Talk:PlanetPhysics/Index of Category Theory

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\section{Index of Category Theory} \subsection{Foundamental Concepts and Axioms}

\subsubsection{Basic Definitions} \begin{enumerate} \item \htmladdnormallink{category theory}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item \htmladdnormallink{identity}{http://planetphysics.us/encyclopedia/Cod.html} \item arrow \item \htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item \htmladdnormallink{composition law}{http://planetphysics.us/encyclopedia/Identity2.html} \item \htmladdnormallink{commutativity}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item associativity \item axioms of category theory \item \htmladdnormallink{ETAC}{http://planetphysics.us/encyclopedia/ETACAxioms.html} \item triples \item \htmladdnormallink{Abelian group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} \item \htmladdnormallink{semigroup}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item \htmladdnormallink{monoid}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item ring \item \htmladdnormallink{diagram}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item precategory \item \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} \item \htmladdnormallink{alternative definition of category}{http://planetphysics.us/encyclopedia/AlternativeDefinitionOfCategory.html} \item subcategory \item \htmladdnormallink{small category}{http://planetphysics.us/encyclopedia/Cod.html} \item category of sets ($Set$, $Ens$) \item automorphism \item \htmladdnormallink{commutative diagram}{http://planetphysics.us/encyclopedia/Commutativity.html} \item concrete category \item dual category \item \htmladdnormallink{duality principle}{http://planetphysics.us/encyclopedia/DualityAndTriality.html} \item endomorphism \item epi \item monic \item extremal \htmladdnormallink{monomorphism}{http://planetphysics.us/encyclopedia/InjectiveMap.html} \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 \htmladdnormallink{types}{http://planetphysics.us/encyclopedia/Bijective.html} of morphisms \item wellpowered category \item zero object \item $\mathcal{U}$-small \item equalizer \item subobject \item \htmladdnormallink{quotient object}{http://planetphysics.us/encyclopedia/DenseSubcategory.html} \item direct product \item direct sum \item pullback \item \htmladdnormallink{pushout}{http://planetphysics.us/encyclopedia/Pushout.html} \item direct limit \item limiting cone \item cocone and colimit \htmladdnormallink{functor}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item complete category \item groupoid (category theoretic) \end{enumerate}

\subsection{2-Categories and Generalizations} \begin{enumerate} \item functor \item autofunctor \item category \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} \item diagonal functor \item endofunctor \item forgetful functor \item \htmladdnormallink{identity functor}{http://planetphysics.us/encyclopedia/IdentityMorphism.html} \item isomorphism \item multifunctor \item \htmladdnormallink{natural transformation}{http://planetphysics.us/encyclopedia/VariableCategory2.html} \item \htmladdnormallink{surjective}{http://planetphysics.us/encyclopedia/BCConjecture.html} maps \item univalent functors \item faithful functor \item full functor \item \htmladdnormallink{natural equivalence}{http://planetphysics.us/encyclopedia/IsomorphismClass.html} \item \htmladdnormallink{adjoint functor}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} \item equivalence of categories \item isomorphic categories \item universal property \item \htmladdnormallink{representable functor}{http://planetphysics.us/encyclopedia/CategoryOfLogicAlgebras.html} \item Equivalent definition of a Representable Functor \item \htmladdnormallink{simplicial}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} object \end{enumerate}

\subsection{Fundamental Theorems} \begin{itemize} \item Yoneda-Grothendieck lemma \item properties of monomorphisms and \htmladdnormallink{epimorphisms}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} \item properties of \htmladdnormallink{regular}{http://planetphysics.us/encyclopedia/CoIntersections.html} 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 \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} spaces \item category of pointed topological spaces \item simplicial category \item \htmladdnormallink{category of groupoids}{http://planetphysics.us/encyclopedia/GroupoidCategory.html} \item category associated to a partial order \item category of \htmladdnormallink{matrices}{http://planetphysics.us/encyclopedia/Matrix.html} \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 \htmladdnormallink{enriched category}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item \htmladdnormallink{algebraic category}{http://planetphysics.us/encyclopedia/CategoryOfLogicAlgebras.html} \item Logic category \item \htmladdnormallink{quantum logic}{http://planetphysics.us/encyclopedia/LQG2.html} category \item \htmladdnormallink{quantum topos}{http://planetphysics.us/encyclopedia/QuantumCategories.html} \end{enumerate}

\subsection{Algebraic categories} \begin{enumerate} \item \htmladdnormallink{algebra formed from a category}{http://planetphysics.us/encyclopedia/AlgebraFormedFromACategory.html} \item \htmladdnormallink{monad}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} \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 \htmladdnormallink{additive category}{http://planetphysics.us/encyclopedia/DenseSubcategory.html} \item \htmladdnormallink{abelian category}{http://planetphysics.us/encyclopedia/AbelianCategory2.html} \item supplemental axioms for an Abelian category \item exact sequence \item exact functor \item Grothendieck spectral sequence \item enough projectives \item enough \htmladdnormallink{injectives}{http://planetphysics.us/encyclopedia/BCConjecture.html} \item \htmladdnormallink{projective object}{http://planetphysics.us/encyclopedia/ProjectiveObject.html} \item injective object \item derived functor \item derived category \item \htmladdnormallink{algebraic K-theory}{http://planetphysics.us/encyclopedia/PAdicMeasure.html} \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 proof of snake lemma \item chain \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} \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 \htmladdnormallink{presheaf}{http://planetphysics.us/encyclopedia/CommutativeRingWithUnit.html} \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 \htmladdnormallink{topos}{http://planetphysics.us/encyclopedia/GrothendieckTopos.html} \item cosmos \item Heyting logic algebra \item subobject classifier \item well-pointed topos \item \htmladdnormallink{power}{http://planetphysics.us/encyclopedia/Power.html} object \item exponential object \item Cartesian closed category \item natural numbers object \end{enumerate}

{\bf This is a contributed entry in progress...}

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