Lecture 1: Basics
Categories
- Category $𝒞$:
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STRUCTURE/DATA (Graph):
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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:
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identity maps:
\[\ast \xto { {\rm id}_x} \Hom_𝒞(x, x)\] -
composition maps:
\[\circ: \Hom_𝒞(y, z) × \Hom_𝒞(x, y) ⟶ \Hom_𝒞(x, z)\]
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LAWS/AXIOMS:
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${\rm id}_x$ is an identity for $\circ$
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the following diagram commutes:
\[\begin{xy} \xymatrix{ 𝒞(z, w) × 𝒞(y, z) × 𝒞(x, y)\ar@{->}[d]_{\circ × 1} \ar@{->}[r]^-{1 × \circ} & 𝒞(y, w) × 𝒞(x, y) \ar@{->}[d]^{\circ} \\ 𝒞(z, w) × 𝒞(x, z) \ar@{->}[r]_{\circ} & 𝒞(x, w) } \end{xy}\]
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- Metacategory:
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arbitrary class of objects and morphisms, interpretation satisfying the axioms/laws.
Ex: metacategories of all sets, groups, topological spaces, etc…
- Category:
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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 $A$:
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denoted by $G ⟶ A$, is a homomorphism $φ: G ⟶ Bij(A)$
- $φ$ is faithful:
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if $Ker \, φ = \lbrace e_G \rbrace$
- $φ$ is free:
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$∃x ∈ A; \; φ(g)(x)=x ⟹ g = e_G$
- $φ$ is transitive:
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$∀x, y ∈ A, \; ∃g; \, φ(g)(x)=y$
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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