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Categories of Matroids

Vaia Patta

March 6, 2015

What is a matroid?

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What is a matroid?

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A matroid describes dependence.

x = ax1 + bx2 ...etc

What is a matroid?

3 / ??

A matroid describes dependence.

x = ax1 + bx2 ...etc

What is a matroid?

3 / ??

A matroid describes dependence.

x = ax1 + bx2 ...etc

What is a matroid?

3 / ??

A matroid describes dependence.

x = ax1 + bx2 ...etc

What is a matroid?

3 / ??

A matroid describes dependence.

x = ax1 + bx2

...etc

What is a matroid?

3 / ??

A matroid describes dependence.

x = ax1 + bx2 ...etc

Interesting classes of matroids

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■ 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

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■ 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

Categories of matroids

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What are the morphisms?

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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?

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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?

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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?

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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

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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

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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

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Matroids•a��

Geometric Lattices

TT

Functors and adjunctions

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Matroids•

a��

Geometric Lattices

TT

V ectk

iiSSSSSSSSSSSSSS

Free matroids : Every set is independent.

Matroids• >// Matroidskk ⊥

// Setii

Functors and adjunctions

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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

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Matroids• Loopless Matroids•_?oo

Simple Matroids•?�

OO

& �

44hhhhhhhhhhhhhhhhhh

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

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