1. Show which graphs are the same.

2. Show which graphs are the same.

1. Show \(D_8\) is not the same as \(Q_8\).

1. Show which graphs are the same.

Moral

• For isomorphism we prefer a construction.
• For non-isomorphism we look for invariants.
• Set aside the fact that invariants often need isomorphisms internally, e.g. "without-loss-of-generality" is isomorphism you refuse to write.

structrualism

Things are different when they have different "structure", i.e. logical operations separate them.

Minimal assumptions

Need isomorphisms, i.e. a category

• Transitive/Product: \((x\cong y) \wedge (y\cong z)\to (x\cong z)\)
• Reflexive/Identity: \(x\to (x\cong x)\)
• Anti-symmetry/Inverses: \((x\cong y)\to (y\cong x)\)

Back to basics

• Equality means can replace like with like \[(x=y) \Rightarrow (\forall P)(P(x)\Leftrightarrow P(Y))\] (elimination/discharge rule of equality)
• Leibniz Law : Difference is about logical structure. \[X\neq Y \;\Rightarrow\; (\exists P)(P(X) \wedge \neg P(Y))\]

• Structuralism: "sameness is isomorphism"   \[X\cong Y \;\Rightarrow\; X=Y\] (introduction rule of equality).
• Tarski-Thesis: Logical Operators = Isomorphism invariants.
• Fact. Logicism = Leibniz + Tarski Thesis \(\Longrightarrow \) Structuralism.

Invariants Are often misleading

It is often said:

"The character table is an invariant of finite groups."

What is "the" character table?  This is non unique.

Reality 

• Replacing isomorphism? \[X\cong Y \Rightarrow I(X)\sim I(Y)\]
  • becomes a tautology in the extreme
  • needs complexity comparison
• Isomorphism on more than the data?
  • Assuming parameters. E.g. Jordan Normal Form classifies conjugacy of square matrices...but only if you agree on an order.
  • In modal logic confusing \(\diamond P\) with \(\Box P\)?

Invariants

Definiton.  In a category \(C\) an invariant is a functor \(I:C\to P\) into a partial order \(P=\langle P,\leq \rangle\)

Claim: Isomorphism invariant 

\[X\cong Y \Rightarrow I(X)=I(Y)\]

is alway implied.

Invariants

Invariants

*Univalence/ Cantor-Schroder-Bernstein

Corollary.  Isomorphism invariants define a distance function (pseudo-matric):

• \(d(u,u)=0\)
• \(d(u,v)=d(v,u)\)
• \(d(u,v)\leq d(u,w)+d(w,v)\)

\(u\cong v\Rightarrow d(u,v)=0\)

Distance induced by chromatic number of graph

Definition. An invariant \(I:C\to P\) is characteristic if for every automorphism \(\varphi\) of \(P\),

\[\varphi(I(X)) = X\]

A characteristic invariant is the only family of invariant that is solely about C (no parameters) and isn't replacing isomorphism.

Theorem (B-M-O'B-W) There are categories with

• objects \(|X|=|Y|\) and \(X\not\cong Y\)
• for every characteristic invariant \(I:C\to P\), \[I(X)=I(Y)\]

• Examples with graphs
• Examples with groups
• Examples with rings

Toy Example: category of 4 vertex graphs <3 edges under inclusion

Easy invariants:

• Components?  Not defining
• Degree sequence?
• ...not characteristic!

Externality

Perhaps "structure" comes from outside.

Break symmetry of invariants by extending context

What about groups?

Characteristic: \(H\leq G\) for every automorphism \(\varphi:G\to G\), \[\varphi(H)=H\]

Looks structural.

Rottlaender Type Groups:

\[\left\{\begin{bmatrix} \alpha & & u \\ & \alpha^u & v \\ & & 1 \end{bmatrix}\right\}\]

Has characteristic subgroups \[\left\{\begin{bmatrix} 1 & & u\\ & 1 & 0 \\ & & 1\end{bmatrix}\right\},\left\{\begin{bmatrix}1& & 0 \\ & 1 & v\\ & & 1 \end{bmatrix}\right\}\] but which is which?  This is non-structural!

What about groups?

Characteristic: \(H\leq G\) for every automorphism \(\varphi:G\to G\), \[\varphi(H)=H\]

Looks structural.

Good Characteristic, e.g. commutator, come from external structure.

• Largest ABELIAN quotient.
• For free groups all char. come this way.
• What about the rest of group theory?

Cheat-sheet to 2-categories

Characteristic Subgroups as Category Theory

Theorem (B-D-M-O'B-W)

\[\{\iota:H\hookrightarrow G \mid \iota(H)\text{ char. } G\}=\left\{\iota_G \mid \iota \in \text{Counital}\left(\overset{\longleftrightarrow}{Grp},Grp\right) \right\}\]

Furthermore, every counital is induced by a counit (e.g. all adjoints).

Category \(E\) with subcategory \(A\), \(I:A\hookrightarrow E\)

\[\text{Counital}\left(A,B\right)=\left\{\iota:C\Rightarrow I\right\}\]

Counit = counital with \(A=E, I=id_E\)

Corollary.  All characteristic structure is external.

I.e. why a group \(G\) fixes a subgroup is the influence of some category.

Most of categories today is written by topology and geometry.

We think algebra offers something different, and so did Freyd, Scedrov, and few others.

(Objective) Categories

          bases Objects \(ob(C)\) a type of data

                       Morphisms: disjoint sets \(\hom_C(X,Y)\) for objects \((X,Y)\)

 operators \(\circ:\hom_C(X,Y)\times \hom_C(Y,Z)\to \hom_C(X,Z)\)

                      \(1_X\in\hom_C(X,X)\)

   rewriting for \(f:\hom_(X,Y), g\in \hom_C(Y,Z), h\in \hom_C(Z,W)\)

\[\begin{aligned} f1_X & = X \\ 1_Y f & = f\\  f(gh) & = (fg)h \end{aligned}\]

2 sorted second order theory

Objects have no axioms in category theory, 

they are aren't needed.

Essentially Algebraic Structures (Eastern algebras)

           base \(A\) a type of data

 operators \([\ldots]:A^n\to A^?\)  read as "maybe \(A\)"

guard rails polynomial formulas \(\Phi,\Upsilon\) in the operators where

                      \(\Phi(a_1,\ldots,a_n)=\Upsilon(a_1,\ldots,a_n)\;\Longrightarrow\;[a_1,\ldots,a_n]\text{ defined.}\)

   rewriting e.g.\ associative when defined, identity when defined.

(Abstract) Categories

           base \(A\) a type of data

 operators \(\circ:A\times A\to A^?\) a partial product (\(A^?\) reads "maybe A")

                       \(\lhd(-):A\to A\) and \((-)\lhd:A\to A\)

guard rails \(a\lhd=\lhd\acute{a}\;\Longrightarrow\;a\dot{a}\text{ defined.}\)

   rewriting

Summary: Set \(\mathbf{1}_{A}=\{a\lhd \mid a:A\}=\{\lhd a\mid a:A\}\) then

\[\begin{aligned} a(\dot a\ddot a) & = (a\dot a)\ddot a & \mathbf{1}_Aa & = \{a\}=a\mathbf{1}_A\end{aligned}\]

Categories = Monoids with some undefined products

\[\begin{aligned}\lhd(a\lhd) &=a\lhd & (\lhd a)\lhd & = \lhd a\\ \lhd(a\dot a) & =\lhd a & (a \dot a)\lhd & = \dot a\lhd\\ a(a\lhd) & = a & (\lhd a)a & = a \\ a(\dot a\ddot a)& =(a \dot a)\ddot a \end{aligned}\]

Category Theory's hidden Addition \(\sqcup\)

Given \(f\in\hom_C(X,Y)\) and \(g\in hom_C(X',Y')\) define 

\[f\sqcup g:=\{f,g\}\subset \hom_C(X,Y)\cup \hom_C(X',Y')\]

*In a hierarchy of sets or types.

Category Theory's hidden Addition \(\sqcup\)

Consequences. Categories \(A\) are much like rings.

• Identities \(e=f_Y, f=1_X\) are idempotents \(e=e^e, f=f^2\)
• Peirce decompositions \[eAf\cdot e'Af'  \subset \begin{cases} eAf' & f=e'\\ \emptyset  \end{cases}\]

\[\mathbf{1}_A = \bigsqcup_{X:ob(C)} 1_X = \begin{bmatrix} 1 & & & \\ & 1 & & \\ &  & \ddots & \\ & & & 1 \end{bmatrix}\]

Slice category

Coslice category

Hom-set

As rings have modules,

categories \(A\) have "capsules" \(X\)

\[\cdot:A\times X\to X^?\]

• \(a(\dot a x)=(a\dot a)x\)
• \(\mathbf{1}_A x=\{x\}\)

An \(A\)-morphism \[M:X\to Y^?\] where when the left-hand side is defined then \[M(ax)=aM(x)\]

In Peirce decomposition terms

\[\begin{bmatrix} & \vdots & \\ \cdots & A_{ij} & \cdots \\ & \vdots & \end{bmatrix} \begin{bmatrix} \vdots \\ x_j \\ \vdots\end{bmatrix}\]

Facts.

• Natural transformations \(\rho:G\Rightarrow F\) are bicapsules \(A\rho A\)
• Counits \(\rho:FG\Rightarrow I\) are \(A,B\)-morphisms \(M:A\to B\)
• Adjoints are pseudo-inverses \(M:A\to B\), \(N:B\to A\) \[MNM=M\qquad NMN=N\]

Extension Theorem in Category Theory Language.

In a category \(E\) of eastern algebras, given a counital \(\rho:C\Rightarrow I\) on a subcategory \(A\leq E\).  If \(A\leq C\leq E\) and \(A\) is full in \(C\), then there exists a counital \(\sigma:D\Rightarrow J\) on \(C\).

Extension Theorem in Algebra Terms.

Set \(\Delta=A\rho A\).  Then there is capsule \(\Upsilon\) where

\[\Delta\cong \text{Res}_A^C(\text{Ind}_A^C(\Delta))\otimes_A \Upsilon\]

Corollary. All characteristic subgroups come from externalities.

Theorem.

1. Internal counitals come from counits.
2. Isoceles counitals extend to internal counits.
3. Counitals of groups extend to isoceles counitals.

Commutator subgroup is the kernel of abelian adjunction (external)

Maringal subgroups externalized by Varieties

why are things different?

1. Structure?  Consistent \(\Leftrightarrow\) Incomplete
2. Externalities? Ramifies (splits into opposing reasons)
3. Depends on your logic?

All our theorems have been done with minimal logic (Intuitionistic Martin-Lof).   Consequence?

1. Can explore alternative fixes, e.g. keep structuralims but remove Leibniz Law.
2. Make all the work computational, e.g. certify characterisitc.
3. Can pursue consistency in different extensions, e.g. 3-morphic.
4. Leads to modeling isomorphism problems in computation with a 3-category