index of algebraic topology IndexOfAlgebraicTopology 2009-05-02 15:49:11 2009-05-02 16:03:08 Topic HGvkT index algebraic topology \section{Foundations} \subsection{Basic Definitions} \begin{itemize} \item open and closed sets \item map \item arrow \item morphism \item path \item cycle \item boundary \item torus, n-tori \item sphere, n-sphere \item square \item cube \item dimension \item vector space \item CW-complex \item graph \item simplicial complex \item spin network \item topological dynamics \item qualitative dynamics \item dense space \item spin foam \item map \item function \item analytical space \item identity \item commutativity \item associativity \item object space \item source space \item target space \item thin square \item topological space \item homeomorphism \item topological groups \item Lie groups and Lie algebras \item graded lie algebras/Lie superalgebras \item supergroups \item groupoid \item groupoid homomorphism \item groupoid categories \item compact space/Hausdorff space \item double groupoid \item double algebroid \item Hamiltonian algebroid \item Polish space \item Polish group \item Polish G-space \item homotopy \item homotopy groups \item homotopy category \item homology cycle \item cohomology cocycle \item fundamental groups \item homotopy lemma and corollary \item topological groups \item fundamental groupoids \item fundamental groupoid functor \item homotopy double groupoid of a Hausdorff space \item homology theory \item axiomatic homology theory \item cohomology groups \item cohomology groupoids \item cohomology theory \item cohomology theorems \item topological categories \end{itemize} \subsection{Category theory concepts in algebraic topology} \begin{itemize} \item category \item topos \item subspace \item subcategory \item automorphism \item commutative diagram \item concrete category \item dual category \item duality principle \item endomorphism \item epimorphism \item monic \item monomorphism \item source \item pushout \item pullback \item cones and cocones \item limit and colimit functors \item direct sum \item open covers \item isomorphism-closed subcategory \item locally finite category \item preimage of category \item product of categories \item types of morphisms \item wellpowered category \item null object \item zero object \item \PMlinkid{$\mathcal{U}$-small}{5658} \item equalizer \item subobject \item quotient object \item categorical direct product \item categorical direct sum \item direct limit \item limiting cone \item complete category \item groupoid (category theoretic) \item enriched category \item double category \item 2-category \item n-category \end{itemize} \subsection{Fundamental Theorems} \begin{enumerate} \item Hurewicz theorem \item cohomology theorems \item homotopy lemma and corollary \item topological space approximation theorem \item Yoneda-Grothendieck Lemma \item van Kampen theorem \item Higher dimensional, generalized van Kampen theorem (HGvKT) \item properties of monomorphisms and epimorphisms \item properties of regular and extremal monomorphisms \item monomorphisms are pullback stable \item proof that an equalizer is a monomorphism \item categorical direct product is an inverse limit \item kernel is an inverse limit \item Grothendieck theorem \end{enumerate} \subsection{2-Categories and Supercategories} \begin{itemize} \item functor \item endofunctor \item category isomorphism \item diagonal functor \item representable functors \item categorical representations and supergroups \item forgetful functor \item identity functor \item isomorphism \item multifunctor \item natural transformation \item essentially surjective \item faithful functor \item full functor \item adjoint functor \item natural equivalence \item equivalence of categories \item \PMlinkname{isomorphic categories}{CategoryIsomorphism} \item universal properties \item representable functors \item Equivalent definition of a Representable Functor \item simplicial objects \end{itemize} \section{Additive Categories and Homology} \begin{enumerate} \item preadditive category \item additive category \item Abelian category \item Non-Abelian categories and non-Abelian Algebraic Topology \item supplemental axioms for an Abelian category \item exact sequence \item exact functor \item Grothendieck spectral sequence \item enough projectives \item enough injectives \item projective object \item injective object \item derived functor \item derived category \item Algebraic K-theory \item examples of algebraic K-theory groups \item Grothendieck group \item Grothendieck category \item delta functor \item horseshoe lemma \item syzygy \item Ext \item Tor \item projective dimension \item 5-lemma \item proof of 5-lemma \item 9-lemma \item snake lemma \item \PMlinkname{proof of snake lemma}{ProofOfSnakeLemma} \item chain homotopy \item chain homotopy equivalence \item chain map \item homology of a chain complex \item Leray spectral sequence \item spectral sequence \end{enumerate} \section{Sheaves, Topoi, Generalizations} \begin{enumerate} \item presheaf \item sheaf \item sheafification \item presheaf of a topological basis \item stalk \item \'Etal\'e space \item resolution of a sheaf \item gerbes \item site \item small site on a scheme \item topos \item cosmos \item subobject classifier \item well-pointed topos \item power object \item natural numbers object \item Cartesian closed category \item exponential object \end{enumerate} \subsection{Examples of Categories} \begin{enumerate} \item discrete category \item category example (arrow category) \item category associated to a partial order \item category of matrices \item Category of pseudomorphisms \item Category of intermorphisms \item examples of initial objects and terminal objects and zero objects \item category of sets \item monomorphisms of category of sets \item monoid as a category \item comma category \item category of pointed topological spaces \item simplicial category \end{enumerate} \subsection{Algebraic Categories} \begin{enumerate} \item algebras formed from a category \item monad \item comonad \item monoidal category \item group object \item nerve \end{enumerate} \subsection{Generalized Categorical Galois theory} \subsubsection{Categorical Galois theory and and Topological greoupoid category} \begin{itemize} \item category of topological groupoids \item Fundamental groupoid functors \end{itemize} \subsection{Nonabelian Algebraic Topology (NAAT)} \subsection{Nonabelian Quantum Algebraic Topology (NA-QAT)} \begin{itemize} \item \emph{Clifford algebras} \item \emph{Hopf algebras} \item \emph{Weak-Hopf algebras} \item \emph{Grasssmann-Hopf algebras} \item \end{itemize} \item Hamiltonian algebroids \item Lie algebroids \item Quantum algebroids \item Quantum Groups and Quantum Compact Groups (QCGs) \item Quantum Groupoids \subsubsection{Noncommutative Geometry (non-commutative geometry)}