We present well known concepts of category theory: natural transofmations and functor categories, and prove propositions related to. Because of the formalization it proved to be convenient to introduce some auxiliary notions, for instance: transformations. We mean by a transformation of a functor F to a functor G, both covariant functors from A to B, a function mapping the objects of A to the m...

The Hom complex of homomorphisms between two graphs was originally introduced to provide topological lower bounds on the chromatic number of graphs. In this paper we introduce new methods for understanding the topology of Hom complexes, mostly in the context of Γ-actions on graphs and posets (for some group Γ). We view the Hom(T, •) and Hom(•, G) as functors from graphs to posets, and introduce...

Consider a coring with exact rational functor, and a finitely generated and projective right comodule. We construct a functor (coinduction functor) which is right adjoint to the hom-functor represented by this comodule. Using the coinduction functor, we establish a bijective map between the set of representative classes of torsion simple right comodules and the set of representative classes of ...

We develop an Auslander-Reiten theory for triangulated categories which is based on Brown’s representability theorem. In a fundamental article [3], Auslander and Reiten introduced almost split sequences for the category of finitely generated modules over an artin algebra. These are short exact sequences which look almost like split exact sequences, but many authors prefer to call them Auslander...

For a Hopf algebra H over a commutative ring k, the category MH of right Hopf modules is equivalent to the category Mk of k-modules, that is, the comparison functor −⊗k H : Mk → MH is an equivalence (Fundamental theorem of Hopf modules). This was proved by Larson and Sweedler via the notion of coinvariants McoH for any M ∈ MH . The coinvariants functor (−) coH : MH → Mk is right adjoint to the ...

1.1 The main result. Given a map f : K → L of pointed CW complexes, let homf (K,L) denote the pointed space of pointed maps K → L, with f as the base point. Recall that a Πalgebra is a (≥ 1)-graded group with an action of the primary homotopy operations (for example, for any pointed topological space M there is a homotopy Π-algebra π∗M = {πiM}i=1). Given a map t : X → Y of Π-algebras, let homt(...

The homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint C are important to the study of (∞, 1)-categories because they provide a means for comparing two models of their respective homotopy theories, giving a Quillen equivalence between the model structures for quasi-categories and simplicial categories. The functor C also gives a cofibrant replacement for ...

We investigate the problem when tensor functor by a bimodule yields singular equivalence. It turns out that this is equivalent to one Hom given same induces triangle equivalence between homotopy categories of acyclic complexes injective modules. give conditions on appears in pair bimodules, defines with level. construct an explicit combinatorial manner, which quadratic monomial algebra and its ...

Given two algebraic groups G, H over a field k, we investigate the representability of functor morphisms (of schemes) Hom(G,H) and subfunctor homomorphisms groups) Homgp(G,H). We show that is represented by group scheme, locally finite type, if k-vector space O(G) finite-dimensional; converse holds not étale. When G linearly reductive smooth, Homgp(G,H) smooth scheme M; moreover, every orbit ac...