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category theory is 'native 2-dimensional' math. i.e. category theory explains everything in terms of graphs, where a graph is made from two different sorts of '

category theory is 'native 2-dimensional' math. i.e. category theory explains everything in terms of graphs, where a graph is made from two different sorts of 'entities', nodes and vertices i.e. categories and morphisms this being math, I wonder to which extent can category theory be re-expressed in terms of sets. perhaps a better question is if category theory can be re-expressed (or founded on) functions? lastly, I wonder if category theory can be expressed in terms of functions (i think maybe it can, without sets?) why shouldn't it be expressible in terms of sets (for some reason I don't think just sets are sufficient, may have to define functions (which possible in terms of sets) before 'expressing' categories starting with set theory)?

voxl 4y ago
Set theory is fine, see the (Stack Project)[https://stacks.math.columbia.edu/browse https://stacks.math.columbia.edu/browse] which develops a ton of modern Category Theory on ZFC (Zermelo-Fraenkel Set Theory with the Axiom of Choice) alone. Alternative foundations of mathematics (Set Theory, Category Theory, Type Theory, and all their variations) can all mutually interpret the other by just postulating sufficiently large universes. You don't pick or advocate one based off its ability to encode mathematics, but instead based on its ability to express your intention and ideas. Really its no different from programming language preference in my book.
Koshkin 4y ago
> i.e. categories and morphisms Objects, not categories.
mydogcanpurr 4y ago
> for some reason I don't think just sets are sufficient The reason you're looking for is that the category of sets is not a set.
- vishal0123 4y ago
  While category of sets technically could not be expressed as a ZFC set, the idea behind the set theory is enough. Also you could add an axiom[0] in ZFC to make category of set a set. [0]: https://en.wikipedia.org/wiki/Grothendieck_universe https://en.wikipedia.org/wiki/Grothendieck_universe
- ginnungagap 4y ago
  This is routinely dealt with through Grothendieck universes. Those are a fancy name for what is pretty much an inaccessible level of the cumulative hierarchy, indeed ZFC+"every set belongs to a Grothendieck universe" is equiconsistent with ZFC+"there is a proper class of inaccessible cardinals". This is not a strong assumption over pure ZFC compared to those set theorists interested in large cardinals work with
justatdotin 4y ago
> I wonder to which extent can category theory be re-expressed in terms of sets... yoneda
- ginnungagap 4y ago
  This also requires universes/inaccessible cardinals to even be stated for categories that are not locally small. But as I mentioned in another comment assuming enough universes exist is not a big deal for set theorists
epgui 4y ago
Set theory is going the way of the dodo. There are modified versions of set theory, but afaik a lot of the more exciting work is happening around type theory and proof assistants these days.
- johnbernier 4y ago
  Set theory is not going anywhere. It may be pushed to the sidelines by new developments in category theory but that is not the same as going extinct like the dodo.
johnbernier 4y ago
In computer science we typically count from zero, not from one. So even though a category has two different sorts of entities: objects and morphisms, since we count from zero, ordinary categories are one-dimensional. Objects are zero dimensional and morphisms are one dimensional. Since you are concerned with sets and functions, the following analogy is helpful to build your intuition for the subject. Dimension 0: sets Dimension 1: functions Dimension 2: commutative squares Dimension 3: commutative cubes The majority of categories, expressed in terms of structured sets and functions, never touch on the second dimension. That is the domain of 2-categories, which although they have three types of elements are nonetheless considered to be two dimensional because we count from zero. That is also where commutative squares come in to play, because as you can imagine squares are quite obviously two dimensional. Functions are 1-dimensional lines or arrows from one place to another. Sets are more analogous to points then anything else, and so naive set theory is zero dimensional. But I think you have the wrong question. You should ask the opposite question: what if functions can be re-expressed or founded on higher dimensional category theory?

voxl 4y ago

Set theory is fine, see the (Stack Project)[https://stacks.math.columbia.edu/browse https://stacks.math.columbia.edu/browse] which develops a ton of modern Category Theory on ZFC (Zermelo-Fraenkel Set Theory with the Axiom of Choice) alone. Alternative foundations of mathematics (Set Theory, Category Theory, Type Theory, and all their variations) can all mutually interpret the other by just postulating sufficiently large universes. You don't pick or advocate one based off its ability to encode mathematics, but instead based on its ability to express your intention and ideas. Really its no different from programming language preference in my book.

Koshkin 4y ago

> i.e. categories and morphisms Objects, not categories.

mydogcanpurr 4y ago

> for some reason I don't think just sets are sufficient The reason you're looking for is that the category of sets is not a set.

vishal0123 4y ago
While category of sets technically could not be expressed as a ZFC set, the idea behind the set theory is enough. Also you could add an axiom[0] in ZFC to make category of set a set. [0]: https://en.wikipedia.org/wiki/Grothendieck_universe https://en.wikipedia.org/wiki/Grothendieck_universe
ginnungagap 4y ago
This is routinely dealt with through Grothendieck universes. Those are a fancy name for what is pretty much an inaccessible level of the cumulative hierarchy, indeed ZFC+"every set belongs to a Grothendieck universe" is equiconsistent with ZFC+"there is a proper class of inaccessible cardinals". This is not a strong assumption over pure ZFC compared to those set theorists interested in large cardinals work with

justatdotin 4y ago

> I wonder to which extent can category theory be re-expressed in terms of sets... yoneda

ginnungagap 4y ago
This also requires universes/inaccessible cardinals to even be stated for categories that are not locally small. But as I mentioned in another comment assuming enough universes exist is not a big deal for set theorists

epgui 4y ago

Set theory is going the way of the dodo. There are modified versions of set theory, but afaik a lot of the more exciting work is happening around type theory and proof assistants these days.

johnbernier 4y ago
Set theory is not going anywhere. It may be pushed to the sidelines by new developments in category theory but that is not the same as going extinct like the dodo.

johnbernier 4y ago

In computer science we typically count from zero, not from one. So even though a category has two different sorts of entities: objects and morphisms, since we count from zero, ordinary categories are one-dimensional. Objects are zero dimensional and morphisms are one dimensional. Since you are concerned with sets and functions, the following analogy is helpful to build your intuition for the subject. Dimension 0: sets Dimension 1: functions Dimension 2: commutative squares Dimension 3: commutative cubes The majority of categories, expressed in terms of structured sets and functions, never touch on the second dimension. That is the domain of 2-categories, which although they have three types of elements are nonetheless considered to be two dimensional because we count from zero. That is also where commutative squares come in to play, because as you can imagine squares are quite obviously two dimensional. Functions are 1-dimensional lines or arrows from one place to another. Sets are more analogous to points then anything else, and so naive set theory is zero dimensional. But I think you have the wrong question. You should ask the opposite question: what if functions can be re-expressed or founded on higher dimensional category theory?