#### Category theory: assumes compositionality of morphisms ... which is essentially computational reducibility.

Category theory: assumes compositionality of morphisms ... which is essentially computational reducibility.

Insofar as category theory models mathematics, that is why mathematics avoids irreducibility.

#### What is “complexity-based category theory”?

What is “complexity-based category theory”?

## Building up to rulial space

Building up to rulial space

Progressively higher-order morphisms limit to rulial space, in which arrows are inevitably reversible; hence it is a topos

Possible claim: the reconstructed “space space” in rulial space is like the homotopy type

But the branchial structure is like the infinity topos (?)

But the branchial structure is like the infinity topos (?)

Grothendieck hypothesis is basically the statement that branchial space is a space...

[Grothendieck hypothesis: why FTL is hard]

GH : true because of comp. irreduc.

[Grothendieck hypothesis: why FTL is hard]

GH : true because of comp. irreduc.

When you do higher-order morphisms ... you’re reaching the PCE level

Either: everything is trivial, and then you can’t get anywhere in the higher-order morphisms .... or eventually you reach the PCE level

Do all non-small categories reach PCE? [Ramseyesque?]

Do all non-small categories reach PCE? [Ramseyesque?]

#### E.g. morphism is successor

E.g. morphism is successor

In[]:=

UndirectedGraph[Table[ii+1,{i,20}]]

Out[]=

{i,i+1}

## Foliation of multiway graph

Foliation of multiway graph

#### Computational boundedness of the foliation process:

related to geometry/finite dimensionality

Computational boundedness of the foliation process:

related to geometry/finite dimensionality

related to geometry/finite dimensionality

There exists a non-Groth. foliation of branchial space, but to find it is not computationally bounded

### Existence of foliations in physics is required for a “state evolving” “mechanistic” model of physics

Existence of foliations in physics is required for a “state evolving” “mechanistic” model of physics

Strong hyperbolicity implies foliatability;

well-posedness : can make it an initial value problem

well-posedness : can make it an initial value problem

### Existence of foliations in metamathematics

Existence of foliations in metamathematics

Hilbert’s program is well-posedness of math

In the theorem theorem interpretation of MW graph...

Foliatability there is a meaningful current global state of math

Cohesiveness of math??

Is there forward progress where all time vectors point to the future?

[Continuity to hypersurface is what prevents random “forward” vectors from pointing backwards]

Is there forward progress where all time vectors point to the future?

[Continuity to hypersurface is what prevents random “forward” vectors from pointing backwards]