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Homotopy groups of small categories and derived functors

Marek Golasiński — 1987

Cahiers de Topologie et Géométrie Différentielle Catégoriques

Homotopies of small categories

Marek Golasiński — 1981

Fundamenta Mathematicae

Componentwise injective models of functors to DGAs

Marek Golasiński — 1997

Colloquium Mathematicae

The aim of this paper is to present a starting point for proving existence of injective minimal models (cf. [8]) for some systems of complete differential graded algebras.

Injective models of $G$ -disconnected simplicial sets

Marek Golasiński — 1997

Annales de l'institut Fourier

We generalize the results by G.V. Triantafillou and B. Fine on $G$ -disconnected simplicial sets. An existence of an injective minimal model for a complete $𝕀$ -algebra is presented, for any $E I$ -category $𝕀$ . We then make use of the $E I$ -category $𝒪 (G, X)$ associated with a $G$ -simplicial set $X$ to apply these results to the category of $G$ -simplicial sets. Finally, we describe the rational homotopy type of a nilpotent $G$ -simplicial set by means of its injective minimal model.

On $G$ -disconnected injective models

Marek Golasiński — 2003

Annales de l’institut Fourier

Let $G$ be a finite group. It was observed by L.S. Scull that the original definition of the equivariant minimality in the $G$ -connected case is incorrect because of an error concerning algebraic properties. In the $G$ -disconnected case the orbit category $𝒪 (G)$ was originally replaced by the category $𝒪 (G, X)$ with one object for each component of each fixed point simplicial subsets $X^{H}$ of a $G$ -simplicial set $X$ , for all subgroups $H \subseteq G$ . We redefine the equivariant minimality and redevelop some results on the rational homotopy...

On generalized “ham sandwich” theorems

Marek Golasiński — 2006

Archivum Mathematicum

In this short note we utilize the Borsuk-Ulam Anitpodal Theorem to present a simple proof of the following generalization of the “Ham Sandwich Theorem”: Let $A_{1}, ..., A_{m} \subseteq ℝ^{n}$ be subsets with finite Lebesgue measure. Then, for any sequence $f_{0}, ..., f_{m}$ of $ℝ$ -linearly independent polynomials in the polynomial ring $ℝ [X_{1}, ..., X_{n}]$ there are real numbers $λ_{0}, ..., λ_{m}$ , not all zero, such that the real affine variety ${x \in ℝ^{n}; λ_{0} f_{0} (x) + \dots + λ_{m} f_{m} (x) = 0}$ simultaneously bisects each of subsets $A_{k}$ , $k = 1, ..., m$ . Then some its applications are studied.

Equivariant cohomology with local coefficients

Marek Golasiński — 1997

Mathematica Slovaca

The homotopy category of chain complexes is a homotopy category

Marek Golasiński; Grzegorz Gromadzki — 1982

Colloquium Mathematicae

A model structure for the homotopy theory of crossed complexes

Ronald Brown; Marek Golasinski — 1989

Cahiers de Topologie et Géométrie Différentielle Catégoriques

Weak homotopy equivalences of mapping spaces and Vogt's lemma

Marek Golasiński; Luciano Stramaccia — 2008

Cahiers de Topologie et Géométrie Différentielle Catégoriques

Comultiplications of the Wedge of Two Moore Spaces

Marek Golasiński; Daciberg Gonçalves — 1998

Colloquium Mathematicae

Co-H-structures on equivariant Moore spaces

Martin Arkowitz; Marek Golasiński — 1994

Fundamenta Mathematicae

Let G be a finite group, $𝕆_{G}$ the category of canonical orbits of G and $A : 𝕆_{G} \to 𝔸$ b a contravariant functor to the category of abelian groups. We investigate the set of G-homotopy classes of comultiplications of a Moore G-space of type (A,n) where n ≥ 2 and prove that if such a Moore G-space X is a cogroup, then it has a unique comultiplication if dim X < 2n - 1. If dim X = 2n-1, then the set of comultiplications of X is in one-one correspondence with $E x t^{n - 1} (A, A \otimes A)$ . Then the case $G = ℤ_{p^{k}}$ leads to an example of infinitely...

Residue class rings of real-analytic and entire functions

Marek Golasiński; Melvin Henriksen — 2006

Colloquium Mathematicae

Let 𝓐(ℝ) and 𝓔(ℝ) denote respectively the ring of analytic and real entire functions in one variable. It is shown that if 𝔪 is a maximal ideal of 𝓐(ℝ), then 𝓐(ℝ)/𝔪 is isomorphic either to the reals or a real closed field that is an η₁-set, while if 𝔪 is a maximal ideal of 𝓔(ℝ), then 𝓔(ℝ)/𝔪 is isomorphic to one of the latter two fields or to the field of complex numbers. Moreover, we study the residue class rings of prime ideals of these rings and their Krull dimensions. Use is made of...

Equivariant Gottlieb groups

Marek Golasiński; Daciberg Lima Gonçalves — 2001

Cahiers de Topologie et Géométrie Différentielle Catégoriques

Equivariant evaluation subgroups and Rhodes groups

Marek Golasiński; Daciberg Gonçalves; Peter Wong — 2007

Cahiers de Topologie et Géométrie Différentielle Catégoriques

Rational homotopy type of nilpotent and complete spaces

Golasiński, Marek — 1989

Proceedings of the Winter School "Geometry and Physics"

On the object-wise tensor product of functors to modules.

Golasiński, Marek — 2000

Theory and Applications of Categories [electronic only]

A note on generalized equivariant homotopy groups

Marek Golasiński; Daciberg L. Gonçalves; Peter N. Wong — 2009

Banach Center Publications

In this paper, we generalize the equivariant homotopy groups or equivalently the Rhodes groups. We establish a short exact sequence relating the generalized Rhodes groups and the generalized Fox homotopy groups and we introduce Γ-Rhodes groups, where Γ admits a certain co-grouplike structure. Evaluation subgroups of Γ-Rhodes groups are discussed.