How is a morphism different from a function

Question

How is a morphism (from category theory) different from a function?

Intuitive explanation + maths would be great

Answer 1

If $G$ is a group, there is a famous example that constructs a category in which morphisms are not functions: you take it to consist of a single object called "$\bullet$" and state that the morphisms (of $\bullet$ to $\bullet$, because there is no other object available) are the elements of $G$. Check for yourself that all the properties defining a category are verified by this (admittedly strange) example.

Answer 2

In algebraic geometry, functions and morphisms are both important concepts, but they serve different purposes and have different definitions:

1. Functions:

   • In algebraic geometry, a function typically refers to a regular function or rational function on an algebraic variety.
   • A regular function on an algebraic variety (V) is a function that is locally given by a quotient of polynomials. It's defined on an open subset of (V), where it behaves smoothly.
   • A rational function on an algebraic variety (V) is a function that is locally given by a quotient of polynomials, but it may have poles where it's not defined.
   • Regular and rational functions are used to study the geometry and topology of algebraic varieties.

2. Morphisms:

   • A morphism in algebraic geometry is a structure-preserving map between algebraic varieties. It's a more general notion than a function.
   • Formally, a morphism (f: V \rightarrow W) between algebraic varieties (V) and (W) is a map that associates to each point (P) in (V) a point (f(P)) in (W), such that there exist polynomial functions (f_1, \ldots, f_m) defined on open subsets of (W) such that the composition of (f) with the projection map from (W) to its affine space is given by (f_i \circ f) for each (i).
   • In simpler terms, a morphism preserves the algebraic structure of the varieties involved. It's defined globally on the entire variety.
   • Morphisms are fundamental in algebraic geometry for studying maps between varieties, including embeddings, projections, and more.

In summary, while functions describe local behavior on algebraic varieties, morphisms describe global structure-preserving maps between them. Both concepts are crucial in algebraic geometry for understanding the geometric properties and relationships between algebraic varieties.

Answer 3

I'm not really sure but here is my thought. I think that is a generalization of relations (at least that the idea stemmed from them). A relation is a subset of the cartesian product of two sets, consisting of couples of elements belonging to the two sets. A relation is more general (or generic perhaps) than a function; functions are relations satisfying the condition that the elements in the domain have at most one image element in the codomain. You could picture both relations and functions with a set of arrows pointing from an element to the other from domain to codomain (which is exactly what we are thought to do in elementary math courses). A morphism in the categorical sense does pretty much the same (and indeed is also called an arrow), except for the fact that relations are defined on sets whereas in categories you work with classes and for the fact that a relation establishes a specific "set of arrows" whereas a morphism refers to all possibile arrows between two distinct objects. Also, a Functor seems to me to do pretty much the same as a "standard" morphism (one not in the categorical sense). My feeling is that, in categories, switching the names morphism and functor would have generated less confusion. Functor sounds like a parent of function, though not being the same, whereas morphism would be called the same.