We study fibration category structures induced by enrichments over symmetric monoidal categories that are also fibration categories. Let $\mathcal V$ be a monoidal category that is also a fibration category. Assume that $\mathcal V$ has an interval object. We show that the fibration category structure on $\mathcal V$ can be transferred over any $\mathcal V$-enriched category through corepresentable functors provided that certain power objects exists. We also give its $G$-equivariant extension for a group $G$, so that under mild conditions the category of $G$-objects in a $\mathcal V$-enriched category admits a (non-trivial) fibration category structure. We later show that several categories of topological algebras and associative algebras, and their $G$-equivariant analogues, can be made into fibration categories obtained in this way. We also present some applications of our results by recovering some existing results on (equivariant) $K$ and $E$-theories of operator algebras.

Let M be a monoid and G : Mon -> Grp be the group completion functor from monoids to groups. Given a collection X of submonoids of M and for each N is an element of X a collection Y_N of subgroups of G(N), we construct a model structure on the category of M-spaces and M-equivariant maps, called the (X, Y)-model structure, in which weak equivalences and fibrations are induced from the standard YN-model structures on G(N)-spaces for all N is an element of X . We also show that for a pair of collections (X, Y) there is a small category 0((X,Y)) whose objects are M-spaces M x_N G(N)/H for each N is an element of X and H is an element of Y_N and morphisms are M-equivariant maps, such that the (X, Y)-model structure on the category of M-spaces is Quillen equivalent to the projective model structure on the category of contravariant O(X,Y)-diagrams of spaces. 

Let G be discrete group and F be a collection of subgroups of G. We show that there exists a left induced model structure on the category of right G-simplicial sets, in which the weak equivalences and cofibrations are the maps that induce weak equivalences and cofibrations on H-orbits for all H in F. This gives a model categorical criterion for maps that induce weak equivalences on H-orbits to be weak equivalences in the F-model structure. 

In this paper we study M(X), the set of diffeomorphism classes of smooth manifolds with the simple homotopy type of X, via a map psi from M(X) into the quotient of K(X) = [X, BSO] by the action of the group of homotopy classes of simple self equivalences of X. The map psi describes which bundles over X can occur as normal bundles of manifolds in M(X). We determine the image of psi when X belongs to a certain class of homology spheres. In particular, we find conditions on elements of K(X) that guarantee they are pullbacks of normal bundles of manifolds in M(X). 

In this paper we discuss some enlargements of the category of sets with semigroup actions and equivariant functions. We show that these enlarged categories possess two idempotent endofunctors. In the case of groups these enlarged categories are equivalent to the usual category of group actions and equivariant functions, and these idempotent endofunctors reverse a given action. For a general semigroup we show that these enlarged categories admit homotopical category structures defined by using these endofunctors and show that up to homotopy these categories are equivalent to the usual category of sets with semigroup actions. We finally construct the Burnside ring of a monoid by using homotopical structure of these categories, so that when the monoid is a group this definition agrees with the usual definition, and we show that when the monoid is commutative, its Burnside ring is equivalent to the Burnside ring of its Grothendieck group. 

We study actions of monoidal categories on objects in a suitably enriched 2-category, and applications in stable homotopy theory. Given a monoidal category I and an I-object A, the (co)stabilization of A is obtained by universally forcing the I-action to be reversible so that every object of I acts on A by auto-equivalences. We introduce a notion of I-equivariance for morphisms between I-objects and give constructions of stabilization and costabilization in terms of weak ends and coends in an enriched I-category of I-objects and I-equivariant morphisms. We observe that the stabilization of a relative category with respect to an action coincides with the usual notion of stabilization in stable homotopy theory when the action is defined by loop space functors. We show that several examples that exist in the literature, including various categories of spectra, fit into our setting after fixing A and the I-action on it. In particular, categories of sequential spectra, coordinate free spectra, genuine equivariant spectra, parameterized spectra indexed by vector bundles are obtained in terms of weak ends in the I-category of relative categories. On the other hand, the costabilization of a relative category with respect to an action gives a stable relative category akin to a version Spanier-Whitehead category. In particular, we establish a form of duality between constructions of stable homotopy categories by spectra and by Spanier-Whitehead like categories. 

For a graph G=(V,E) a set of  vertices D is called a dominating set if every vertex  in  V\D is adjacent to a vertex in D. A domatic-2-partition of G is a partition of its vertices into two disjoint dominating sets. In this paper, for a finite simple connected graph G, we construct a graph dynamical system F and show that the set of dominating sets of G are in one-to-one correspondence with the image of F. Moreover, we obtain the set of all domatic-2-partitions of G from the set of all periodic orbits of F. Finally, we extended actions of two dynamical systems  to an action of a free semigroup on two letters, and determine independent dominating sets and idomatic partitions using its maximal invariant subset with a reversible action.

Description: We first study actions of monoids on finite sets and define Burnside ring of a monoid by using homotopy theory, which we applied to theory of finite automata. Later we study actions of monoidal categories on categories and their applications on stabilization, cohomology and homology theories. 

Description: By comparing Adams and James spectral sequences, we compute cobordism groups of manifolds that have normal bundles lifting representation bundles over BZ/3 up to dimension 10. We also determine manifolds representing the generators of these cobordism groups.