Exercises 1 : Introduction
EX 1: Categories and functors
1.
: sets and functions : topological spaces and continuous maps : vector spaces and linear maps : groups and homomorphisms , etc… : posets and monotonous maps : 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
- there exist unique maps
and by terminality by universality of- idem for
4 / 5.
It’s “the” (up to unique isomorphism) terminal object in the category of cones.
6.
There exist:
We check that
is an isomorphism:
indeed, by universality:
since
7.
Same demonstration as before.
8.
We show that
M2.
With the Yoneda lemma:
9.
Coproducts: disjoint union of sets
Initial object in
10.
- Category of relations
: -
- objects: sets
- morphisms: relations
It is cartesian:
- terminal object:
since the only morphism is - products: the disjoint union:
is defined as , and if ,
What are coproducts in this category? A product in the opposite category.
But as
They are the same.
11.
Products are direct sums, whose basis are disjoint unions of basis of
12.
- terminal objet: unit category
- product: product category
EX 3: Pullbacks
1.
It is a product.
2.
It is the set
EX 4: Dual notions
1.
Coproducts in:
: disjoint unions : disjoint unions : product of spaces : 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
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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