PlanetPhysics/Category

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The concept of category emerged in 1943-1945 from work in algebraic topology and Homological Algebra by S. Eilenberg and S. Mac Lane, as a generalization of the algebraic concepts of semigroup, monoid, group, groupoid, etc., as well as an extension of topological concepts and diagrams employed in algebraic topology and homological algebra. Thus many properties of mathematical systems can be unified by a presentation with diagrams of arrows that may represent functions, transformations, distributions, operators, etc., and that-- in the case of concrete categories-- may also include objects such as class elements, sets, topological spaces, etc. ; the usefulness of such diagrams comes from the composition of the arrows and the (fundamental) axioms that define any category which allow mathematical constructions to be represented by universal properties of diagrams.

Definitions
To introduce the modern concept of category, according to S. MacLane without using any set theory, one needs to introduce first the notions of metagraph and metacategory.

A concrete metagraph $$\mathcal{M}_G$$ consists of objects, $$A, B, C,$$... and arrows $$f, g, h,$$... between objects, and two operations as follows:


 * a \htmladdnormallink{domain {http://planetphysics.us/encyclopedia/Bijective.html} operation}, $$dom$$, which assigns to each arrow $$f$$ an object $$A~ =~dom ~f$$
 * a \htmladdnormallink{codomain {http://planetphysics.us/encyclopedia/Bijective.html} operation}, $$cod$$, which assigns to each arrow $$f$$ an object $$B~ = ~cod ~f,$$ represented as $$f: A \to B$$ or $$A \stackrel{f}{\longrightarrow} B$$

A metacategory $$\mathbb{C}$$ is a metagraph with two additional operations:


 * Identity, $$id$$ or {\mathbf 1}, which assigns to each object $$A$$ a unique arrow $$id_A$$, or $$1_A$$;
 * Composition, $$\circ$$, which assigns to each pair of arrows $$$$ with $$dom~ g = cod~ f$$ a unique arrow $$g \circ f$$ called their composite , such that $$g \circ f : dom f \to cod g,$$

that are subject to two axioms:


 * c1. (Unit law) : for all arrows $$f: A \to B$$ and $$g:B \to C$$ the composition with the identity arrow $$1_B$$ results in $$ 1_B \circ f = f$$ and $$g \circ 1_B = g ;$$
 * c2. Associativity : for given objects and arrows in the (categorical) sequence: $$A \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow}  C \stackrel{h}{\longrightarrow}  D,  $$ one always the equality  $$ h \circ(g \circ f) =  (h \circ g) \circ f , $$ whenever the composition $$\circ$$ is defined.

A category $$\mathcal{C}$$ is an interpretation of a metacategory within set theory. Thus, a category is a graph  -- defined by a set $$Ob \mathcal{C}:=\mathbb{O}$$, a set of arrows* (called also morphisms) $$Mor\mathcal{C}:= \mathbb{A}$$, and two functions:

$$ dom: Mor \mathcal{C} \to Ob \mathcal{C}$$ and

$$cod: Mor\mathcal{C} \to Ob \mathcal{C},~$$ -- that also has two additional functions: $$id: Ob \mathcal{C} \to Mor \mathcal{C}$$ defined by the assignments $$\mathbb{A} \times_\mathbb{O} \mathbb{A} \longrightarrow \mathbb{A}$$ called identity, and a composition $$c = "\circ"$$, that is, $$ c \longmapsto id_c$$, defined by the assignments $$(g,f) \longmapsto g \circ f$$, such that: $$ dom(id_A) = A = cod(id_A), dom(g \circ f) = domf, cod (g \circ f)= codg,$$ for all objects $$A \in Ob \mathcal{C}$$ and all composable pairs of arrows (morphisms) $$(g,f) \in \mathbb{A} \times_\mathbb{O} \mathbb{A}, $$ and also such that the unit law and associativity axioms c1 and c2  hold.


 * Note that the set of all morphisms $$Mor~ \mathcal{C}$$ of a category $$\mathcal{C}$$ is sometimes denoted as $$\mathcal{M}$$, or in French publications as $$Fl ~ \mathcal{C}$$.

For convenience one also defines a $$Hom$$ (or $$hom$$) set as: $$Hom(B,C) := [f|f \in \mathcal{C}, dom f= B, cod f = C],$$ which is also denoted as $$[B,C]_\mathcal{C}$$, or simply $$[B,C].$$

Alternative definitions
There are several alternative definitions of a category. Thus, as defined by W.F. Lawvere, a category is an interpretation of the ETAC axioms from his elementary theory of abstract categories. For small categories-- whose $$Ob \mathcal{C}$$ is a set and also $$Mor\mathcal{C}$$ is a set-- one has a \htmladdnormallinkdirect definition. {http://planetmath.org/encyclopedia/AlternativeDefinitionOfSmallCategory.html}

If, on the other hand, $$Hom_\mathcal{C} (X,Y)$$ is a class rather than a set then the category $$\mathcal{C}$$ is called large.