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FinSet
In the mathematical field of category theory, FinSet is the category whose objects are all finite sets and whose morphisms are all functions between them. FinOrd is the category whose objects are all finite ordinal numbers and whose morphisms are all functions between them. Properties FinSet is a full subcategory of Set, the category whose objects are all sets and whose morphisms are all functions. Like Set, FinSet is a large category. FinOrd is a full subcategory of FinSet as by the standard definition, suggested by John von Neumann, each ordinal is the well-ordered set of all smaller ordinals. Unlike Set and FinSet, FinOrd is a small category. FinOrd is a skeleton of FinSet. Therefore, FinSet and FinOrd are equivalent categories. Topoi Like Set, FinSet and FinOrd are topoi. As in Set, in FinSet the categorical product of two objects ''A'' and ''B'' is given by the cartesian product , the categorical sum is given by the disjoint union , and the exponential object ''B''''A' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PRO (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. Because of “symmetric”, for each ''n'', the symmetric group on ''n'' letters is given as a subgroup of the automorphism group of ''n''. The name PROP is an abbreviation of "PROduct and Permutation category". The notion was introduced by Adams and MacLane; the topological version of it was later given by Boardman and Vogt. Following them, J. P. May then introduced the notion of “operad”, a particular kind of PROP. There are the following inclusions of full subcategories: pg 45 :\mathsf \subset \tfrac\mathsf \subset \mathsf where the first category is the category of (symmetric) operads. Examples and variants An important ''elementary'' class of PROPs are the sets \mathcal^ of ''all'' matrices (regardless of number of rows and colum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the '' cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an ''infinite set''. For example, the set of all positive integers is infinite: :\. Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set is called finite if there exists a bijection :f\colon S\to\ for some natural number . The number is the set's cardinality, denoted as . The empty set o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lawvere Theory
In category theory, a Lawvere theory (named after United States, American mathematician William Lawvere) is a category (mathematics), category that can be considered a categorical counterpart of the notion of an equational theory. Definition Let \aleph_0 be a skeleton (category_theory), skeleton of the category FinSet of finite sets and function (mathematics), functions. Formally, a Lawvere theory consists of a small category ''L'' with (strictly associativity, associative) finite product (category theory), products and a strict identity-on-objects functor (category theory), functor I:\aleph_0^\text\rightarrow L preserving finite products. A model of a Lawvere theory in a category ''C'' with finite products is a finite-product preserving functor . A morphism of models where ''M'' and ''N'' are models of ''L'' is a natural transformation of functors. Category of Lawvere theories A map between Lawvere theories (''L'', ''I'') and (''L''′, ''I''′) is a finite-produc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skeleton (category Theory)
In mathematics, a skeleton of a category is a subcategory that, roughly speaking, does not contain any extraneous isomorphisms. In a certain sense, the skeleton of a category is the "smallest" equivalent category, which captures all "categorical properties" of the original. In fact, two categories are equivalent if and only if they have isomorphic skeletons. A category is called skeletal if isomorphic objects are necessarily identical. Definition A skeleton of a category ''C'' is an equivalent category ''D'' in which no two distinct objects are isomorphic. It is generally considered to be a subcategory. In detail, a skeleton of ''C'' is a category ''D'' such that: * ''D'' is a subcategory of ''C'': every object of ''D'' is an object of ''C'' :\mathrm(D)\subseteq \mathrm(C) for every pair of objects ''d''1 and ''d''2 of ''D'', the morphisms in ''D'' are morphisms in ''C'', i.e. :\mathrm_D(d_1, d_2) \subseteq \mathrm_C(d_1, d_2) and the identities and compositions in ''D'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Sum
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products. Definition Let C be a category and let X_1 and X_2 be objects of C. An object is called the coproduct of X_1 and X_2, written X_1 \sqcup X_2, or X_1 \oplus X_2, or sometimes simply X_1 + X_2, if there exist morphisms i_1 : X_1 \to X_1 \sqcup X_2 and i_2 : X_2 \to X_1 \sqcup X_2 satisfying the following universal property: for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number Object
In category theory, a natural numbers object (NNO) is an object endowed with a recursive structure similar to natural numbers. More precisely, in a category E with a terminal object 1, an NNO ''N'' is given by: # a global element ''z'' : 1 → ''N'', and # an arrow ''s'' : ''N'' → ''N'', such that for any object ''A'' of E, global element ''q'' : 1 → ''A'', and arrow ''f'' : ''A'' → ''A'', there exists a unique arrow ''u'' : ''N'' → ''A'' such that: # ''u'' ∘ ''z'' = ''q'', and # ''u'' ∘ ''s'' = ''f'' ∘ ''u''. In other words, the triangle and square in the following diagram commute. The pair (''q'', ''f'') is sometimes called the ''recursion data'' for ''u'', given in the form of a recursive definition: # ⊢ ''u'' (''z'') = ''q'' # ''y'' ∈E ''N'' ⊢ ''u'' (''s'' ''y'') = ''f'' (''u'' (''y'')) The above definition is the universal property of NNOs, meaning they are defined up to canonical isomorphism. If the arrow ''u'' as defined above merely has to exist, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Set Theory
General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms. Ontology The ontology of GST is identical to that of ZFC, and hence is thoroughly canonical. GST features a single primitive ontological notion, that of set, and a single ontological assumption, namely that all individuals in the universe of discourse (hence all mathematical objects) are sets. There is a single primitive binary relation, set membership; that set ''a'' is a member of set ''b'' is written ''a ∈ b'' (usually read "''a'' is an element of ''b''"). Axioms The symbolic axioms below are from Boolos (1998: 196), and govern how sets behave and interact. As with Z, the background logic for GST is first order logic with identity. Indeed, GST is the fragment of Z obtained by omitting the axioms Union, Power Set, Elementary Se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subobject Classifier
In category theory, a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object ''X'' in the category correspond to the morphisms from ''X'' to Ω. In typical examples, that morphism assigns "true" to the elements of the subobject and "false" to the other elements of ''X.'' Therefore, a subobject classifier is also known as a "truth value object" and the concept is widely used in the categorical description of logic. Note however that subobject classifiers are often much more complicated than the simple binary logic truth values . Introductory example As an example, the set Ω = is a subobject classifier in the category of sets and functions: to every subset ''A'' of ''S'' defined by the inclusion function '' j '' : ''A'' → ''S'' we can assign the function ''χA'' from ''S'' to Ω that maps precisely the elements of ''A'' to 1 (see characteristic function). Every function from ''S'' to Ω arises in this fashion from prec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Exponentiation
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations. Addition The union of two disjoint well-ordered sets ''S'' and ''T'' can be well-ordered. The order-type of that union is the ordinal that results from adding the order-types of ''S'' and ''T''. If two well-ordered sets are not already disjoint, then they can be replaced by order-isomorphic disjoint sets, e.g. replace ''S'' by × ''S'' and ''T'' by × ''T''. This way, the well-ordered set ''S'' is written "to the left" of the well-ordere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Addition
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations. Addition The union of two disjoint well-ordered sets ''S'' and ''T'' can be well-ordered. The order-type of that union is the ordinal that results from adding the order-types of ''S'' and ''T''. If two well-ordered sets are not already disjoint, then they can be replaced by order-isomorphic disjoint sets, e.g. replace ''S'' by × ''S'' and ''T'' by × ''T''. This way, the well-ordered set ''S'' is written "to the left" of the well-ordered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Of Ordinals
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations. Addition The union of two disjoint well-ordered sets ''S'' and ''T'' can be well-ordered. The order-type of that union is the ordinal that results from adding the order-types of ''S'' and ''T''. If two well-ordered sets are not already disjoint, then they can be replaced by order-isomorphic disjoint sets, e.g. replace ''S'' by × ''S'' and ''T'' by × ''T''. This way, the well-ordered set ''S'' is written "to the left" of the well-ordered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
