed model category structure for the homotopy theory of diagrams. We show this Quillen type homotopy theory is equivalent to the homotopy theory of J-CW complexes. In Section 6 we apply our constructions and results to prove a useful result in equivariant homotopy theory originally proved by Elmendorf by a different method.

Stability in pro-homotopy theory

If is a category, an object of pro- is stable if it is isomorphic in pro- to an object of . A local condition on such a pro-object, called strong-movability, is defined, and it is shown in various contexts that this condition is equivalent to stability. Also considered, in the case is a suitable model category, is the stability problem in the homotopy category Ho(pro-), where pro- has the induced closed model category structure defined by Edwards and Hastings [6].

Remarks on the closed model category structure of simplicial spectra

Remarks on the closed model category structure of simplicial spectra