The notion of normal subobject having an intrinsic meaning in any proto-modular category, we introduce the notion of normal functor, namely left exact conservative functor which reeects normal subobjects. The point is that for the category Gp of groups the change of base functors, with respect to the bration of pointed objects, are not only conservative (this is the deenition of a protomodular ...

This paper is devoted to connections between trace monoids and cubical sets. We prove that the category of trace monoids is isomorphic to the category of generalized tori and it is a reflective subcategory of the category of cubical sets. Adjoint functors between the categories of cubical sets and trace monoid actions are constructed. These functors carry independence preserving morphisms in th...

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 give a formula for the relative Deligne tensor product of two indecomposable finite semisimple module categories over pointed braided fusion category an algebraically closed field.

lthough fuzzy set theory and  sheaf theory have been developed and studied independently,  Ulrich Hohle shows that a large part of fuzzy set  theory  is in fact a subfield of sheaf theory. Many authors have studied mathematical structures, in particular, algebraic structures, in both  categories of these generalized (multi)sets. Using Hohle's idea, we show that for a (universal) algebra $A$, th...

This paper explicitly constructs cofree coalgebras over operads in the category of DG-modules. It is shown that the existence of an operadaction on a coalgebra implies a “generalized coassociativity” that facilitates the construction. Special cases are considered in which the general expression simplifies (such as the pointed, irreducible case). In the pointed irreducible case, this constructio...

The notion of ×-homotopy from [Doca] is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space Hom∗(G,H) with the homotopy groups of Hom∗(G,H ). Here Hom∗(G,H) is a space which parameterizes pointed graph maps from G to H (a pointed version of the usual Hom complex), and H is the graph of based...