categories of matroids - university of oxford
TRANSCRIPT
Interesting classes of matroids
4 / ??
■ Pointed matroids : distinguished element •,where {•} is dependent (i.e. • is a loop)
■ Loopless matroids : • is the only loop
■ Simple matroids : loopless with no minimaldependent sets (circuits) of cardinality 2
■ Representable matroid : those arising fromlinear dependence of vectors
Interesting classes of matroids
4 / ??
■ Pointed matroids : distinguished element •,where {•} is dependent (i.e. • is a loop)
■ Loopless matroids : • is the only loop
■ Simple matroids : loopless with no minimaldependent sets (circuits) of cardinality 2
■ Representable matroid : those arising fromlinear dependence of vectors
Interesting classes of matroids
4 / ??
■ Pointed matroids : distinguished element •,where {•} is dependent (i.e. • is a loop)
■ Loopless matroids : • is the only loop
■ Simple matroids : loopless with no minimaldependent sets (circuits) of cardinality 2
■ Representable matroid : those arising fromlinear dependence of vectors
Interesting classes of matroids
4 / ??
■ Pointed matroids : distinguished element •,where {•} is dependent (i.e. • is a loop)
■ Loopless matroids : • is the only loop
■ Simple matroids : loopless with no minimaldependent sets (circuits) of cardinality 2
■ Representable matroid : those arising fromlinear dependence of vectors
What are the morphisms?
6 / ??
Rank of a set X: Size of the largest independentset contained in XClosed sets (or flats): Maximal sets of a fixedrank r
Strong maps : Functions between matroids forwhich the preimage of a closed set is closed
What are the morphisms?
6 / ??
Rank of a set X: Size of the largest independentset contained in X
Closed sets (or flats): Maximal sets of a fixedrank r
Strong maps : Functions between matroids forwhich the preimage of a closed set is closed
What are the morphisms?
6 / ??
Rank of a set X: Size of the largest independentset contained in XClosed sets (or flats): Maximal sets of a fixedrank r
Strong maps : Functions between matroids forwhich the preimage of a closed set is closed
What are the morphisms?
6 / ??
Rank of a set X: Size of the largest independentset contained in XClosed sets (or flats): Maximal sets of a fixedrank r
Strong maps : Functions between matroids forwhich the preimage of a closed set is closed
Limits and Colimits
7 / ??
Matroids• Matroids Loopless• Simple Simple• Repr. over k Repr. over k•
Limits:
Products ✗ ✗ ✗ ✗ ✗ ✗ ✗Exponentials ✗ ✗ ✗ ✗ ✗ ✗ ✗Pullbacks ✗ ✗ ✗ ✗ ✗ ✗ ✗Equalisers ✓ ✓ ✓ ✓ ✓ ✓ ✓
Colimits:
Coproducts ✓ ✓ ✓ ✓ ✓ ✓ ✓Coequalisers ✗ ✗ ✗ ✗ ✗ ✗ ✗Pushouts ✗ ✗ ✗ ✗ ✗ ✗* ✗*
Limits and Colimits
7 / ??
Matroids• Matroids Loopless• Simple Simple• Repr. over k Repr. over k•
Limits:
Products ✗ ✗ ✗ ✗ ✗ ✗ ✗Exponentials ✗ ✗ ✗ ✗ ✗ ✗ ✗Pullbacks ✗ ✗ ✗ ✗ ✗ ✗ ✗Equalisers ✓ ✓ ✓ ✓ ✓ ✓ ✓
Colimits:
Coproducts ✓ ✓ ✓ ✓ ✓ ✓ ✓Coequalisers ✗ ✗ ✗ ✗ ✗ ✗ ✗Pushouts ✗ ✗ ✗ ✗ ✗ ✗* ✗*
Limits and Colimits
7 / ??
Matroids• Matroids Loopless• Simple Simple• Repr. over k Repr. over k•
Limits:
Products
✗ ✗ ✗ ✗ ✗ ✗ ✗Exponentials ✗ ✗ ✗ ✗ ✗ ✗ ✗Pullbacks ✗ ✗ ✗ ✗ ✗ ✗ ✗Equalisers ✓ ✓ ✓ ✓ ✓ ✓ ✓
Colimits:
Coproducts ✓ ✓ ✓ ✓ ✓ ✓ ✓Coequalisers ✗ ✗ ✗ ✗ ✗ ✗ ✗Pushouts ✗ ✗ ✗ ✗ ✗ ✗* ✗*
Limits and Colimits
7 / ??
Matroids• Matroids Loopless• Simple Simple• Repr. over k Repr. over k•
Limits:
Products ✗ ✗ ✗ ✗ ✗ ✗ ✗
Exponentials ✗ ✗ ✗ ✗ ✗ ✗ ✗Pullbacks ✗ ✗ ✗ ✗ ✗ ✗ ✗Equalisers ✓ ✓ ✓ ✓ ✓ ✓ ✓
Colimits:
Coproducts ✓ ✓ ✓ ✓ ✓ ✓ ✓Coequalisers ✗ ✗ ✗ ✗ ✗ ✗ ✗Pushouts ✗ ✗ ✗ ✗ ✗ ✗* ✗*
Limits and Colimits
7 / ??
Matroids• Matroids Loopless• Simple Simple• Repr. over k Repr. over k•
Limits:
Products ✗ ✗ ✗ ✗ ✗ ✗ ✗Exponentials ✗ ✗ ✗ ✗ ✗ ✗ ✗Pullbacks ✗ ✗ ✗ ✗ ✗ ✗ ✗
Equalisers ✓ ✓ ✓ ✓ ✓ ✓ ✓
Colimits:
Coproducts ✓ ✓ ✓ ✓ ✓ ✓ ✓Coequalisers ✗ ✗ ✗ ✗ ✗ ✗ ✗Pushouts ✗ ✗ ✗ ✗ ✗ ✗* ✗*
Limits and Colimits
7 / ??
Matroids• Matroids Loopless• Simple Simple• Repr. over k Repr. over k•
Limits:
Products ✗ ✗ ✗ ✗ ✗ ✗ ✗Exponentials ✗ ✗ ✗ ✗ ✗ ✗ ✗Pullbacks ✗ ✗ ✗ ✗ ✗ ✗ ✗Equalisers
✓ ✓ ✓ ✓ ✓ ✓ ✓
Colimits:
Coproducts ✓ ✓ ✓ ✓ ✓ ✓ ✓Coequalisers ✗ ✗ ✗ ✗ ✗ ✗ ✗Pushouts ✗ ✗ ✗ ✗ ✗ ✗* ✗*
Limits and Colimits
7 / ??
Matroids• Matroids Loopless• Simple Simple• Repr. over k Repr. over k•
Limits:
Products ✗ ✗ ✗ ✗ ✗ ✗ ✗Exponentials ✗ ✗ ✗ ✗ ✗ ✗ ✗Pullbacks ✗ ✗ ✗ ✗ ✗ ✗ ✗Equalisers ✓ ✓ ✓ ✓ ✓ ✓ ✓
Colimits:
Coproducts ✓ ✓ ✓ ✓ ✓ ✓ ✓Coequalisers ✗ ✗ ✗ ✗ ✗ ✗ ✗Pushouts ✗ ✗ ✗ ✗ ✗ ✗* ✗*
Limits and Colimits
7 / ??
Matroids• Matroids Loopless• Simple Simple• Repr. over k Repr. over k•
Limits:
Products ✗ ✗ ✗ ✗ ✗ ✗ ✗Exponentials ✗ ✗ ✗ ✗ ✗ ✗ ✗Pullbacks ✗ ✗ ✗ ✗ ✗ ✗ ✗Equalisers ✓ ✓ ✓ ✓ ✓ ✓ ✓
Colimits:
Coproducts ✓ ✓ ✓ ✓ ✓ ✓ ✓Coequalisers ✗ ✗ ✗ ✗ ✗ ✗ ✗Pushouts ✗ ✗ ✗ ✗ ✗ ✗* ✗*
Limits and Colimits
7 / ??
Matroids• Matroids Loopless• Simple Simple• Repr. over k Repr. over k•
Limits:
Products ✗ ✗ ✗ ✗ ✗ ✗ ✗Exponentials ✗ ✗ ✗ ✗ ✗ ✗ ✗Pullbacks ✗ ✗ ✗ ✗ ✗ ✗ ✗Equalisers ✓ ✓ ✓ ✓ ✓ ✓ ✓
Colimits:
Coproducts
✓ ✓ ✓ ✓ ✓ ✓ ✓Coequalisers ✗ ✗ ✗ ✗ ✗ ✗ ✗Pushouts ✗ ✗ ✗ ✗ ✗ ✗* ✗*
Limits and Colimits
7 / ??
Matroids• Matroids Loopless• Simple Simple• Repr. over k Repr. over k•
Limits:
Products ✗ ✗ ✗ ✗ ✗ ✗ ✗Exponentials ✗ ✗ ✗ ✗ ✗ ✗ ✗Pullbacks ✗ ✗ ✗ ✗ ✗ ✗ ✗Equalisers ✓ ✓ ✓ ✓ ✓ ✓ ✓
Colimits:
Coproducts ✓ ✓ ✓ ✓ ✓ ✓ ✓
Coequalisers ✗ ✗ ✗ ✗ ✗ ✗ ✗Pushouts ✗ ✗ ✗ ✗ ✗ ✗* ✗*
Limits and Colimits
7 / ??
Matroids• Matroids Loopless• Simple Simple• Repr. over k Repr. over k•
Limits:
Products ✗ ✗ ✗ ✗ ✗ ✗ ✗Exponentials ✗ ✗ ✗ ✗ ✗ ✗ ✗Pullbacks ✗ ✗ ✗ ✗ ✗ ✗ ✗Equalisers ✓ ✓ ✓ ✓ ✓ ✓ ✓
Colimits:
Coproducts ✓ ✓ ✓ ✓ ✓ ✓ ✓Coequalisers
✗ ✗ ✗ ✗ ✗ ✗ ✗Pushouts ✗ ✗ ✗ ✗ ✗ ✗* ✗*
Limits and Colimits
7 / ??
Matroids• Matroids Loopless• Simple Simple• Repr. over k Repr. over k•
Limits:
Products ✗ ✗ ✗ ✗ ✗ ✗ ✗Exponentials ✗ ✗ ✗ ✗ ✗ ✗ ✗Pullbacks ✗ ✗ ✗ ✗ ✗ ✗ ✗Equalisers ✓ ✓ ✓ ✓ ✓ ✓ ✓
Colimits:
Coproducts ✓ ✓ ✓ ✓ ✓ ✓ ✓Coequalisers ✗ ✗ ✗ ✗ ✗ ✗ ✗
Pushouts ✗ ✗ ✗ ✗ ✗ ✗* ✗*
Limits and Colimits
7 / ??
Matroids• Matroids Loopless• Simple Simple• Repr. over k Repr. over k•
Limits:
Products ✗ ✗ ✗ ✗ ✗ ✗ ✗Exponentials ✗ ✗ ✗ ✗ ✗ ✗ ✗Pullbacks ✗ ✗ ✗ ✗ ✗ ✗ ✗Equalisers ✓ ✓ ✓ ✓ ✓ ✓ ✓
Colimits:
Coproducts ✓ ✓ ✓ ✓ ✓ ✓ ✓Coequalisers ✗ ✗ ✗ ✗ ✗ ✗ ✗Pushouts
✗ ✗ ✗ ✗ ✗ ✗* ✗*
Limits and Colimits
7 / ??
Matroids• Matroids Loopless• Simple Simple• Repr. over k Repr. over k•
Limits:
Products ✗ ✗ ✗ ✗ ✗ ✗ ✗Exponentials ✗ ✗ ✗ ✗ ✗ ✗ ✗Pullbacks ✗ ✗ ✗ ✗ ✗ ✗ ✗Equalisers ✓ ✓ ✓ ✓ ✓ ✓ ✓
Colimits:
Coproducts ✓ ✓ ✓ ✓ ✓ ✓ ✓Coequalisers ✗ ✗ ✗ ✗ ✗ ✗ ✗Pushouts ✗ ✗ ✗ ✗ ✗ ✗* ✗*
Functors and adjunctions
9 / ??
Matroids•
a��
Geometric Lattices
TT
V ectk
iiSSSSSSSSSSSSSS
Free matroids : Every set is independent.
Matroids• >// Matroidskk ⊥
// Setii
Functors and adjunctions
9 / ??
Matroids•
a��
Geometric Lattices
TT
V ectk
iiSSSSSSSSSSSSSS
Free matroids : Every set is independent.
Matroids• >// Matroidskk ⊥
// Setii
Functors and adjunctions
9 / ??
Matroids•
a��
Geometric Lattices
TT
V ectk
iiSSSSSSSSSSSSSS
Free matroids : Every set is independent.
Matroids• >// Matroidskk ⊥
// Setii
Functors and adjunctions
10 / ??
Matroids• Loopless Matroids•_?oo
Simple Matroids•?�
OO
& �
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Free Matroids•_?oo
X8
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OO
Functors and adjunctions
11 / ??
Matroids•
a��
Loopless Matroids•_?oo
a��
Simple Matroids•?�
OO
& �
44hhhhhhhhhhhhhhhhhh
Free Matroids•_?oo
X8
jjVVVVVVVVVVVVVVVVVV?�
OO
a
bb
Matroids as functors:
Geometric Lattices // Sub