PLETHYSTIC ALGEBRA Introduction Consider an example from
... In section four, we discuss the restriction, induction, and co-induction functors for a morphism P → Q of plethories, and we state the reconstruction theorem. As always, the content of such a theorem is entirely category theoretic (Beck’s theorem). All the same, the result is worth stating: Theorem. ...
... In section four, we discuss the restriction, induction, and co-induction functors for a morphism P → Q of plethories, and we state the reconstruction theorem. As always, the content of such a theorem is entirely category theoretic (Beck’s theorem). All the same, the result is worth stating: Theorem. ...
On the continuity of the inverses of strictly monotonic
... and embed the results of the preceding section into a more general framework in order to supply the topological background underlying these results. In particular, we want to explore whether there exists a reasonable generalisation of Proposition 2.1. As we shall see, it turns out that there is inde ...
... and embed the results of the preceding section into a more general framework in order to supply the topological background underlying these results. In particular, we want to explore whether there exists a reasonable generalisation of Proposition 2.1. As we shall see, it turns out that there is inde ...
FIBRATIONS AND HOMOTOPY COLIMITS OF
... the functor obtained by applying Kan’s Ex∞ functor [7] at each b ∈ B, and L2 denotes the simplicial prolongation sPshB → sShB of the sheafification functor. (2) a local fibration if p∗ f (b) : p∗ X(b) → p∗ Y (b) is a Kan fibration for each b ∈ B. It should be pointed out that local fibrations are no ...
... the functor obtained by applying Kan’s Ex∞ functor [7] at each b ∈ B, and L2 denotes the simplicial prolongation sPshB → sShB of the sheafification functor. (2) a local fibration if p∗ f (b) : p∗ X(b) → p∗ Y (b) is a Kan fibration for each b ∈ B. It should be pointed out that local fibrations are no ...
- Departament de matemàtiques
... category with two compatible strict monoidal structures, and the Eckmann-Hilton argument shows that such are commutative. It is natural to consider the non-strict version of this situation, i.e. a category equipped with two (non-strict) monoidal structures which are compatible up to coherent isomorp ...
... category with two compatible strict monoidal structures, and the Eckmann-Hilton argument shows that such are commutative. It is natural to consider the non-strict version of this situation, i.e. a category equipped with two (non-strict) monoidal structures which are compatible up to coherent isomorp ...
Leon Henkin and cylindric algebras. In
... The purely algebraic theory of cylindric algebras, exclusive of set algebras and representation theory, is fully developed in [71]. The parts of this theory due at mainly to Henkin are as follows. If A is a CAα and Γ = {ξ(0), . . . , ξ(m − 1)} is a finite subset of α, then we define c(Γ) a = cξ(0) · ...
... The purely algebraic theory of cylindric algebras, exclusive of set algebras and representation theory, is fully developed in [71]. The parts of this theory due at mainly to Henkin are as follows. If A is a CAα and Γ = {ξ(0), . . . , ξ(m − 1)} is a finite subset of α, then we define c(Γ) a = cξ(0) · ...
www.math.uwo.ca
... defined explicitly in purely analytic terms, without reference to, but with guidance from, ...
... defined explicitly in purely analytic terms, without reference to, but with guidance from, ...
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.