model categories of diagram spectra
... category of finite based sets, and W is the category of based spaces homeomorphic to finite CW complexes. We often use D generically to denote such a domain category for diagram spectra. When D = F or D = W , there is no distinction between Dspaces and D-spectra, DT = DS . The functors U are forgetf ...
... category of finite based sets, and W is the category of based spaces homeomorphic to finite CW complexes. We often use D generically to denote such a domain category for diagram spectra. When D = F or D = W , there is no distinction between Dspaces and D-spectra, DT = DS . The functors U are forgetf ...
Hopf algebras
... 1. The category Set whose objects are sets, and where the set of morphisms between two sets is given by all mappings between those sets. 2. Let k be a commutative ring, then Mk denotes the category with as objects all (right) kmodules, and with as morphisms between two k-modules all k-linear mapping ...
... 1. The category Set whose objects are sets, and where the set of morphisms between two sets is given by all mappings between those sets. 2. Let k be a commutative ring, then Mk denotes the category with as objects all (right) kmodules, and with as morphisms between two k-modules all k-linear mapping ...
The bounded derived category of an algebra with radical squared zero
... and a cycle if w is non-trivial, reduced and closed. The degree ∂(w) of a walk w is defined as follows. We first define ∂(w) = 0, 1, or −1 in case w is a trivial path, an arrow, or the inverse of an arrow respectively, and then extend this definition to all walks in Q by ∂(uv) = ∂(u) + ∂(v) whenever ...
... and a cycle if w is non-trivial, reduced and closed. The degree ∂(w) of a walk w is defined as follows. We first define ∂(w) = 0, 1, or −1 in case w is a trivial path, an arrow, or the inverse of an arrow respectively, and then extend this definition to all walks in Q by ∂(uv) = ∂(u) + ∂(v) whenever ...
Weakly Perfect Generalized Ordered Spaces
... nice spaces, e.g., among Lindelöf spaces or compact Hausdorff spaces. Kočinac [8] used the set-theoretic principle ♦ to construct a compact Hausdorff space that is weakly perfect but not perfect. Subsequently, R.W. Heath [7] showed that there is a ZFC example of a Lindelöf quasi-developable space ...
... nice spaces, e.g., among Lindelöf spaces or compact Hausdorff spaces. Kočinac [8] used the set-theoretic principle ♦ to construct a compact Hausdorff space that is weakly perfect but not perfect. Subsequently, R.W. Heath [7] showed that there is a ZFC example of a Lindelöf quasi-developable space ...
Introduction The following thesis plays a central role in deformation
... The following thesis plays a central role in deformation theory: (∗) If X is a moduli space over a field k of characteristic zero, then a formal neighborhood of any point x ∈ X is controlled by a differential graded Lie algebra. This idea was developed in unpublished work of Deligne, Drinfeld, and F ...
... The following thesis plays a central role in deformation theory: (∗) If X is a moduli space over a field k of characteristic zero, then a formal neighborhood of any point x ∈ X is controlled by a differential graded Lie algebra. This idea was developed in unpublished work of Deligne, Drinfeld, and F ...
HHG-published (pdf, 416 KiB) - Infoscience
... is an element of M.B; C /. A similar construction works in the case of left C –coactions. Using the universal property of the product, one can easily show that for any right C –coaction W B ! B C , D .B pr2 /: It is also immediately obvious that pr2 .B f / D f for all f 2 M.B; C /. Hence ...
... is an element of M.B; C /. A similar construction works in the case of left C –coactions. Using the universal property of the product, one can easily show that for any right C –coaction W B ! B C , D .B pr2 /: It is also immediately obvious that pr2 .B f / D f for all f 2 M.B; C /. Hence ...
The Coarse Baum-Connes Conjecuture for Relatively Hyperbolic
... Objects: Coarse equivalence classes of metric spaces. Morphisms: Hom(X, Y) = {f : X → Ycoarse map}/close. Definition The coarse K-homology KX∗ (−) is a coarse version of the K-homology which is a covariant functor from the category of coarse spaces to the category of abelian groups. Let X be a metri ...
... Objects: Coarse equivalence classes of metric spaces. Morphisms: Hom(X, Y) = {f : X → Ycoarse map}/close. Definition The coarse K-homology KX∗ (−) is a coarse version of the K-homology which is a covariant functor from the category of coarse spaces to the category of abelian groups. Let X be a metri ...
Group action
In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object. A group action formalizes of the relationship between the group and the symmetries of the object. It relates each element of the group to a particular transformation of the object.In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices. It is the same as a group action of G on an ordered basis of a vector space.A group action is an extension to the notion of a symmetry group in which every element of the group ""acts"" like a bijective transformation (or ""symmetry"") of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron.If G is a group and X is a set, then a group action may be defined as a group homomorphism h from G to the symmetric group on X. The action assigns a permutation of X to each element of the group in such a way that the permutation of X assigned to the identity element of G is the identity transformation of X; a product gk of two elements of G is the composition of the permutations assigned to g and k.The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Despite this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.