Thread: difference between morphism and function

Hi.

What is the difference between morphism and function?

Originally Posted by vernal

What is the difference between morphism and function?

In what context do you ask this question?

Originally Posted by Plato

In what context do you ask this question?

yes,

Originally Posted by vernal

yes,

You did not answer the question about context!

a morphism is a categorical concept: lots of different "kinds" of things could be morphisms, depending on the context.

in the context of groups, rings or modules, a morphism is a group, ring or module homomorphism.

in the context of partially-ordered sets, a morphism is an order-preserving map.

in the context of topology with respect to homotopy classes, a morphism is an (induced) map between sets of homotopy classes on two spaces.

in the special case where we consider the category of all (locally-small) sets, a morphism is a function.

in many categories, the objects are "sets with additional structure" and the morphisms are "structure-preserving functions". this is not always the case, however:

one such "strange" category, is where the objects are (non-zero) natural numbers, and a morphism m→n is an mxn matrix with entries from a commutative ring.

another example is any monoid, which can be regarded as a category with one object M, and a morphism is simply an element x of M.

still another example is the category whose objects are elements of the set of all divisors of a positive integer, where there is a unique arrow a→b iff a divides b.

a related example to the above is the category n, where n is a non-negative natural number, whose objects are the natural numbers ≤ n, and we have an morphism a→b,

iff a ≤ b. these categories are in a 1-1 correspondence with the finite ordinal numbers.

Originally Posted by Deveno

a morphism is a categorical concept: lots of different "kinds" of things could be morphisms, depending on the context.

in the context of groups, rings or modules, a morphism is a group, ring or module homomorphism.

in the context of partially-ordered sets, a morphism is an order-preserving map.

in the context of topology with respect to homotopy classes, a morphism is an (induced) map between sets of homotopy classes on two spaces.

in the special case where we consider the category of all (locally-small) sets, a morphism is a function.

in many categories, the objects are "sets with additional structure" and the morphisms are "structure-preserving functions". this is not always the case, however:

one such "strange" category, is where the objects are (non-zero) natural numbers, and a morphism m→n is an mxn matrix with entries from a commutative ring.

another example is any monoid, which can be regarded as a category with one object M, and a morphism is simply an element x of M.

still another example is the category whose objects are elements of the set of all divisors of a positive integer, where there is a unique arrow a→b iff a divides b.

a related example to the above is the category n, where n is a non-negative natural number, whose objects are the natural numbers ≤ n, and we have an morphism a→b,

iff a ≤ b. these categories are in a 1-1 correspondence with the finite ordinal numbers.

Thanks for your help.
if f be a morphism , then f(0)=0 ?? or what case is true? just f is homomorphism?

again, the answer depends on context. IF a category has a "zero object", that is: an object with is both initial AND terminal (in that there is just ONE possible morphism 0→A for any object A, and ONE possible morphism A→0) then the only possible map 0→0 is called the "zero morphism". in the category of groups, for example, the 1-element group (or trivial group) (often just called "{e}") is a zero object (up to isomorphism, as is the case with all "categorical constructions"). in the category of vector spaces over a given field, the "zero object" is a 0-dimensional vector space.

in general categories, we may not HAVE a 0-object. for example, the null set is an initial object in the category of (locally-small) sets, but the terminal object is a (any) singleton set (isomorphisms in this category are just bijective functions, and clearly the sets {x} and {y} are set-isomorphic (just define f(x) = y)) so we cannot say for a function f:Ø→A that f(Ø) = Ø, although we CAN say that for a function g:A→{x} that g(a) = x, for all a in A.

for many algebraic structures (many of which form categories, with a corresponding notion of (homo-)morphism), 0-objects DO exist, and it IS true that f(0) = 0. this is true for:

monoids (but NOT semi-groups)
groups
abelian groups
rings
fields
modules
vector spaces
algebras

which although being just a very few types of structure, cover a lot of ground.

notably absent from the list above, is the category of topological spaces, with continuous maps as morphisms. in fact, it is easy to see that even if X = Y possesses an element 0 (for example, they might both be R, the real numbers), it is not necessarily true that a continuous function maps 0 to 0, in fact: x→x+1 is a counter-example.