We define Anderson-Brown-Cisinski (ABC) cofibration categories, and construct homotopy colimits of diagrams of objects in ABC cofibraction categories. Homotopy colimits for Quillen model categories are obtained as a particular case. We attach to each ABC cofibration category a right derivator. A dual theory is developed for homotopy limits in ABC fibration categories and for left derivators. Th...

In this short note we show that the homotopy category of smooth compactifications of smooth algebraic varieties is equivalent to the homotopy category of smooth varieties over a field of characteristic zero. As an application we show that the functor sending a variety to the p-th step of the Hodge filtration of its complex cohomology is representable in the homotopy category of simplicial presh...

The purpose of this paper is to collect the homotopical methods used in the development of the theory of flows initialized by author’s paper “A model category for the homotopy theory of concurrency”. It is presented generalizations of the classical Whitehead theorem inverting weak homotopy equivalences between CW-complexes using weak factorization systems. It is also presented methods of calcul...

The paper presents the optimal homotopy perturbation method, which is a new method to find approximate analytical solutions for nonlinear partial differential equations. Based on the well-known homotopy perturbation method, the optimal homotopy perturbation method presents an accelerated convergence compared to the regular homotopy perturbation method. The applications presented emphasize the h...

The present paper deals with the homotopy perturbation method. The question of whether the homotopy perturbation method is simply the conventional Taylor series expansion is examined. It is proven that under particular choices of the auxiliary parameters the homotopy perturbation method is indeed the Taylor series expansion of the sought solution of nonlinear equations.

Over a monoidal model category, under some mild assumptions, we equip the categories of colored PROPs and their algebras with projective model category structures. A BoardmanVogt style homotopy invariance result about algebras over cofibrant colored PROPs is proved. As an example, we define homotopy topological conformal field theories and observe that such structures are homotopy invariant.

We recall a group-theoretic description of the first non-vanishing homotopy group of a certain (n+1)-ad of spaces and show how it yields several formulae for homotopy and homology groups of specific spaces. In particular we obtain an alternative proof of J. Wu’s group-theoretic description of the homotopy groups of a 2-sphere.

A b str act . Let Σ be the image of a topological embedding of Sn−2 into Sn. In this paper the homotopy groups of the complement Sn−Σ are studied. In contrast with the situation in the smooth and piecewise linear categories, it is shown that the first nonstandard homotopy group of the complement of such a topological knot can occur in any dimension in the range 1 through n − 2. If the first non...

The concept of the homotopy theory of modules was discovered by Peter Hilton as a result of his trip in 1955 to Warsaw, Poland, to work with Karol Borsuk, and to Zurich, Switzerland, to work with Beno Eckmann. The idea was to produce an analog of homotopy theory in topology. Yet, unlike homotopy theory in topology, there are two homotopy theories of modules, the injective theory, πn(A,B), and t...

In this work, we have applied Elzaki transform and He's homotopy perturbation method to solvepartial dierential equation (PDEs) with time-fractional derivative. With help He's homotopy per-turbation, we can handle the nonlinear terms. Further, we have applied this suggested He's homotopyperturbation method in order to reformulate initial value problem. Some illustrative examples aregiven in ord...