4 ms·

Right, so can someone explain what the hell these slides mean? I studied a lot of abstract mathematics at uni but I wasn't exposed to category theory, and these

by aidos 12y ago

Right, so can someone explain what the hell these slides mean? I studied a lot of abstract mathematics at uni but I wasn't exposed to category theory, and these slides aren't helpful. They start off with objects, what are they? Are they sets? I'm going to assume they're sets. Then the arrows represent mappings, something like a function that takes an input and maps to stuff in one of those other objects (set of objects?). Then the codomain is the set of objects you can get to by calling that function on the domain of objects. Am I in the right ball-park here?

tel 12y ago
Heh, not quite. Treat objects and arrows abstractly like you're building a graph. Actual meaning can be applied later. A category is thus a set (or proper class) of objects, O, and a set (or proper class) of arrows, A along with two mappings dom and cod from A -> O. The laws are first that for any object (o in O) there exists an arrow (id(o) in A) such that dom(id(o)) = o and cod(id(o)) = o and second that if you have two arrows (f, g in A) such that cod(f) = dom(g) then there is defined their composition (compose(f,g) in A) such that dom(compose(f,g)) = dom(f) and cod(compose(f,g)) = cod(g). You can draw all that as the objects and arrows and things these slides display. So far this just gives you a graph-like structure. The "category laws" ensure coherence of the category and state that (compose(f, id(x)) = f) when defined, (compose(id(x), g) = g) when defined, and that compose is associative. From an abstract algebraic POV this is almost like saying that A is a monoid which ought to (correctly) suggest that most of the interesting structure of a category is carried by the arrows. Now, if you want to talk about specific categories you can do what you were doing. For instance, the category of sets, Set, is the category where you take O to be the proper class of all sets and A to be the functions between them. Immediately, you'll find that dom(.)/cod(.) are the domain and codomain of your functions, id(.) locates the identity function on a set, and compose(.,.) composes functions. As a more interesting example you could consider Grp the category where O is all groups and A is all group homomorphisms. This is a subcategory of Set obviously and it also comes with much more interesting structure. For instance, an important category theoretic statement about Grp is that Grp has a "zero object", which is to say that (a) there exists an object init such that there is a unique arrow from init to any other object, (b) there exists an object fin such that there is a unique arrow from any other object to fin, and (c) init = fin. http://en.wikipedia.org/wiki/Zero_object_%28algebra%29 http://en.wikipedia.org/wiki/Zero_object_%28algebra%29
- deleted 12y ago
  [deleted]
xyzzyz 12y ago
Ah, the joys of learning math from slides without the talk the slides were helpers to. The objects here are not necessarily sets, and the arrows are not necessarily functions -- this is actually one of the points of category theory, to abstract the notion of morphism away from set theory. This becomes necessary, when in your regular mathematical life you encounter morphisms which are not strictly functions, but nevertheless in many ways behave as such. Best example would be the homotopy classes of maps between topological spaces. Homotopy classes do not map points from domain space to codomain space, but nevertheless you can compose them, there's a homotopy class of identity, which acts as an identity on composition, sometimes there are homotopy classes that are inverses and so on. If this sounds complicated and unnatural, that's because it is: it is pointless to learn category theory before learning where and how it is useful. This is completely opposite order than in which it was conceived, and totally misses the purpose the category theory, which is to make everything clearer, to reduce the number of notions, to make it easier to notice analogous behavior between separate concepts, to reduce number of repeating arguments. Learning what a morphism or functor or natural transformation is, without having tens of examples of these notions, only increases the number of concepts without a corresponding increase in understanding. My advice is, if you want to learn category theory, learn algebraic topology first, or at least abstract algebra (groups, rings, modules, fields). Otherwise, you'll see very little purpose, and very little use.
- nrinaudo 12y ago
  I disagree with your last point, to some extent. While not strictly necessary, learning a bit of category theory makes FP abstractions such as functors and monoids a bit more accessible. I can't disagree with the fact that learning category for that sole purpose sounds like a lot of work for too little benefit, though.
- ww2 12y ago
  Good point but examples can also be drawn from other fields like programming rather than from mathematics. So far I found that the link below is very enlightening. http://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/ http://bartoszmilewski.com/2014/10/28/category-theory-for-pr...
enord 12y ago
They are arrows. Arrows and objects are the primitives of category theory, just like sets and set membership are the primitives of set theories (well most of them anyway). A category that captures the notion of sets and maps can be described in terms of arrows, objects and some semantic rules about which arrows (here interpreted as maps) connect which objects (here interpreted as sets). Wiki on the "Category of sets [and maps]" -> http://en.wikipedia.org/wiki/Category_of_sets http://en.wikipedia.org/wiki/Category_of_sets Other rules about arrows between objects describe categories that capture other mathematical concepts.

by aidos 12y ago

tel 12y ago

Heh, not quite. Treat objects and arrows abstractly like you're building a graph. Actual meaning can be applied later. A category is thus a set (or proper class) of objects, O, and a set (or proper class) of arrows, A along with two mappings dom and cod from A -> O. The laws are first that for any object (o in O) there exists an arrow (id(o) in A) such that dom(id(o)) = o and cod(id(o)) = o and second that if you have two arrows (f, g in A) such that cod(f) = dom(g) then there is defined their composition (compose(f,g) in A) such that dom(compose(f,g)) = dom(f) and cod(compose(f,g)) = cod(g). You can draw all that as the objects and arrows and things these slides display. So far this just gives you a graph-like structure. The "category laws" ensure coherence of the category and state that (compose(f, id(x)) = f) when defined, (compose(id(x), g) = g) when defined, and that compose is associative. From an abstract algebraic POV this is almost like saying that A is a monoid which ought to (correctly) suggest that most of the interesting structure of a category is carried by the arrows. Now, if you want to talk about specific categories you can do what you were doing. For instance, the category of sets, Set, is the category where you take O to be the proper class of all sets and A to be the functions between them. Immediately, you'll find that dom(.)/cod(.) are the domain and codomain of your functions, id(.) locates the identity function on a set, and compose(.,.) composes functions. As a more interesting example you could consider Grp the category where O is all groups and A is all group homomorphisms. This is a subcategory of Set obviously and it also comes with much more interesting structure. For instance, an important category theoretic statement about Grp is that Grp has a "zero object", which is to say that (a) there exists an object init such that there is a unique arrow from init to any other object, (b) there exists an object fin such that there is a unique arrow from any other object to fin, and (c) init = fin. http://en.wikipedia.org/wiki/Zero_object_%28algebra%29 http://en.wikipedia.org/wiki/Zero_object_%28algebra%29

deleted 12y ago
[deleted]

xyzzyz 12y ago

Ah, the joys of learning math from slides without the talk the slides were helpers to. The objects here are not necessarily sets, and the arrows are not necessarily functions -- this is actually one of the points of category theory, to abstract the notion of morphism away from set theory. This becomes necessary, when in your regular mathematical life you encounter morphisms which are not strictly functions, but nevertheless in many ways behave as such. Best example would be the homotopy classes of maps between topological spaces. Homotopy classes do not map points from domain space to codomain space, but nevertheless you can compose them, there's a homotopy class of identity, which acts as an identity on composition, sometimes there are homotopy classes that are inverses and so on. If this sounds complicated and unnatural, that's because it is: it is pointless to learn category theory before learning where and how it is useful. This is completely opposite order than in which it was conceived, and totally misses the purpose the category theory, which is to make everything clearer, to reduce the number of notions, to make it easier to notice analogous behavior between separate concepts, to reduce number of repeating arguments. Learning what a morphism or functor or natural transformation is, without having tens of examples of these notions, only increases the number of concepts without a corresponding increase in understanding. My advice is, if you want to learn category theory, learn algebraic topology first, or at least abstract algebra (groups, rings, modules, fields). Otherwise, you'll see very little purpose, and very little use.

nrinaudo 12y ago
I disagree with your last point, to some extent. While not strictly necessary, learning a bit of category theory makes FP abstractions such as functors and monoids a bit more accessible. I can't disagree with the fact that learning category for that sole purpose sounds like a lot of work for too little benefit, though.
ww2 12y ago
Good point but examples can also be drawn from other fields like programming rather than from mathematics. So far I found that the link below is very enlightening. http://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/ http://bartoszmilewski.com/2014/10/28/category-theory-for-pr...

enord 12y ago

They are arrows. Arrows and objects are the primitives of category theory, just like sets and set membership are the primitives of set theories (well most of them anyway). A category that captures the notion of sets and maps can be described in terms of arrows, objects and some semantic rules about which arrows (here interpreted as maps) connect which objects (here interpreted as sets). Wiki on the "Category of sets [and maps]" -> http://en.wikipedia.org/wiki/Category_of_sets http://en.wikipedia.org/wiki/Category_of_sets Other rules about arrows between objects describe categories that capture other mathematical concepts.