Difference between revisions of "TOP (category)"

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Definition

$\mathrm{TOP}$ is the category of all topological spaces, the objects are tuples of a set $X$ and a topology $\mathcal{J}_X$ on $X$ and the arrows, or morphisms of the category are continuous functions^[1]. More explicitly.

The objects of $\mathrm{TOP}$ are all topological spaces, $(X,\mathcal{J}_X)$
The arrows/morphisms of $\mathrm{TOP}$ are the continuous functions between spaces.

Discussion

TODO: Discuss as a subcategory of $\mathrm{SET}$ , remember it must first go under the forgetful functor to discard the topological structure and distill it to just sets and mappings

References

<cite_references_link_accessibility_label> ↑ An Introduction to Category Theory - Harold Simmons - 1st September 2010 edition

v • d • e Category Theory

Overview of the concepts of Category Theory

Key objects	Category, Functor (Covariant, Contravariant), Subcategory	$\xymatrix{ & \text{Arrow} \\ \text{Monic} \ar@{^{(}->}[ur] & & \text{Epic} \ar@{^{(}->}[ul] \\ & \text{Bimorphism} \ar@{^{(}->}[ur] \ar@<-0.5ex>@{^{(}->}[ul] \\ {\begin{array}{c}\text{Section}\\ \text{(Split monic)} \end{array} } \ar@{^{(}->}[uu] & & {\begin{array}{c}\text{Retraction}\\ \text{(Split epic)} \end{array} } \ar@<-0.75ex>@{^{(}->}[uu] \\ & \text{Isomorphism} \ar@{^{(}->}[ur] \ar@<-0.5ex>@{^{(}->}[ul] \ar@{^{(}->}[uu] }$

Typical morphism types (see diagram on right)	Arrow (AKA: Morphism), Monic, Epic, Bimorphism, Section (AKA: Split monic), Retraction (AKA: Split epic), Isomorphism

Key objects	$\text{Initial}$ , $\text{Final}$ (compared)

Primitive constructs	Wedge (AKA Cone & Cocone)

Key constructs	Product / Coproduct (Product and coproduct compared), Limit / Colimit, Equaliser / Coequaliser

Important examples	Demonstrating why category arrows are best thought of as arrows and not functions

Trivial category examples	Category induced by a monoid, Category induced by a poset

Common categories	$\mathrm{SET}$ , $\mathrm{Pfn}$ , $\mathrm{GROUP}$

[Expand] v • d • e Topology

Overview of the structures studied in topology

Primitives	topological space, open set, neighbourhood

Global properties of topological spaces	Compactness, Connectedness (Path-Connectedness), Hausdorff-ness

Local properties of topological spaces

Constructs	product (see also box, difference between), subspace (AKA: induced), quotient, disjoint union, adjunction, cone over a topological space

Subtypes of topological spaces	inner product spaces $\subset$ normed spaces $\subset$ metric spaces $\subset$ topological spaces

Maps between topological spaces	Continuous map (AKA: topological homomorphism), topological homeomorphism

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Categories:

Hidden categories:

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Difference between revisions of "TOP (category)"

Latest revision as of 20:10, 20 February 2016

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