A Crash Course in Topological Groups
... The underlying additive group of a Banach space (or more generally, topological vector space): Rn , Lp (X , µ), C0 , Cc (X ), ... Groups of homeomorphisms of a ‘nice’ topological space, or diffeomorphisms of a smooth manifold, can be made into topological groups. Any group taken with the discrete to ...
... The underlying additive group of a Banach space (or more generally, topological vector space): Rn , Lp (X , µ), C0 , Cc (X ), ... Groups of homeomorphisms of a ‘nice’ topological space, or diffeomorphisms of a smooth manifold, can be made into topological groups. Any group taken with the discrete to ...
Cones on homotopy probability spaces
... of notation, this coalgebra automorphism will also be denoted ϕ. In [6, 4], ϕ is called the cumulant map. Most concepts and calculations for an L∞ algebra V can be transported by ϕ. Definition 2.4. Let V be a graded vector space with an associative product. Let f be a C-linear map SV → SV . We call ...
... of notation, this coalgebra automorphism will also be denoted ϕ. In [6, 4], ϕ is called the cumulant map. Most concepts and calculations for an L∞ algebra V can be transported by ϕ. Definition 2.4. Let V be a graded vector space with an associative product. Let f be a C-linear map SV → SV . We call ...
Topology A chapter for the Mathematics++ Lecture Notes
... The usual definition of continuity of a mapping from introductory courses uses the notion of distance: a mapping is continuous if the images of sufficiently close points are again close. This can be formalized for mappings between metric spaces. We recall that a metric space is a pair (X, dX ), wher ...
... The usual definition of continuity of a mapping from introductory courses uses the notion of distance: a mapping is continuous if the images of sufficiently close points are again close. This can be formalized for mappings between metric spaces. We recall that a metric space is a pair (X, dX ), wher ...
LOVELY PAIRS OF MODELS: THE NON FIRST ORDER CASE
... If this holds (the “good” case), then T P is simple as well, and provides an elegant means for the study of certain independence-related properties of T itself, as mentioned above. If this fails (the “bad” case), then the first order theory T P is pretty much useless. This was noticed by Poizat in t ...
... If this holds (the “good” case), then T P is simple as well, and provides an elegant means for the study of certain independence-related properties of T itself, as mentioned above. If this fails (the “bad” case), then the first order theory T P is pretty much useless. This was noticed by Poizat in t ...
Basic Category Theory
... (U is \upwards closed", or an \upper set"). This is a topology, called the Alexandro topology w.r.t. the order . If (X; ) and (Y; ) are two partially ordered sets, a function f : X ! Y is monotone for the orderings if and only if f is continuous for the Alexandro topologies. This gives an impor ...
... (U is \upwards closed", or an \upper set"). This is a topology, called the Alexandro topology w.r.t. the order . If (X; ) and (Y; ) are two partially ordered sets, a function f : X ! Y is monotone for the orderings if and only if f is continuous for the Alexandro topologies. This gives an impor ...
4. Sheaves Definition 4.1. Let X be a topological space. A presheaf
... f : X −→ Y and a sheaf morphism f # : OY −→ f∗ OX . A locally ringed space, is a ringed space (X, OX ) such that in addition every stalk OX,x of the structure sheaf is a local ring. A morphism of locally ringed spaces is a morphism of ringed spaces, such that for every point x ∈ X, the induced map f ...
... f : X −→ Y and a sheaf morphism f # : OY −→ f∗ OX . A locally ringed space, is a ringed space (X, OX ) such that in addition every stalk OX,x of the structure sheaf is a local ring. A morphism of locally ringed spaces is a morphism of ringed spaces, such that for every point x ∈ X, the induced map f ...
Lecture notes on Witt vectors
... w : WS (A) → AS is a natural transformation of functors from rings to rings. Proof. Let A be the polynomial ring Z[an , bn | n ∈ S]. Then the unique ring homomorphism φp : A → A that maps an to apn and bn to bpn satisfies that φp (f ) = f p modulo pA. Let a and b be the sequences (an | n ∈ S) and (b ...
... w : WS (A) → AS is a natural transformation of functors from rings to rings. Proof. Let A be the polynomial ring Z[an , bn | n ∈ S]. Then the unique ring homomorphism φp : A → A that maps an to apn and bn to bpn satisfies that φp (f ) = f p modulo pA. Let a and b be the sequences (an | n ∈ S) and (b ...
Topology of Open Surfaces around a landmark result of C. P.
... of a finite set of polynomials in n variables with coefficients in C. The set of all C-valued maps on X which can be represented by polynomials is denoted by k[X] and is called the coordinate ring of X. One may say that the geometry of X is completely encoded in the algebraic structure of the ring k ...
... of a finite set of polynomials in n variables with coefficients in C. The set of all C-valued maps on X which can be represented by polynomials is denoted by k[X] and is called the coordinate ring of X. One may say that the geometry of X is completely encoded in the algebraic structure of the ring k ...
Which spheres admit a topological group structure?
... G acts freely on X). Observation 2. Every topological group acts freely on itself. Indeed, since translations are homeomorphisms we can define the map L : G → Homeo(G) such that g 7→ Lg , which is an homomorphism. Moreover, Lg is a translation, so if g 6= e then Lg has no fixed points. The following ...
... G acts freely on X). Observation 2. Every topological group acts freely on itself. Indeed, since translations are homeomorphisms we can define the map L : G → Homeo(G) such that g 7→ Lg , which is an homomorphism. Moreover, Lg is a translation, so if g 6= e then Lg has no fixed points. The following ...
Separation of Variables and the Computation of Fourier
... Example 2.8 Note that the quiver of Figure 1 is not a Bratteli diagram. However, by removing the top arrow and adding an arrow from vertex 0 to the bottom vertex of grading 1, we produce a Bratteli diagram. Consider a group algebra chain C[Gn ] > C[Gn−1 ] > · · · > C[G1 ] > C[G0 ] = C. To associate ...
... Example 2.8 Note that the quiver of Figure 1 is not a Bratteli diagram. However, by removing the top arrow and adding an arrow from vertex 0 to the bottom vertex of grading 1, we produce a Bratteli diagram. Consider a group algebra chain C[Gn ] > C[Gn−1 ] > · · · > C[G1 ] > C[G0 ] = C. To associate ...
Algebraic models for higher categories
... In order to have a more algebraic model for fibrant objects we want to fix fillers for all diagrams. Definition 2.1. An algebraic fibrant object (of C) is an object X ∈ C together with a distinguished filler for each morphism h : Aj → X with j ∈ J. That means a morphism F (h) : Bj → X rendering diag ...
... In order to have a more algebraic model for fibrant objects we want to fix fillers for all diagrams. Definition 2.1. An algebraic fibrant object (of C) is an object X ∈ C together with a distinguished filler for each morphism h : Aj → X with j ∈ J. That means a morphism F (h) : Bj → X rendering diag ...
An algebraically closed field
... 4. Relative completeness. With the notation of §2, let s4 be a field-family with respect to F, and define a function v: ET{s4) -> Fu{oo} by setting v(x) equal to the first element of S(x) for x # 0, and by setting v(0) = oo. Under the conventions that oo = oo +00 = 00 + y > y for all y e F, v is a v ...
... 4. Relative completeness. With the notation of §2, let s4 be a field-family with respect to F, and define a function v: ET{s4) -> Fu{oo} by setting v(x) equal to the first element of S(x) for x # 0, and by setting v(0) = oo. Under the conventions that oo = oo +00 = 00 + y > y for all y e F, v is a v ...
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.