Simplex category, the Glossary
In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps.[1]
Table of Contents
16 relations: Abstract simplicial complex, Adjoint functors, Category theory, Empty set, Functor category, Mathematics, Monad (category theory), Monoid (category theory), Monoidal category, Monotonic function, Morphism, Ordinal number, Presheaf (category theory), PROP (category theory), Simplicial category, Simplicial set.
- Categories in category theory
- Free algebraic structures
- Simplicial sets
Abstract simplicial complex
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. Simplex category and abstract simplicial complex are algebraic topology and simplicial sets.
See Simplex category and Abstract simplicial complex
Adjoint functors
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories.
See Simplex category and Adjoint functors
Category theory
Category theory is a general theory of mathematical structures and their relations.
See Simplex category and Category theory
Empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.
See Simplex category and Empty set
Functor category
In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in the category). Simplex category and functor category are categories in category theory.
See Simplex category and Functor category
Mathematics
Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
See Simplex category and Mathematics
Monad (category theory)
In category theory, a branch of mathematics, a monad is a triple (T, \eta, \mu) consisting of a functor T from a category to itself and two natural transformations \eta, \mu that satisfy the conditions like associativity.
See Simplex category and Monad (category theory)
Monoid (category theory)
In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category is an object M together with two morphisms. Simplex category and monoid (category theory) are categories in category theory.
See Simplex category and Monoid (category theory)
Monoidal category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor that is associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism.
See Simplex category and Monoidal category
Monotonic function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.
See Simplex category and Monotonic function
Morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces.
See Simplex category and Morphism
Ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
See Simplex category and Ordinal number
Presheaf (category theory)
In category theory, a branch of mathematics, a presheaf on a category C is a functor F\colon C^\mathrm\to\mathbf.
See Simplex category and Presheaf (category theory)
PROP (category theory)
In category theory, a branch of mathematics, a PROP is a symmetric strict monoidal category whose objects are the natural numbers n identified with the finite sets \ and whose tensor product is given on objects by the addition on numbers.
See Simplex category and PROP (category theory)
Simplicial category
In mathematics, simplicial category may refer to.
See Simplex category and Simplicial category
Simplicial set
In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplex category and simplicial set are algebraic topology, Homotopy theory and simplicial sets.
See Simplex category and Simplicial set
See also
Categories in category theory
- Axiomatic foundations of topological spaces
- Category of Markov kernels
- Category of abelian groups
- Category of compactly generated weak Hausdorff spaces
- Category of groups
- Category of manifolds
- Category of matrices
- Category of measurable spaces
- Category of metric spaces
- Category of modules
- Category of preordered sets
- Category of rings
- Category of sets
- Category of small categories
- Category of topological spaces
- Category of topological vector spaces
- Comma category
- Connected category
- Cyclic category
- Derived category
- Differential graded category
- Discrete category
- FinSet
- FinVect
- Fukaya category
- Functor category
- Graded vector space
- Homotopy category
- Kleisli category
- Monoid (category theory)
- Pointed space
- Regular category
- Simplex category
- Small set (category theory)
- Super vector space
Free algebraic structures
- Free Boolean algebra
- Free Lie algebra
- Free abelian group
- Free algebra
- Free category
- Free convolution
- Free group
- Free independence
- Free lattice
- Free module
- Free monoid
- Free object
- Free product
- History monoid
- Polynomial ring
- Simplex category
- Stably free module
- Term algebra
- Trace monoid
Simplicial sets
- Abstract simplicial complex
- Clique complex
- Delta set
- Dendroidal set
- Discrete differential geometry
- Dold–Kan correspondence
- Fibration of simplicial sets
- Independence complex
- Kan fibration
- Nerve (category theory)
- Nerve complex
- Pro-simplicial set
- Regge calculus
- Segal space
- Simplex category
- Simplex tree
- Simplicial approximation theorem
- Simplicial complex
- Simplicial complex recognition problem
- Simplicial group
- Simplicial homotopy
- Simplicial manifold
- Simplicial map
- Simplicial presheaf
- Simplicial set
- Subdivision (simplicial complex)
- Symmetric spectrum
- Vietoris–Rips complex
References
[1] https://en.wikipedia.org/wiki/Simplex_category
Also known as Ordinal category.