Maths - Category Theory - Weak Equivilance

Weak equivilance provides a simpler way to determine equivilance.

On the equivilance page we discussed how to determine equivilance by finding two functors. F and G, which give the identity when composed in both directions. equivilance
Weak equivilance gives us a way to do this using only a functor in one direction F. However this only tells us that an equivilance exists, not the functors which give the equivilance. weak equivilance

Given this functor F : C -> D there is an equivalence if and only if the following are all true:

Weak Factorisation System

Example in Set

In a set category where morphisms are functions. Each function can be factored into injective and surjective (epi and monic) functions.

Or we can look at this as fibre and co-fibre. More about fibres on page here.

  diagramThe square commutes, the function from A to D factors so there must be a unique arrow (shown dotted) to connect them.

Example in Simplical Set

Here all functions are (weakly) order preserving. These can be factored into degenerate arrows (which increase dimension) and arrows which remove a dimension.

More about simplical sets on page here.


Further Information

Weak equivalence was first used in algebraic topology, in particular, in model category theory as described on the page here.

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see also:

Adjunctions from Morphisms

Correspondence about this page

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Terminology and Notation

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