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String topology and the based loop space.
String topology and the based loop space.

... A in terms of a model category structure on the category of A-modules, and we use twosided bar constructions as models for this homological algebra. Via Rothenberg-Steenrod constructions, we relate these algebraic constructions to the topological setting. Additionally, we state the key properties of ...
Aspects of topoi
Aspects of topoi

... A cartesian closed category is a finitely bicomplete category such that for every pair of objects representable. ...
Topology in the 20th century
Topology in the 20th century

... Jones (1990), Kontsevich (1998), whose central mathematical contributions during those years relate to topology; also Kodaira (1950), Grothendieck (1966), Mumford (1974), Deligne (1978), Yau (1982) and Voevodsky (2002), whose work is at the crossroads of the ideas of topology, algebraic geometry and ...
Lecture 4 Super Lie groups
Lecture 4 Super Lie groups

... Theorem 3.4. The category of super Lie groups is equivalent to the category of super Harish Chandra pairs. Roughly speaking this theorem says that each problem in the category of super Lie groups can be reformulated as an equivalent problem in the language of SHCP. We shall not prove this theorem h ...
derived smooth manifolds
derived smooth manifolds

... general cup product formula could be trivially attained. For example, one could extend Man by including nontransverse intersections which were given no more structure than their underlying space, and the derived cobordism relation could be chosen to be maximal (i.e., one equivalence class); then the ...
Topology Proceedings - topo.auburn.edu
Topology Proceedings - topo.auburn.edu

... closed subgroups and to those quotients which are complete, and it has a demonstrably good Lie theory. It is therefore indeed surprising that this class of groups has been little investigated in a systematic fashion. The first to recognize that a Lie algebra can be attached to a locally compact grou ...
Charged Spaces
Charged Spaces

... see §2). A solution Y to the unbased version of the fiberwise suspension problem need not admit a section B → Y , whereas of course a solution to the based version comes equipped with a section. Even in the case when X ∈ T (B × S 0 → B) is in the image of the forgetful functor R(B) → T (B × S 0 → B) ...
Homological algebra
Homological algebra

... MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to do this first so that grad students will be more familiar with the ideas when they are applied to a ...
Topological types of Algebraic stacks - IBS-CGP
Topological types of Algebraic stacks - IBS-CGP

... 1.2.11. For example, we can compute the topological type of the classifying stack BGm where Gm is the multiplicative group scheme over C. After profinite completion, the topological type h(BGm ) is weakly equivalent to the classifying space BS 1 of the unit circle, which is in turn weakly equivalent ...
Iterated Bar Complexes of E-infinity Algebras and Homology
Iterated Bar Complexes of E-infinity Algebras and Homology

... of linear homological algebra in the context of operads. Let R be any operad. In [20], we show that a functor SR (M, −) : R C → C is naturally associated to any right R-module M and all functors on R-algebras which are defined by composites of colimits and tensor products have this form. Let R = E o ...
COMPLETION FUNCTORS FOR CAUCHY SPACES
COMPLETION FUNCTORS FOR CAUCHY SPACES

... modification functors which are subject to the following additional conditions. ...
Soergel diagrammatics for dihedral groups
Soergel diagrammatics for dihedral groups

... planar strip R × [0, 1]. The edges of this graph are labelled by elements of S, which we call “colors.” The only vertices appearing are univalent vertices, trivalent vertices joining three edges of the same color, and if m is finite, vertices of valence 2m whose edge labels alternate between the two ...
EQUIVARIANT SYMMETRIC MONOIDAL STRUCTURES 1
EQUIVARIANT SYMMETRIC MONOIDAL STRUCTURES 1

... that H/K ⊂ F 0 and any X ∈ F 0 can be expressed as a disjoint union of copies of various H/K. These parameterize the norms that an F 0 -commutative monoid has. The remaining axioms for an indexing coefficient system guarantee that (1) M is a commutative monoid (in a traditional sense) and (2) norm m ...
Generalization at Higher Types
Generalization at Higher Types

... These examples show that while G may be more a suitable category in which to find maximal generalizations than T, it is not ideal. We can improve on G by restricting attention to only those generalizations which are relevant, where relevance means that each subterm is useful in forming the generaliz ...
Cloak and dagger
Cloak and dagger

... Frobenius law in physics Quantum field theory: replace particles by fields; state space varies over space-time I ...
Finite flat group schemes course
Finite flat group schemes course

... The example to bear in mind if you’re worrying about set-theoretic issues is the category of sets: there is no “set of all sets” because its subset, the set of all sets that don’t contain themselves as elements, gives an easy contradiction. However there is, as far as I am concerned, a category of a ...
Inverse semigroups and étale groupoids
Inverse semigroups and étale groupoids

... Pseudogroups of transformations are pseudogroups of partial homeomorphisms between the open subsets of a topological space. They play an important rôle in geometry. See page 110 of Three-dimensional geometry and topology by William P. Thurston. They are the origin of inverse semigroup theory. Impor ...
Syntactic categories and types: Ajdukiewicz and modern categorial
Syntactic categories and types: Ajdukiewicz and modern categorial

... 33 atomic types, e.g. π (subject), πk for k = 1, 2, 3 (k−th person subject), s (statement), s1 (statement in present tense), s2 (statement in past tense), n (name), n0 (mass noun), n1 (count noun), n2 (plural noun), n̄ (complete noun phrase), n̄k (k−th person complete noun phrase), and others. The ...
When are induction and conduction functors isomorphic
When are induction and conduction functors isomorphic

... arises : “if the functors Ind and Coind are isomorphic, does it follow that the ring R is strongly graded ?” A simple example (see Remark 3.3) shows that the answer to this question is negative. So we may ask this other question : “if R is a graded ring and the functors Ind and Coind are isomorphic, ...
a Quandle?
a Quandle?

Model Theory - Wilfrid Hodges
Model Theory - Wilfrid Hodges

... [203], Tarski tells us that by 1930 he had defined the relation of elementary equivalence, in modern symbols ≡: A ≡ B if the same first-order sentences are true in A as in B.) Nevertheless the quantifier elimination itself involves only one structure at a time, and any comparisons come later. There are ...
Duality between modal algebras and neighbourhood frames
Duality between modal algebras and neighbourhood frames

DIRECTED HOMOTOPY THEORY, II. HOMOTOPY CONSTRUCTS
DIRECTED HOMOTOPY THEORY, II. HOMOTOPY CONSTRUCTS

... The category of d-spaces is written as dTop. It has all limits and colimits, constructed as in Top and equipped with the initial or final d-structure for the structural maps; for instance a path I → ΠXi is directed if and only if all its components I → Xi are so. The forgetful functor U : dTop → Top ...
THE PICARD GROUP OF EQUIVARIANT STABLE HOMOTOPY
THE PICARD GROUP OF EQUIVARIANT STABLE HOMOTOPY

FORMAL PLETHORIES Contents 1. Introduction 3 1.1. Outline of the
FORMAL PLETHORIES Contents 1. Introduction 3 1.1. Outline of the

... Johnson, and Wilson made an attempt at incorporating the composition structure into the definition of a Hopf ring and called it an enriched Hopf ring [BJW95, Chapter 10], but stopped short of describing the full algebraic structure. For this, one needs the language of plethories. Plethories were fir ...
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Category theory



Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows (also called morphisms). A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Category theory can be used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Several terms used in category theory, including the term ""morphism"", are used differently from their uses in the rest of mathematics. In category theory, a ""morphism"" obeys a set of conditions specific to category theory itself. Thus, care must be taken to understand the context in which statements are made.
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