Maths - Category Theory - Comparing Objects

When comparing objects it is often more interesting when two sets are not identical but preserve some form of structure when mapping between them.

adjunction

relative strength type and category notation description
tightest Equality C and D are the same in these terms
  Isomorphism
1d = GF
FG = 1c
There must be an arrow and inverse between the objects and when composed these give the identity map for every element.
  Equivalence
1d≡GF
FG≡1c

equivalence triangle

For equivalence there is a arrow in both directions between the two objects. It is not necessary that GF and FG are the identity elements but they must be isomorphic to the identity elements.

loosest

Adjunctions

adjunctions
unit:
μ : 1c =>GF
co-unit:
ξ : FG => 1d

triangle identities: as 2-cells:
adjunction triangle identity adjunction 2-cell

For adjunctions there is a arrow in both directions between the two objects. It is not necessary that GF and FG are the identity elements but only that they have natural transformations to the identity elements.

FG is unit (does not change object - injective followed by surjective) but GF does change object (surjective followed by injective). Like equivalence but in one direction.

Alternative definition: there is an isomorphism:
φ: D(FA , X) → C(A, GX)
for every A∈C and X∈D

Equivalence Relations

In a less category specific approach to binary relations they may, or may not, have the following properties:

  Equivalence Relation Group
  x≡y equivilance group
transitive
x≡y   y≡z
x≡z
equivilance group
reflexive x≡x equivilance group
symmetric
x≡y
y≡x
equivilance group

Comparison of Sets

Here we investigate this comparisons with the simplest concrete categories - sets.

type of
comparison

Sets GF
Equality equality of sets sets equality GF
Isomorphism
1d = GF
FG = 1c
compare sets isomorphism compare sets isomorphism GF
Equivalence
1d≡GF
FG≡1c
sets equivalence

sets equivalence GF

Adjunction

 

 

 

 


metadata block
see also:

Adjunctions from Morphisms

Correspondence about this page

Book Shop - Further reading.

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flag flag flag flag flag flag The Princeton Companion to Mathematics - This is a big book that attempts to give a wide overview of the whole of mathematics, inevitably there are many things missing, but it gives a good insight into the history, concepts, branches, theorems and wider perspective of mathematics. It is well written and, if you are interested in maths, this is the type of book where you can open a page at random and find something interesting to read. To some extent it can be used as a reference book, although it doesn't have tables of formula for trig functions and so on, but where it is most useful is when you want to read about various topics to find out which topics are interesting and relevant to you.

 

Terminology and Notation

Specific to this page here:

 

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