Difference between revisions of "Isomorphism (category theory)"

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1 Definition
2 See also
3 References

Definition

Given a pair of arrows, $A\mathop{\longrightarrow}^fB$ and $B\mathop{\longrightarrow}^g A$ in a category $\mathscr{C}$ , we say that $f$ and $g$ are isomorphisms^[1] if:

$g\circ f=\text{Id}_A$ and $f\circ g=\text{Id}_B$

We may also call $f$ and $g$ an inverse pair of isomorphisms. For clarity I say again: both $f$ and $g$ are themselves isomorphisms

References

Jump up ↑ 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	$\mathrm{SET}$ , $\mathrm{Pfn}$ , $\mathrm{GROUP}$

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Difference between revisions of "Isomorphism (category theory)"

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Revision as of 14:17, 13 March 2016 (view source) Alec (Talk \| contribs) (Created page with "{{Stub page\|Needs proof-reading and checking, created to reduce the clutter in the "wanted pages" list}} __TOC__ ==Definition== Given a pair of arrows, {{M\|A\mathop{...")	Latest revision as of 14:19, 13 March 2016 (view source) Alec (Talk \| contribs) m (Alec moved page Isomorphism to Isomorphism (category theory) without leaving a redirect: There are loads of different isomorphisms, while this superceeds all, they're still needed)
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