SubsubsectionTopology from the categorical viewpoint
One of the primary insights of twentieth century mathematics is that objects should not be studied in isolation. Rather, to understand objects we must also understand relationships between objects. Topologies offer one notion of “relation”, organizing the points of a set into neighborhoods. As an abstraction of geometry, topology is immensely successful, but it fails to capture a second, more structural notion of relation.
Category theory organizes objects by their transformations. Once these notions of object and relation are abstracted, it becomes possible to compare and contrast different fields of mathematics. Theorems about wildly different mathematical objects are often identical in their categorical content. In this manner, category theory becomes a meta-mathematical tool for both identifying and conjecturing structural results.
Our present goal is to use the language of category theory to motivate definitions and interpret theorems. Moving into the algebro-topological portion of the course, this language will become even more important as we use the homology functor to compare the categories of topological spaces and vector spaces.
We begin with a motivating example. Sets are a type of mathematical object. Sets are related by functions. Each function has a domain (source) and codomain (target). In standard notation, \(f:A\to B\) denotes a function with domain \(A\) and codomain \(B\text{.}\) Of course, functions admit composition: given \(g:B\to C\) and \(f:A\to B\) we can form \(g\circ f:A\to C\) by assigning \(g(f(a))\) to \(a\in A\text{.}\) This composition is associative: if \(f:A\to B\text{,}\)\(g:B\to C\text{,}\) and \(h:C\to D\) are functions, then
Moreover, each set (even the empty set!) admits an identity function \(\operatorname{id}_A:A\to A\text{.}\) This function takes each \(a\in A\) to \(a\) and satisfies the following property: for each \(g:A\to B\) and each \(f:C\to A\text{,}\)
In the following definition, we will see that categories consist of objects, morphisms (with source and target objects), composition, and identity morphisms satisfying associativity and identity properties. As such, sets and functions form our first example of a category.
A category \(\mathcal{C}\) consists of a collection of objects \(\mathrm{Ob}\,\mathcal{C}\) and a collection of morphisms \(\mathrm{Mor}\,\mathcal{C}\) along with assignments \(s,t:\mathrm{Mor}\,\mathcal{C}\to \mathrm{Ob}\,\mathcal{C}\) (called the source and target maps). Let \(\mathcal{C}(x,y)\subseteq \mathrm{Mor}\,\mathcal{C}\) denote the collection of morphisms with source \(x\in \mathrm{Ob}\,\mathcal{C}\) and target \(y\in \mathrm{Ob}\,\mathcal{C}\text{.}\) Then for each \(x,y,z\in \mathrm{Ob}\,\mathcal{C}\text{,}\)\(\mathcal{C}\) is also equipped with a composition
Additionally, for each \(x\in \mathrm{Ob}\,\mathcal{C}\) there is an identity morphism \(\operatorname{id}_x\in \mathcal{C}(x,x)\text{.}\) This data must satisfy the following properties:
It is useful to think about categories diagrammatically. These diagrams use letters to represent objects, and labelled arrows to represent morphisms. So if \(x,y\in \mathrm{Ob}\,\mathcal{C}\) and \(f\in \mathcal{C}(x,y)\text{,}\) we may draw an arrow from \(x\) to \(y\) labelled \(f\) in order to represent that \(f\) is a morphism with source \(x\) and target \(y\text{.}\) (Note that we have dropped \(\mathcal{C}\) from our notation here: usually it will be clear from context which category we are working in.) We may also write \(f\colon x\to y\) to represent \(f\in \mathcal{C}(x,y)\text{.}\)
Now suppose that in addition to \(f\) we also have a morphism \(g\colon y\to z\text{.}\) Composition tells us that we then get a new morphism \(g\circ f\colon x\to z\text{.}\) We can put all of this information into a single commutative diagram.
A triangle with objects \(x\text{,}\)\(y\text{,}\)\(z\) and arrows \(f\colon x\to y\text{,}\)\(g\colon y\to z\text{,}\) and \(g\circ f\colon x\to z\text{.}\) The arrows form a triangle with indicated sources and targets.
When we say that a diagram like the one above commutes, we are saying precisely that the two paths from \(x\) to \(z\) determine the same morphism, i.e. the bottom arrow equals the composite of the two slanted arrows. The geometric presentation of the diagram is unimportant; we can rearrange the nodes without changing the meaning.
We can also use diagrams to express the axioms for a category. Associativity can be pictured as the statement that two different ways of composing \(f\text{,}\)\(g\text{,}\) and \(h\) agree.
A four-object diagram showing arrows \(f\text{,}\)\(g\text{,}\)\(h\) and the composites \(g\circ f\text{,}\)\(h\circ g\text{.}\)
We can interpret this axiom as saying that we can paste together commutative triangles to produce commutative quadrilaterals. The identity axiom also admits a simple diagrammatic expression.
We have already mentioned that sets and functions form a category. We denote this category \(\mathsf{Set}\text{.}\) Similarly, there is a category \(\mathsf{FinSet}\) with objects finite sets and morphisms functions between finite sets.
Example2.2.3.The category of vector spaces and linear transformations.
Let \(k\) be a field and let \(\mathsf{Vect}_k\) have objects \(k\)-vector spaces and morphisms \(k\)-linear transformations. Since linear transformations compose (in the set-theoretic sense) to give new linear transformations, \(\mathsf{Vect}_k\) is also a category. (It is obvious that the identity function \(1_V:V\to V\) is linear.) We can also consider the category \(\mathsf{FinVect}_k\) of finite-dimensional \(k\)-vector spaces and \(k\)-linear transformations.
The empty category \(\varnothing\) has no objects (i.e. \(\mathrm{Ob}\,\varnothing=\varnothing\)) and no morphisms (\(\mathrm{Mor}\,\varnothing=\varnothing\)). The source, target, and composition functions are all the empty function \(\varnothing\to\varnothing\) and all properties are satisfied vacuously!
The trivial category \(\bullet\) has a singleton set \(\{*\}\) for its objects and a single morphism (necessarily the identity on \(*\)), \(\operatorname{id}_*:* \to *\text{.}\) The only composition to define is \(\operatorname{id}_*\circ \operatorname{id}_*=\operatorname{id}_*\) and this is enough to check the associativity and identity properties as well.
Example2.2.6.A category whose morphisms are not functions.
Not every category has special classes of functions as morphisms. Consider \(\mathsf{Mat}_k\) whose objects are the natural numbers \(\mathbb{N}=\{0,1,2,\ldots\}\) and whose morphisms \(\mathsf{Mat}_k(m,n)\) are \(n\times m\) matrices with entries in a field \(k\text{.}\) Composition is given by matrix multiplication (check compatibility!) and \(\operatorname{id}_k\) is the \(k\times k\) identity matrix. Since matrix multiplication is associative, this forms a category.
If you are suspicious that this category is eerily similar to \(\mathsf{FinVect}_k\text{,}\) worry not: it is! After studying equivalences of categories and skeleta, you will understand exactly how similar.
The following two examples should seem completely obvious if you have already taken a course in abstract algebra. If you have not, there is no harm in skipping them.
There are categories \(\mathsf{Gp}\text{,}\)\(\mathsf{FinGp}\text{,}\) and \(\mathsf{AbGp}\) of groups, finite groups, and Abelian groups, respectively. In each case, morphisms are group homomorphisms.
There are categories \(\mathsf{Ring}\) and \(\mathsf{CommRing}\) of rings and commutative rings, respectively. In both cases, morphisms are ring homomorphisms. There is also a category \(\mathsf{Field}\) of fields and field homomorphisms.
Example2.2.9.The category of topological spaces and continuous functions.
The category \(\mathsf{Top}\) has topological spaces as its objects and continuous functions as its morphisms. In order to check this, we must see that the composition of continuous functions is continuous and that identity functions are continuous. If \(X\) is a topological space and \(\operatorname{id}_X:X\to X\) is the identity function, then for any \(U\subseteq X\) open, \(\operatorname{id}_X^{-1}(U)=U\) is open in \(X\text{,}\) so \(\operatorname{id}\operatorname{id}_X\) is continuous. The associativity and identity axioms hold because they hold for functions.
Of course, there is also a category whose objects are topological spaces and morphisms are arbitrary functions between underlying sets. And while we are free to define such a category, it is not of particular interest. A category is a tool for studying relationships between objects. If we choose the wrong set of relationships (i.e. the wrong morphisms), then we end up with an uninteresting — or, worse yet, misleading — category.
If category theory is all about relations between mathematical objects, and categories are themselves mathematical objects then we need a way to compare — or relate — categories. This is what functors do.
A functor from a category \(\mathsf C\) to a category \(\mathsf D\text{,}\) denoted \(F\colon \mathsf C\to \mathsf D\) is an assignment on objects \(F\colon \operatorname{Ob} \mathsf C\to \operatorname{Ob} \mathsf D\) together with an assignment on morphisms \(F\colon \mathsf C(x,y)\to \mathsf D(Fx,Fy)\) for each \(x,y\in \operatorname{Ob} \mathsf C\) satisfying the following properties:
for all \(x\in\operatorname{Ob} \mathsf C\text{,}\)\(F\operatorname{id}_x = \operatorname{id}_{Fx}\text{,}\) and
