Lecture 1: Basics

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Categories

Category $𝒞$:
  • STRUCTURE/DATA (Graph):

    • a class of objects:

      \underbrace{\vert 𝒞 \vert}_{\rlap{\text{also denoted by } {\rm ob}\, 𝒞 \text{, or } 𝒞_0}} : Set
    • a “Hom” map, associating to each every pair of objects a morphism/arrow:

      \underbrace{\Hom_𝒞}_{\rlap{\text{ also denoted by } 𝒞(\bullet) \text{, or } 𝒞_1}}: \vert 𝒞 \vert ⟶ \vert 𝒞 \vert ⟶ Set
    • For all objects:

      • identity maps:

        \ast \xto { {\rm id}_x} \Hom_𝒞(x, x)
      • composition maps:

        \circ: \Hom_𝒞(y, z) × \Hom_𝒞(x, y) ⟶ \Hom_𝒞(x, z)
  • LAWS/AXIOMS:

    • ${\rm id}_x$ is an identity for $\circ$

    • the following diagram commutes:

      \begin{xy} \xymatrix{ 𝒞(z, w) × 𝒞(y, z) × 𝒞(x, y)\ar@{->}[d]_{\circ × 1} \ar@{->}[r]^-{1 × \circ} & 𝒞(y, w) × 𝒞(x, y) \ar@{->}[d]^{\circ} \\ 𝒞(z, w) × 𝒞(x, z) \ar@{->}[r]_{\circ} & 𝒞(x, w) } \end{xy}
Metacategory:

arbitrary class of objects and morphisms, interpretation satisfying the axioms/laws.

Ex: metacategories of all sets, groups, topological spaces, etc…

Category:

interpretation of the category axioms within set theory

NB: a category is a monoid for the product defined as

E ×_V E ≝ \lbrace \pair g f \; \mid \; g, f ∈ E \text{ and } t f = s g\rbrace

where

\begin{xy} \xymatrix{ E \ar@<-3pt>[r]_{ s } \ar@<+3pt>[r]^{ t } & V } \end{xy}

is a graph.

$2$-Categories:

  • $\Hom_𝒞$ has its values in $Cat$
  • the composition and identity maps are functors
  • ${\rm id}x$ and ${\rm id}{ {\rm id}_x}$ are identities for the composition functor.

Arrows-only version

We can do without objects, as identity arrows are in one-to-one correspondence with objects.

Arrows-only version:

  • STRUCTURE:
    • arrows $f, g, …$
    • certain pairs $\pair g f$
    • composition operation: $\pair g f ⟼ gf$
  • LAWS:
    • $k(gf)$ is defined iff $(kg)f$ is, and when they are, they are equal (denoted by $kgf$)
    • $kgf$ is defined iff $kg$ and $gf$ are
    • for each $f$, there exist identity arrows (i.e. arrows such $u$ that $fu = u$ whenever $fu$ is defined and $uf = u$ whenever $uf$ is defined) $u$ and $u’$ such that $fu$ and $u’f$ are defined.
  • the arrows of a metacategory of objects and arrows satisfy the arrow-only axioms

  • conversely: the identity arrows of an arrows-only metacategory, taken as objects, satisfy the objects-and-arrows axioms.

Natural transformations

  • Why commutativity? For horizontal composition
  • Naturality (defined by the same formula for all objects): example of $V ≅ V^\ast$ (not natural) and $V ≅ (V^\ast)^\ast$ (natural)
  • Equivalence of Categories

Exercises

Functors

Faithfulness/Fullness. Example: the forgetful functor $U: Ab ⟶ Set$ is faithful but not full.

Let $Grp \xto F Ab$ be a function on objects such that $F(G) = Z(G)$ for all $G$ (center of $G$). Can we extend $F$ to a functor?

Solution: No.

By contradiction: Consider $𝔖_3 xto {ε} (\lbrace ± 1\rbrace, ×)$ (signature) and $(\lbrace ± 1\rbrace, ×) \xto {σ} 𝔖_3$ where $σ(-1) = (1, 2)$.

Take the image by $F$:

  • $\lbrace1\rbrace \xto {F ε} (\lbrace ± 1\rbrace, ×)$ (as $𝔖_3$ has a trivial center and $\lbrace ±1 \rbrace$ is abelian)
  • $(\lbrace ± 1\rbrace, ×) \xto {F σ} \lbrace1\rbrace$

And:

ε \circ σ = id_{\lbrace ± 1\rbrace}

But

F ε \circ F σ = id_{\lbrace ± 1\rbrace}

cannot hold, because $F σ$ is not injective (or: $F ε$ is not surjective).

Group action

An action of a group $G$ on a set $A$:

denoted by $G ⟶ A$, is a homomorphism $φ: G ⟶ Bij(A)$

$φ$ is faithful:

if $Ker \, φ = \lbrace e_G \rbrace$

$φ$ is free:

$∃x ∈ A; \; φ(g)(x)=x ⟹ g = e_G$

$φ$ is transitive:

$∀x, y ∈ A, \; ∃g; \, φ(g)(x)=y$


Exercise: $F ⊣ G: B ⟶ A$ $G$ is full iff ∀b, ∃ f; f \circ ε_b = id_{FG b}

Exercise: $F ⊣ G: B ⟶ A$ $G$ is faithful iff $ε_B$ is epi


Exercise (Example of Stone Duality): FinSet^{op} ≅ FinBoolAlg

Exercise: id, (\_)^{\ast \ast}: FdVect ⟶ FdVect

id \overset{α}{⟹} (\_)^{\ast \ast}

Exercise: Determinant as a natural transformation.

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