Let ~ C be a category whose objects are semigroups with topology and morphisms are closed semigroup relations, in particular, continuous homomorphisms. An object X of the category ~ C is called ~ C-closed if for each morphism Φ ⊂ X×Y in the category ~ C the image Φ(X) = {y ∈ Y : ∃x ∈ X (x, y) ∈ Φ} is closed in Y. In the paper we survey existing and new results on topological groups, which are ~...

Definition 1.1. A weak factorization system (L,R) in a category K consists of two classes L and R of morphisms of K such that (1) R = L , L = R and (2) any morphism h of K has a factorization h = gf with f ∈ L and g ∈ R. Definition 1.2. A model category is a complete and cocomplete category K together with three classes of morphisms F , C and W called fibrations, cofibrations and weak equivalen...

For the category $\mathscr V$ of complex algebraic varieties, Grothendieck group commutative monoid isomorphism classes correspondences $X \xleftarrow f M \xrightarrow g Y$ with a proper morphism $f$ and smooth $g$ (such

Let D1, D2, D be non empty sets and let x be an element of [: [:D1, D2 :], D :]. Then x1,1 is an element of D1. Then x1,2 is an element of D2. Let D1, D2 be non empty sets and let x be an element of [:D1, D2 :]. Then x2 is an element of D2. Next we state the proposition (1) Let C, D be category structures. Suppose the category structure of C = the category structure of D. If C is category-like,...

For objects A and B belonging to a category C we denote by C(A,B) the set of all morphisms from A to B in C and, for any f ∈ C(A,B), the objects A and B are called, respectively, the domain and codomain of f . For example, if A and B are abelian groups, and we denote by U(A) and U(B) the underlying sets of A and B, then Ab(A,B) is the set of all group homomorphisms having domain A and codomain ...

This paper concerns a number of diagram categories, namely the partition, planar Brauer, partial Motzkin and Temperley–Lieb categories. If [Formula: see text] denotes any these if is fixed morphism, then an associative operation may be defined on by text]. The resulting semigroup called sandwich semigroup. We conduct thorough investigation semigroups, with emphasis structural combinatorial prop...

We prove a coherence theorem for invertible objects in a symmetric monoidal category. This is used to deduce associativity, skewcommutativity, and related results for multi-graded morphism rings, generalizing the well-known versions for stable homotopy groups.

Then SetF(Ω) is a category with Ω-sets as objects and maps f : A → B such that γ(f(x), f(y)) ≥ δ(x, y) for all x, y ∈ A as morphisms f : (A, δ) → (B, γ). Moreover, by SetR(Ω) we understand a category which is an analogy of a category of sets with relations between sets as morphisms. Objects of this category SetR(Ω) are the same as in the category SetF(Ω) and morphism f : (A, δ) → (B, γ) are map...

We establish model category structures on algebras and modules over operads in symmetric spectra, and study when a morphism of operads induces a Quillen equivalence between corresponding categories of algebras (resp. modules) over operads.

We establish sufficient and necessary conditions for a natural sink to be a colimit in DCPO. Based on these conditions we show how to construct a colimit for any functor F from a small category „ into the category DCPO. This demonstrates that the category DCPO is cocomplete. We also investigate under what conditions the colimit object is algebraic. 0 Introduction Colimits play an important role...