User:Lethe/list of categories

In category theory, categories are the main object of study. The following is a list of important categories, and a glossary of named categories.

what are the abbreviated names for these categories? I could take a guess

 * category of presheaves ( a functor category) and the category of sheaves Sh(X) and Psc(X)
 * category of CW complexes
 * category of complex manifolds
 * measure spaces
 * Cauchy spaces
 * Riemannian manifolds
 * projective spaces over K
 * Affine spaces over K
 * stein manifolds
 * category of structures for a given language
 * varieties (affine, quasi-affine, projective, quasi-projective). dually equivalent to fgDom

to be inserted

 * Tych
 * Kähler
 * Rel category of sets and relations between them (not concrete)
 * comUnC*Alg commutative unital C*-algebras with unital *-homomorphisms
 * compTopGrp compact topological groups
 * fdVectK finite dimensional vector spaces
 * CoAlgK coalgebras
 * BiAlgK bialgebras
 * FRL Fröhlicher spaces (ccc) a full subcategory of DSP
 * Frm
 * G-Set of group actions. a topos
 * TVS
 * LCTVS
 * CPO complete partial orders
 * DCPO
 * CABA complete atomic boolean algebras. dually equivalent to Set by some version of Stone
 * Prof the category of categories, profunctors, and natural tranformations
 * GRPO G-relative pushouts and RPO relative pushouts
 * Bun bunch contexts
 * Algτ algebras (in the sense of universal algebra) with signature τ.
 * Chu(V,k) Chu spaces over V valued in k

specific categories

 * Ab:  abelian groups
 * Adj: small categories and adjunctions between them
 * K-Alg: algebras over field K
 * AlgSet/K: algebraic sets with regular maps
 * Bool: Boolean algebras and their homomorphisms
 * CAb: compact topological abelian groups
 * Cat: all small categories with functors
 * CGHaus: compactly generated Hausdorff spaces
 * nCob: (n&minus;1)-dimensional manifolds with n-dimensional cobordisms
 * Comp: chain complexes
 * CompBool: complete Boolean algebras
 * CompHaus: compact Hausdorff spaces
 * Compmet: complete metric spaces
 * CRng: commutative rings
 * $$\mathfrak{Dif}$$: diffeological spaces
 * Diff or Smooth: smooth manifolds with smooth maps
 * Div: divisible abelian groups
 * Dom: integral domains
 * Domm: integral domains with monomorphisms
 * EnsV: sets and functions within a given universal set V
 * Euclid: Euclidean vector spaces with orthogonal transformations
 * Fin: The skeletal category of finite sets
 * Finord: finite ordinals
 * FinSet: finite sets with functions
 * Fld: fields with field homomorphisms
 * Grp: all groups with their group homomorphisms
 * Grph: directed graphs
 * Haus: Hausdorff spaces
 * Hilb: Hilbert spaces with linear maps
 * LCA: locally compact abelian groups with continuous homomorphisms
 * Lconn: locally connected spaces
 * LieGrp: Lie groups
 * Mag: The category consisting of all magmas with their homomorphisms
 * R-Mod: left R-modules
 * Mod-S: right S-modules
 * R-Mod-S: bimodules
 * MatrK: matrices over field (or sometimes ring) K
 * Med: The category consisting of all medial magmas with their homomorphisms
 * Met: all metric spaces with short maps
 * Mon: monoids with monoid homomorphisms
 * Moncat: monoidal categories and strict morphisms
 * Ord: all preordered sets with monotonic functions
 * Rng: rings
 * Sch: schemes
 * Ses-A: short exact sequences of A-modules
 * Set: sets with functions
 * Set*: pointed sets with base preserving functions
 * Smgrp: semigroups
 * StrAlgSet/K: structured algebraic sets
 * Top: all topological spaces with continuous functions
 * Toph: topological spaces with homotopy classes
 * TOP(X): open sets in the topological space X with inclusions
 * Uni: all uniform spaces with uniformly continuous functions
 * VectK or K-Vect: vector spaces over the field K (which is held fixed) with their K-linear maps
 * Vect(K,Z/2Z): Z2-graded vector spaces with graded maps
 * 0: The empty category
 * 1: The one object category with one morphism
 * 2: The two object category with one morphism not an identity between the distinct objects (this is the ordinal number 2 viewed as a category)
 * 3: The three object with one morphisms between each distinct pair of objects (this is the ordinal number 3 viewed as a category)
 * &darr;&darr;: The two object category with two morphisms from one object to the other

classes of categories

 * any preordered set is a category with elements for objects and "<" as the morphisms. It follows that any ordinal is a category.
 * any monoid is a category with one object and elements as morphisms
 * consequently any group is a category with one object with elements as morphisms and has the categorical property that all its morphisms are isomorphisms
 * a category is generated by any graph
 * given any set, the discrete category on that set has the elements as objects and only identity morphisms
 * given any category C, we may form the dual category Cop
 * given two categories C and D, we may form the product category CxD
 * given two categories C and D, we may form the functor category DC
 * given a category C and an object b of C, the comma category (b &darr; C) of objects under b is arrows from b.  The comma category (C &darr; b) of objects over b is  arrows to b.  More generally, given two functors F and G to C, one may form the comma category (F &darr; G)
 * assuming the axiom of choice, every category has a skeleton, whose objects are representatives of the isomorphism classes of the category.