Category Theory (Lecture)

I followed this course by, Prof. Benno van den Berg during my 5th Semester at the ILLC Amsterdam for my M.Sc. Logic (UvA). A full description of the course can be found in the MasterMath website at link.

Assignments

• Overleaf: CatTh, 1
• Overleaf: CatTh, 2
• Overleaf: CatTh, 3
• Overleaf: CatTh, 4
• Overleaf: CatTh, 5
• Overleaf: CatTh, 6

I use https://q.uiver.app to draw the diagrams.

Lecture Notes

Category

• A cath. $\mathcal{C}$ is a tuple of:
  1. $Ob(\mathcal{C})$ collection of objects of $\mathcal{C}$
  2. $Mor(\mathcal{C})$ of morphisms/arrows in $\mathcal{C}$
     • hence, for $f\in Mor(\mathcal{C})$, then $dom(f),cod(f)\in Ob(\mathcal{C})$
     • $f:A\to B:=f\in Mor(\mathcal{C})\wedge dom(f)=A\in Ob(\mathcal{C})\wedge cod(f)=B\in OB(\mathcal{C})$
  • hence, $\mathcal{C}$ is a directed-multi-graph
  3. s.t. ${\forall }_{A\in Ob(\mathcal{C})}{\exists }_{1_A\in Mor(\mathcal{C})}{1}_A:A\to A$
  4. if ${\exists }_{f:A\to B}{\exists }_{g:B\to C}\top \to {\exists }_{g\circ f:A\to C}\top$
     1. s.t. $f\circ 1_A=f=1_B\circ f$ and $h\circ (g\circ f)=(h\circ g)\circ f$
• A category is uniquely defined by:
  1. $Ob(C)$ a class
  2. $Mor(\mathcal{C})$ a class of the form $\{(A_1,B_1),...\}$ for $A_i\in Ob(\mathcal{C})$
  3. $\circ :Mor(\mathcal{C})\times Mor(\mathcal{C})\to Mor(\mathcal{C})$
     • by ex.1 specifying what is the identity of each object, is not necessary, since they’re unique
• $\mathcal{C}$ is loc.small if $ZFC\models {\forall }_{A,B\in Ob(\mathcal{C})}{\exists }_{x}x=Mor(A,B)$
• $\mathcal{C}$ is small if $Ob(\mathcal{C})$ and $Mor(\mathcal{C})$ are sets.
• a Monoid is a one-obj. loc.small category
• a small category without parallel arrows is a preorder
  • write $A\le B$ if ${\exists }_{f:A\to B}\top$.
  • by def. of cat. given above, "$\le$" is trans. and refl.
• for $\mathcal{C}$ cat., $f:A\to B$ is iso. if ex. $g:B\to A$ s.t. $g\circ f=1_A\wedge f\circ g=1_B$
  • $g$ is unique and called the inverse of $f$
• $Ho{m}_{\mathcal{C}}(A,B)$ all arrows from $A$ to $B$, (homeset)
• $f:A\to B$ is split.epi if ex $g:B\to A$ s.t. $fg=1_B$
  • ($g$ is called a section of $f$, and $f$ a retraction of $g$)

Functor

• Fundamentally categories can be as generalisations of:
  • Monoid
    • in fact, cart.prod. of categories is a category.
  • Preorders
• Product Category: $\mathcal{C}\times \mathcal{D}$
  1. $Ob(\mathcal{C}\times \mathcal{D}):=\{(C_0,D_0):C_0\in Ob(\mathcal{C})\wedge D_0\in \mathcal{D}\}$
  2. $Ho{m}_{\mathcal{C}\times \mathcal{D}}((C_0,D_0),(C_1,D_1)):=Ho{m}_{\mathcal{C}}(C_0,C_1)\times Ho{m}_{\mathcal{D}}(D_0,D_1))$
  3. $(g_0,g_1)\circ (f_0,f_1):=(g_0\circ f_0,g_1\circ f_1)$
• Opposite Category: ${\mathcal{C}}^{op}$
  1. $Ob({\mathcal{C}}^{op}):=Ob(\mathcal{C})$
  2. $Ho{m}_{{\mathcal{C}}^{op}}(A,B):=Ho{m}_{\mathcal{C}}(B,A)$
  3. $g{\circ }_{{\mathcal{C}}^{op}}f:=f{\circ }_{\mathcal{C}}g$
• Duality principle: of any definition, I can always find a dual (co-)
  • initial / terminal object
    • Terminal objects are unique up to unique isomorphism
  • $f:A\to B$ is a mono ifg. $fg=fh\to g=h$
    • mono are “often” inj.func
      • proved for $\text{Set}$
      • in $\text{Cat}$: $F$ mono iff. $F$ inj.on.obj. and faithful, ex.1
  • $f:A\to B$ is epi iff. $gf=hf\to g=h$
    • epi are “often” surj.func
      • proved for $\text{Set}$
      • in $\text{Cat}$: $F$ surj.onobj. and full $\Rightarrow$ $F$ full., ex.1
• recall $(\mathbb{N},+)$ is a generated by $1$.
• Functor:
  1. $F_0:C_0\to D_0$
  2. ${\forall }_{a,b\in {\mathcal{C}}_0}{\exists }_{F_1^{a,b}}{F}_1^{a,b}:Ho{m}_{\mathcal{C}}(a,b)\to Ho{m}_{\mathcal{D}}(F(a),F(b))$
  3. s.t. ${\forall }_{F(A),F(B)}F(A_A^{\mathcal{C}})=1_{F(A)}^{\mathcal{D}}$
  4. $F(\beta ){\circ }_{\mathcal{D}}F(\alpha )=F(\beta {\circ }_{\mathcal{C}}\alpha )$
• Functorial Structures
  • Example: $R$ a ring, $M_2(R)$ the $2\times 2$ matrixes with components in $R$,
    • is $M_2:\text{Ring}\to \text{Ring}$, $f:R\to S$, $M_2(f):M_2(R)\to M_2(S),M_s(f)\left(\begin{array}{c}a,b\\ c,d\end{array}\right)=\left(\begin{array}{c}f(a),f(b)\\ f(c),f(d)\end{array}\right)$
  • $F:\mathcal{C}\to \mathcal{D}$, if $\alpha \in {\mathcal{C}}_1$ iso, then $F(\alpha )$ iso.
• $F$ full if ${\forall }_{A,B\in {\mathcal{C}}_0}surj.(F^{A,B}:Ho{m}_{\mathcal{C}}(A;B)\to Ho{m}_{\mathcal{D}}(FA,FB))$
• $F$ faithful if ${\forall }_{A,B\in {\mathcal{C}}_0}inj.(F^{A,B}:Ho{m}_{\mathcal{C}}(A;B)\to Ho{m}_{\mathcal{D}}(FA,FB))$
• $F$ fully faithful if full and faithful
• $F$ _fully faithful has reflection on isos, i.e. if $\alpha :A\to B$ s.t. $F_1(\alpha )$ iso, then $\alpha$ iso.

Natural Transformation

• for $\mathcal{C},\mathcal{D}$ ex. $[\mathcal{C},\mathcal{D}]$ cat. s.t. $[\mathcal{C},\mathcal{D}{]}_0=Ho{m}_{\text{Cat}}(\mathcal{C},\mathcal{D})$, $[\mathcal{C},\mathcal{D}{]}_1$ are natural transformations
  • an element of it is ${\mu }_C:F(C)\to G(C)$
  • for $F,G:\mathcal{C}\to \mathcal{D}$ a nat.tr. $\sigma :F\Rightarrow G$ is a fam. of morph. ${\forall }_{C\in {\mathcal{C}}_0}{\sigma }_C:F(C)\to G(C)$ s.t.
    1. for ${\forall }_{C,C^{\prime }\in {\mathcal{C}}_0}{\exists }_{\alpha \in Ho{m}_{\mathcal{C}}(C,C^{\prime })}Diag.Comm.$, (Naturality Condition)
       1. $Diag$: $F(C){\to }^{F(\alpha )}F{C}^{\prime }{\to }^{{\sigma }_{C^{\prime }}}G(C^{\prime })$ and $F(C){\to }^{{\sigma }_C}G(C){\to }^{G(\alpha )}G(C^{\prime })$
  • if $\sigma :F\Rightarrow G$ in $[\mathcal{C},\mathcal{D}]$ is iso. iff ${\forall }_{C\in {\mathcal{C}}_0}iso.({\sigma }_C)$
  • note: $\circ :[\mathcal{D},\mathcal{E}]\times [\mathcal{C},\mathcal{D}]\to [\mathcal{C},\mathcal{E}]$, functor composition, is the object part of a functor, 3.3

Equivalence of Categories

• $\mathcal{C}\simeq \mathcal{D}$ if there are functors $F:\mathcal{C}\to \mathcal{D}$, $G:\mathcal{C}\to \mathcal{D}$ s.t.
  1. $G\circ F\cong 1_{\mathcal{C}}$
  2. $F\circ G\cong 1_{\mathcal{D}}$
  • it is called equivalence and $G$ is the pseudo-inverse
  • also $\cong$ is isomorphism of categories
    • $\mathcal{C}\cong \mathcal{D}$ iff ex. $F^{\prime }:\mathcal{C}\to \mathcal{D}$, $G^{\prime }:\mathcal{D}\to \mathcal{C}$ s.t. (i) $G\circ F=1$, (ii) $F\circ G=1$
• Equivalence preserves and reflects terminal objects (and most other properties)
• $F:\mathcal{C}\to \mathcal{D}$ is an equivalence iff:, 3.6
  1. $F$ is essentially surjective
     • that is: ${\forall }_{D\in {\mathcal{D}}_0}{\exists }_{C\in {\mathcal{C}}_0}F(C)\cong D$
  2. $F$ is fully faithful
     • $F$ is full and faith

Limits

• for $F:\mathcal{C}\to \mathcal{D}$ , a cone is $(D,\mu )$ s.t. $D\in {\mathcal{D}}_0$, $\mu :{\Delta }_D\Rightarrow F$ (cone), 4.1
  • ${\Delta }_D:\mathcal{C}\to \mathcal{D}$, for $g:D\to D^{\prime }$
  • ${\Delta }_g:{\Delta }_D\to {\Delta }_{D^{\prime }}$ s.t. $({\Delta }_g{)}_C=G$.
  • $\Delta :\mathcal{D}\to [\mathcal{C},\mathcal{D}]$
• $(D,\mu )\to (D^{\prime },{\mu }^{\prime })$, is a map $g:D\to D^{\prime }$ s.t. ${\mu }^{\prime }\circ {\Delta }_g=\mu$ (map of cones), 4.1
• a lim.cone for $F$ is term.obj of $Cone(F)$ (limiting cone)
• a lim.con for $!:\text{0}\to \mathcal{D}$ is term.obj. in $\mathcal{D}$
  • $Cone(!)\cong \mathcal{D}$
• These section are better understood in the Lecture Notes
  • 4.1.2
  • 4.1.3
  • 4.1.4
• complete categories:
  • % sets, top, grps, ring, ab.gr, vs. over a fixed field, cat. (fields are not!) %5.2 & 6 skipped!! See email 14.10