Automating Algebraic Methods in Isabelle
... systems, and many system semantics are based on algebra. Many computational logics, for instance temporal, dynamic or Hoare logics, have algebraic siblings. Algebraic approaches offer simple abstract modelling languages, system analysis via equational reasoning, and a well developed meta-theory, nam ...
... systems, and many system semantics are based on algebra. Many computational logics, for instance temporal, dynamic or Hoare logics, have algebraic siblings. Algebraic approaches offer simple abstract modelling languages, system analysis via equational reasoning, and a well developed meta-theory, nam ...
C3.4b Lie Groups, HT2015 Homework 4. You
... 1Recall the centre of a group is Z(G) = {g ∈ G : hg = gh for all h ∈ G} = {g ∈ G : hgh−1 = g for all h ∈ G}. 2meaning continuous loops can always be continuously deformed to a point. ...
... 1Recall the centre of a group is Z(G) = {g ∈ G : hg = gh for all h ∈ G} = {g ∈ G : hgh−1 = g for all h ∈ G}. 2meaning continuous loops can always be continuously deformed to a point. ...
Topology of Rn - Will Rosenbaum
... 1 Basic Notions 1.1 Open sets In this section, we describe the standard topology on Rn . Definition 1. Suppose x ∈ Rn and r ∈ R with r > 0. We define the open ball of radius r centered at x to be B(x, r) = {y ∈ Rn | |y − x| < r} . Definition 2. We call a subset A ⊆ Rn open if for every x ∈ A, there ...
... 1 Basic Notions 1.1 Open sets In this section, we describe the standard topology on Rn . Definition 1. Suppose x ∈ Rn and r ∈ R with r > 0. We define the open ball of radius r centered at x to be B(x, r) = {y ∈ Rn | |y − x| < r} . Definition 2. We call a subset A ⊆ Rn open if for every x ∈ A, there ...
Factorization homology of stratified spaces
... map, one can then construct a link homology theory, via factorization homology with coefficients in this triple. This promises to provide a new source of such knot homology theories, similar to Khovanov homology. Khovanov homology itself does not fit into this structure, for a very simple reason: an ...
... map, one can then construct a link homology theory, via factorization homology with coefficients in this triple. This promises to provide a new source of such knot homology theories, similar to Khovanov homology. Khovanov homology itself does not fit into this structure, for a very simple reason: an ...
Categorically proper homomorphisms of topological groups
... Let us now consider any g : K → G with K c-compact. Then the image g(K) ≤ G is also c-compact and, in fact, compact, since it is Abelian. But G has no non-trivial compact subgroup, so g must be constant. Consequently, the pullback of g along f is the constant morphism K → Gd , and we can conclude th ...
... Let us now consider any g : K → G with K c-compact. Then the image g(K) ≤ G is also c-compact and, in fact, compact, since it is Abelian. But G has no non-trivial compact subgroup, so g must be constant. Consequently, the pullback of g along f is the constant morphism K → Gd , and we can conclude th ...
homotopy types of topological stacks
... one to transport homotopical information back and forth between the diagram and its homotopy type. The above theorem has various applications. For example, it implies an equivariant version of Theorem 1.1 for the (weak) action of a discrete group. It also allows one to define homotopy types of pairs ...
... one to transport homotopical information back and forth between the diagram and its homotopy type. The above theorem has various applications. For example, it implies an equivariant version of Theorem 1.1 for the (weak) action of a discrete group. It also allows one to define homotopy types of pairs ...
Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied both through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.