IndexOfCategoryTheory 2009-05-02 16:11:48 2009-05-02 16:24:55 Topic index of category theory {\bf This is a contributed entry in progress...} \section{Index of Category Theory} \subsection{Foundamental Concepts and Axioms} \subsubsection{Basic Definitions} \begin{enumerate} \item category theory \item object \item identity \item arrow \item morphism \item composition law \item commutativity \item associativity \item axioms of category theory \item ETAC \item triples \item Abelian group \item group \item groupoid \item semigroup \item monoid \item ring \item diagram \item precategory \item category \item alternative definition of category \item subcategory \item small category \item category of sets ($Set$, $Ens$) \item automorphism \item commutative diagram \item concrete category \item dual category \item duality principle \item endomorphism \item epi \item monic \item extremal monomorphism \item source \item sink \item initial source \item final sink \item isomorphism-closed subcategory \item locally finite category \item preimage of category \item product of categories \item types of morphisms \item wellpowered category \item zero object \item \PMlinkid{$\mathcal{U}$-small}{5658} \item equalizer \item subobject \item quotient object \item direct product \item direct sum \item pullback \item pushout \item direct limit \item limiting cone \item cocone and colimit functor \item complete category \item groupoid (category theoretic) \end{enumerate} \subsection{2-Categories and Generalizations} \begin{enumerate} \item functor \item autofunctor \item category isomorphism \item diagonal functor \item endofunctor \item forgetful functor \item identity functor \item isomorphism \item multifunctor \item natural transformation \item surjective maps \item univalent functors \item faithful functor \item full functor \item natural equivalence \item adjoint functor \item equivalence of categories \item \PMlinkname{isomorphic categories}{CategoryIsomorphism} \item universal property \item representable functor \item Equivalent definition of a Representable Functor \item simplicial object \end{enumerate} \subsection{Fundamental Theorems} \begin{itemize} \item Yoneda-Grothendieck lemma \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 \end{itemize} \subsection{Examples of Categories} \begin{enumerate} \item discrete category \item category example (arrow category) \item category of sets \item category of Abelian groups \item category of Topological spaces \item category of pointed topological spaces \item simplicial category \item category of groupoids \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 monoid as a category \item comma category \item enriched category \item Algebraic category \item Logic category \item Quantum Logic category \item Quantum Topos \end{enumerate} \subsection{Algebraic categories} \begin{enumerate} \item algebra formed from a category \item monad \item comonad \item monoidal category \item group object \item nerve \item gerbs \end{enumerate} \section{Additive Categories and Homology} \begin{enumerate} \item preadditive category \item additive category \item Abelian category \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 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, and the like} \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 site \item small site on a scheme \item topos \item cosmos \item Heyting logic algebra \item subobject classifier \item well-pointed topos \item power object \item exponential object \item Cartesian closed category \item natural numbers object \end{enumerate}