Lecture 1: Basics

\[\newcommand\yoneda{ {\bf y}} \newcommand\oppositeName{ {\rm op}} \newcommand\opposite[1]{ {#1}^\oppositeName} \newcommand\id[1][{}]{ {\rm id}_{#1}} \newcommand\Id[1][{}]{ {\rm Id}_{#1}} \newcommand\Cat[1]{\mathcal{#1}\/} \newcommand\Category[1]{ {\mathbf{ #1}}} \newcommand\Set{\Category{Set}} \newcommand\tensor{\otimes} \newcommand\unit{\mathbb I} \newcommand\carrier[1]{\underline{#1}} \newcommand\Con[1][]{\mathrm{Con}_{#1}} \newcommand\ConStar[1][]{\overline{\mathrm{Con}}_{#1}} \newcommand\dc{\mathop{\downarrow}} \newcommand\dcp{\mathop{\downarrow^{\mathrm{p}}}} \newcommand\Dc{\mathop{\Downarrow}} \newcommand\Dcp{\mathop{\Downarrow^{\mathrm{p}}}} \newcommand{\Ddc}{\mathop{\require{HTML} \style{display: inline-block; transform: rotate(-90deg)}{\Rrightarrow}}} \newcommand\CP{\mathfrak{P}} \newcommand\Conf[1][]{\mathcal C^{\mathrm o}#1} \newcommand\ConfStar[1][]{\mathcal C#1} \newcommand\Path{\mathbb{P}} \newcommand\PathPlus{\mathbb{P}_+} \newcommand\ES{\mathcal{E}} \newcommand\PAD{\mathscr{P}} \newcommand\Comma[2]{#1\downarrow#2} \newcommand\lub{\bigvee} \newcommand\glb{\bigwedge} \newcommand\restrict[1]{\left.\vphantom{\int}\right\lvert_{#1}} \newcommand\C{\mathbb C} \newcommand\D{\mathbb D} \newcommand\wC{\widetilde {\Category C}} \newcommand\catA{\Category A} \newcommand\catB{\Category B} \newcommand\catC{\Category C} \newcommand\catD{\Category D} \newcommand\catE{\Category E} \newcommand\Psh[1]{\widehat{#1}} \newcommand\PshStar[1]{\widehat{#1}} \newcommand{\Pfin}{\mathop{ {\mathcal P}_{\rm fin}}\nolimits} \newcommand\eqdef{:=‰} \newcommand\ie{\emph{i.e.~}} \newcommand\conflict{\mathrel{\sim\joinrel\sim}} \newcommand\Cocomp{\mathbf{Cocomp}} \newcommand\const[1]{\Delta_{#1}} \newcommand\Lan[2]{\mathop{\mathrm{Lan}}_{#1}(#2)} \newcommand\Ran[2]{\mathop{\mathrm{Ran}}_{#1}(#2)} \newcommand\Nerve[1]{\mathrm{N}_{#1}} \newcommand{\dinat}{\stackrel{\bullet}{\longrightarrow}} \newcommand{\End}[2][c]{\int_{#1} #2(c, c)} \newcommand{\Coend}[2][c]{\int^{#1} #2(c, c)} \newcommand{\obj}[1]{\vert #1 \vert} \newcommand{\elem}[1]{\int #1} \newcommand{\tens}[2]{#1 \cdot #2} \newcommand{\cotens}[2]{#2^{#1}} \newcommand{\Nat}{\mathop{\rm Nat}\nolimits} \newcommand{\colim}{\mathop{\rm colim}\nolimits} \newcommand{\cancolimAC}[1]{\big((i/#1)\stackrel{U}{→} \catA \stackrel{i}{→} \catC\big)} \newcommand{\canlimAC}[1]{\big((#1\backslash i)\stackrel{U}{→} \catA \stackrel{i}{→} \catC\big)} \newcommand\pair[2]{\left<{#1}, {#2}\right>} \newcommand\triple[3]{\anglebrackets{ {#1}, {#2}, {#3}}} \newcommand\anglebrackets[1]{\left<{#1}\right>} \newcommand\set[1]{\left\{#1\right\}} \newcommand\suchthat{\middle\vert} \newcommand\xto\xrightarrow \newcommand\xfrom\xleftarrow \newcommand\parent[1]{\left({#1}\right)} \newcommand\Hom{\mathord{\mathrel{\rm Hom}}}\]

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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