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
• $(E,\le )$: 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 $1^{\prime }$ are two terminal objects:

• there exist unique maps $u:1⟶1^{\prime }$ and $v:1^{\prime }⟶1$ by terminality
• $v\circ u\in {𝒞}_1(1,1)=\{i{d}_1\}$ by universality of $1$
• idem for $u\circ v=i{d}_{1^{\prime }}$

4 / 5.

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

6.

There exist: ${\pi }_1:1\times A⟶1$, ${\pi }_2:1\times A⟶A$

We check that $⟨{\pi }_2,{\pi }_1⟩:1\times A⟶A\times 1$

is an isomorphism:

indeed, by universality: $⟨i{d}_A,i{d}_1⟩$ is the only arrow in ${𝒞}_1(A\times 1,A\times 1)$.

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

7.

Same demonstration as before.

8.

We show that $(A\times B)\times C$ is a product of $A$ and $B\times C$.

M2.

With the Yoneda lemma: ${𝒞}_1(y((A\times B)\times C),y(A\times (B\times C)))\simeq {𝒞}_1((A\times B)\times C,A\times (B\times C))$, so:

9.

Coproducts: disjoint union of sets

Initial object in $Set$: the empty set $\varnothing$.

10.

Category of relations $Rel$:

• objects: sets
• morphisms: relations $\subseteq A\times B$

It is cartesian:

1. terminal object: $\varnothing$ since the only morphism $\subseteq X\times \varnothing$ is $\varnothing$
2. products: the disjoint union: ${\pi }_X\subseteq X\bigsqcup Y\times X$ is defined as $\{({\iota }_X(x),x)\mid x\in X\}$, and if $f\subseteq C\times X,g\subseteq C\times Y$, $⟨f,g⟩\subseteq \{(c,{\iota }_X(x))\mid (c,x)\subseteq f\}\cup \{(c,{\iota }_Y(y))\mid (c,y)\subseteq g\}$

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 $A$ and $B$.

12.

$Cat$ is cartesian:

• terminal objet: unit category
• product: product category

EX 3: Pullbacks

1.

It is a product.

2.

It is the set $\{(x,y)\in A\times B\mid f(x)=g(y)\}$

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 $f,g:X⟶X$, $f\circ g=g\circ f$

2.

Homomorphism of monoids:

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

3.

A comonoid