The goal of this paper is to explore basic topics in Algebraic Topology. The paper will first begin by exploring the concept of loops on topological spaces and how one can look at these loops and form a group. From there, it moves to discuss covering spaces giving basic definitions and constructing the universal cover. After establishing this, covering spaces are used to calculate the fundament...

Algebraic topology is full of computations with rings, and where we find rings we should seek geometry through methods of algebraic geometry. The geometry of formal varieties turn out to organize many interesting computations in topology, and certain formal varieties called commutative, one-dimensional formal groups give the best global picture of stable homotopy theory currently available. I w...

Algebraic data types and catamorphisms (folds) play a central role in functional programming as they allow programmers to define recursive data structures and operations on them uniformly by structural recursion. Likewise, in object-oriented (OO) programming, recursive hierarchies of object types with virtual methods play a central role for the same reason. There is a semantical correspondence ...

Algebraic data types and catamorphisms (folds) play a central role in functional programming as they allow programmers to define recursive data structures and operations on them uniformly by structural recursion. Likewise, in object-oriented (OO) programming, recursive hierarchies of object types with virtual methods play a central role for the same reason. There is a semantical correspondence ...

For us, G-bundles are taken in the fppf topology (or, equivalently, the étale topology). Remember that BunG depends on the choice of X, even though the notation suppresses this! Last time, Francois proved that BunGLn is an algebraic stack: • It satisfies descent in the fppf (equivalently, étale) topology; i.e., it’s a stack; • Its diagonal morphism is representable by an algebraic space (implyi...

All real algebraic varieties that appear in the present paper are assumed to be affine (that is, isomorphic to algebraic subsets of R for some n). Morphisms between real algebraic varieties are called regular maps. For background material on real algebraic geometry the reader may consult [6]. Every real algebraic variety carries also the Euclidean topology, induced by the usual metric topology ...

Suppose we have an equation with integer coefficients, e.g. y = x + x and we want to understand its solutions over a finite field. If we consider the solutions over C, the set of solutions is a complex manifold and can be studied using the powerful tools of complex analysis and algebraic topology. However, since the set of solutions over a finite field is also finite, and any obvious topology m...

Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff , the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of preshe...