Examples of categories where morphisms are not functions
Can someone give examples of categories where objects are some sort of structure based on sets and morphisms are not functions?
$\endgroup$ 49 Answers
$\begingroup$Let $G$ be a directed graph. Then we can think of $G$ as a category whose objects are the vertices in $G$. Given vertices $a, b \in G$ the morphisms from $a$ to $b$ are the set of paths in the graph $G$ from $a$ to $b$ with composition being concatenation of paths. Note that we allow "trivial" paths that start at a given vertex $a$, traverse no edges, and end at the starting vertex $a$. We have to do this for the category to have identities.
Also we can create categories whose morphisms are not exactly maps but instead equivalence classes of maps. For example given an appropriately nice ring $R$ we can create the stable module category whose objects are $R$-modules. Given two $R$-modules $A$ and $B$ the set of morphisms from $A$ to $B$ is the abelian group of $R$-module homomorphisms from $A$ to $B$ modulo the subgroup of homomorphisms that factor through a projective $R$-module.
$\endgroup$ $\begingroup$Consider the category with exactly two objects $a$ and $b$ (they can be anything you want!) and which has exactly one non-identity morphism —going from $a$ to $b$— which is my chair.
$\endgroup$ 3 $\begingroup$Assuming suitable access to 'large sets', every category is isomorphic to a (possibly large) category of (possibly large) sets and (possibly large) functions.
We can replace each object $A$ with the (possibly large) set of all arrows with codomain $A$. Call this set $|A|$.
Then, we can replace each arrow $f : A \to B$ with the (possibly large) function $|A| \to |B|$ defined by $g \mapsto f \circ g$.
In many categories, we can restrict the sets $|A|$ to include only arrows coming from a particular (small) set of domains. When we can do this, we can avoid making use of large sets/functions.
$\endgroup$ 2 $\begingroup$I think the Fukaya category of a symplectic manifold is an example. The objects are the Lagrangian submanifolds, while the morphisms are the intersection points of the Lagrangians.
$\endgroup$ $\begingroup$Also natural deduction can be considered as a category with formulas as objects and proofs as morphisms.
$\endgroup$ $\begingroup$Algebras/any structures and binary relations or partial functions between them.
$n+1$-manifolds as arrows with $2$ pieces of $n$-dim. boundaries, called cobordism (like a cylinder, between two disks). The objects are the $n$-manifolds.
The objects are points in a topological space, and the arrows are paths. Or, the objects are the paths and the arrows are homotopopies between them. Moreover, we can make higher dimensional categories about this one..
Many many more...
$\endgroup$ 1 $\begingroup$Topological spaces with continuous maps defined up to homotopy.
$\endgroup$ 3 $\begingroup$Suppose that $X$ is a set and$\mathcal{P}(X)$ is the collection of all subsets of $X$. Let the objects objects of the category be the elements of $\mathcal{P}(X)$. Let the morphisms be ordered pairs $(A, B)$ where $A \subseteq B$. Functions are sets of ordered pairs. The composition of $(A,B)$ and $(B, C)$ is $(A, C)$.
$\endgroup$ 2 $\begingroup$Here is another example, much simpler than the previous one I gave: the category of integer matrices of all (finite) dimensions. (Any associative ring of coefficients could be used instead of integers.)
The morphisms of this category are matrices with the usual operation of multiplication (playing the role of composition of morphisms), and the objects are natural numbers. A matrix $m\times n$ is a morphism from $n$ to $m$ (or from $m$ to $n$, passing to the opposite category).
Generally speaking, the true elements of a category are morphisms, and objects are just opaque labels to keep track of when two morphisms can be composed. A category is just a generalisation of a group (where not every element is invertible and not every pair of elements is composable).
However, this may be not a good example, because the question was about objects that "are some sort of structure based on sets," and natural numbers are not exactly "based on sets."
If desired, here is a version for sets as objects: let morphisms between sets $A$ and $B$ be "matrices" $B\times A$ with only finitely many non-zero entries. That is, the objects are arbitrary sets and morphisms $A\to B$ are functions $B\times A\to\mathbf{Z}$ which take non-zero values on finitely many arguments. Define the composition of morphisms as the multiplication of these "matrices."
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