Exercises 1 : Introduction

EX 1: Categories and functors

1.

  • Set: sets and functions
  • Top: topological spaces and continuous maps
  • Vect: vector spaces and linear maps
  • Grp: groups and homomorphisms
  • Mon,Fields,Ring, etc…
  • Poset: posets and monotonous maps
  • (E,): the poset category
  • categories with a single object are monoids
  • logical formuals and implications

EX 2: Cartesian categories

1.

  • terminal object: greatest element
  • product: greatest lower bound

2.

Products of sets are cartesian products, terminal objects are singletons.

3.

If 1 and 1 are two terminal objects:

  • there exist unique maps u:11 and v:11 by terminality
  • vu𝒞1(1,1)={id1} by universality of 1
  • idem for uv=id1

4 / 5.

It’s “the” (up to unique isomorphism) terminal object in the category of cones.

6.

There exist: π1:1×A1, π2:1×AA

We check that π2,π1:1×AA×1

is an isomorphism:

π2,π1π1,π2=π2π1,π1π2=idA,id1

indeed, by universality: idA,id1 is the only arrow in 𝒞1(A×1,A×1).


A1×A

since A is a cartesian product of 1 and A (easy to check).

7.

Same demonstration as before.

8.

We show that (A×B)×C is a product of A and B×C.

M2.

With the Yoneda lemma: 𝒞1(y((A×B)×C),y(A×(B×C)))𝒞1((A×B)×C,A×(B×C)), so:

(A×B)×CA×(B×C)y((A×B)×C)y(A×(B×C))𝒞1(_,(A×B)×C)𝒞1(_,A×(B×C))

9.

Coproducts: disjoint union of sets

Initial object in Set: the empty set .

10.

Category of relations Rel:
  • objects: sets
  • morphisms: relations A×B

It is cartesian:

  1. terminal object: since the only morphism X× is
  2. products: the disjoint union: πXXY×X is defined as {(ιX(x),x)xX}, and if fC×X,gC×Y, f,g{(c,ιX(x))(c,x)f}{(c,ιY(y))(c,y)g}

What are coproducts in this category? A product in the opposite category.

But as

Relop=Rel

They are the same.

11.

Products are direct sums, whose basis are disjoint unions of basis of A and B.

πA:{ABAaBasisAabBasisB0

12.

Cat is cartesian:

  • terminal objet: unit category
  • product: product category

EX 3: Pullbacks

1.

It is a product.

2.

It is the set {(x,y)A×Bf(x)=g(y)}

EX 4: Dual notions

1.

Coproducts in:

  • Set: disjoint unions
  • Rel: disjoint unions
  • Top: product of spaces
  • Vect: product of spaces

EX 5: (Co)monoids in cartesian categories

1.

Monoid:

A category with one single object.

The monoid is commutative whenever for all f,g:XX, fg=gf

2.

Homomorphism of monoids:

a functor from the first category seen as a monoid to the second one

3.

A comonoid

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