5.1 Categories and Functors · Homology & Cohomology

As Eilenberg-Mac Lane first observed, ‘category’ has been defined in order to be able to define ‘functor’ and ‘functor’ has been defined in order to be able to define ‘natural transformation’.

— Saunders Mac Lane, Categories for the Working Mathematician (2nd ed.), p. 18

Look back over this book and you will find two kinds of characters. There were things — sets, groups, vector spaces, spaces, simplicial complexes, chain complexes — and there were arrows between things — functions, homomorphisms, linear maps, continuous maps, simplicial maps, chain maps. The things got the chapter titles, but the arrows did the real work. Brouwer’s fixed-point theorem was not proved by staring at a disk; it was proved by asking what a map with no fixed point would force homology to do (§3.5 Maps, Invariance, and First Triumphs). This chapter is about what all those arrows have in common.

The common pattern has a name: a category. A map that turns one kind of thing-and-arrow world into another, the way homology turns spaces into groups, is a functor. And a way of comparing two functors that involves no arbitrary choices is a natural transformation. These three ideas were invented in the 1940s by a topologist and an algebraist, Samuel Eilenberg and Saunders Mac Lane, to straighten out a puzzle about homology (Eilenberg & Mac Lane 1945). So you are not starting a new subject so much as learning the names of things you have been doing since Part I.

You have been doing category theory all along

Here is a game. For each line below, the left column names some objects and the middle column names the arrows between them. Read down the right column: it is always the same three facts.

Objects	Arrows	What we always used
sets (§1.1 Sets and Functions)	functions	follow one arrow by another; every object has a do-nothing arrow; brackets don’t matter
groups, abelian groups	homomorphisms
vector spaces	linear maps (matrices)
topological spaces	continuous maps
simplicial complexes	simplicial maps
chain complexes	chain maps

In every case you can compose: if $f$ goes from $A$ to $B$ and $g$ goes from $B$ to $C,$ then “first $f,$ then $g ”$ is an arrow $g \circ f$ from $A$ to $C$ (read “ $g$ after $f ”,$ as in §1.1 Sets and Functions). A composite of homomorphisms is a homomorphism; a composite of continuous maps is continuous; a product of matrices is a matrix. In every case there is an identity, the arrow that does nothing. And in every case composition is associative: following $f,$ then $g,$ then $h$ gives the same result however you group the steps.

Those three facts sound too obvious to be worth stating. But think about how much of this book rested on them. When we proved in §3.5 Maps, Invariance, and First Triumphs that a continuous map $f$ induces a homomorphism $f_{*}$ on homology, the two facts that made it useful were

(g \circ f)_{*} = g_{*} \circ f_{*} and (id_{X})_{*} = id_{H_{n} (X)} .

Those equations say: homology respects composition and identities. They are statements about arrows, not about holes. Category theory is what you get when you take statements like these seriously and study them for their own sake. Its subtitle in this book — “the mathematics of mathematics” — is a little grand, but not wrong: it is a language for the patterns that different parts of mathematics share.

“…mathematics is the art of giving the same name to different things. … When language has been well chosen, one is astonished to find that all demonstrations made for a known object apply immediately to many new objects: nothing requires to be changed, not even the terms, since the names have become the same.” — Henri Poincaré, Science and Method, in Francis Maitland’s translation (Poincaré 1914, p. 34)

Category theory takes Poincaré’s remark literally. A bijection of sets, an isomorphism of groups, a homeomorphism of spaces and a homotopy equivalence will all turn out to be “the same thing”: an isomorphism in the appropriate category. Quotients, direct sums, kernels and products will turn out to be instances of one idea, the universal property. Seeing the shared shape does not do the hard work for you — computing $H_{n} (S^{n})$ still needed geometry — but it tells you which work needs doing and which comes for free.

Categories: dots and arrows

Here is the pattern written out in full. It looks long, but every clause is one of the table’s three facts, or the bookkeeping needed to state them. Read it once, then play with the figure after it.

Notice what the definition does not say. It never says that objects have elements, or that arrows are functions. An arrow is anything with a source and a target that can be composed, and that freedom lets some unexpected things qualify, as the figure and the examples below show.

Three small categories Click arrows head to tail · a third arrow tests associativity

Click an arrow to start a path.

Figure 5.1.1 Click an arrow, then an arrow that starts where the first one ends: the composite lights up in gold. In A small category the faint dashed arrows are composites that the axioms force into existence —

g \circ f

is a genuine arrow from

A

C,

different from

k .

Click three arrows in a row to see both bracketings agree, or click an identity loop to watch it change nothing. In Divisors of 12 an arrow

a \to b

means “

a

divides

b ”;

in A group there is one object and the arrows are rotations.

A gallery of categories

Here are the categories this book has quietly been living in. For each, ask: what are the objects, what are the arrows, how do arrows compose, and what are the identities?

$Set$ : objects are sets, arrows are functions, composition is composition of functions, identities are the identity functions $x \mapsto x .$
$Grp$ and $Ab$ : objects are groups (respectively abelian groups, §1.3 Groups), arrows are homomorphisms. That a composite of homomorphisms is again a homomorphism is a small calculation (Exercise 2).
$Vect$ : real vector spaces and linear maps (§1.5 Linear Algebra). There is one such category for each field of scalars; the book used $R$ and $Z /2 .$
$Top$ : topological spaces and continuous maps (§2.1 Spaces and Continuity).
$hTop$ , the homotopy category: the same objects as $Top,$ but an arrow $X \to Y$ is a homotopy class $[f]$ of continuous maps (§2.3 Homotopy). We compose classes by composing representatives, $[g] \circ [f] = [g \circ f] .$ This is well defined — the answer does not depend on which representatives you pick — because if $f ≃ f^{'}$ and $g ≃ g^{'}$ then $g \circ f ≃ g^{'} \circ f^{'} .$ Once again a quotient (“declare homotopic maps equal”) produces a new world in which to work.
$Ch$ : chain complexes of abelian groups and chain maps (§3.2 Chains and the Boundary Operator, §3.5 Maps, Invariance, and First Triumphs). The next chapter starts here.

Those are all “sets with structure, and functions that respect it”. Now three categories of a different flavour.

Example A partially ordered set is a category

A partially ordered set, or poset, is a set with a relation $\leq$ that is reflexive ( $a \leq a),$ transitive ( $a \leq b$ and $b \leq c$ give $a \leq c)$ and antisymmetric ( $a \leq b$ and $b \leq a$ give $a = b)$ — the kind of relation met in §1.2 Equivalence and Quotients, but one-directional. Make it a category: the objects are the elements, and there is exactly one arrow $a \to b$ when $a \leq b,$ and none otherwise. Composition exists because $\leq$ is transitive; identities exist because $\leq$ is reflexive. Associativity is automatic, since between two objects there is at most one arrow.

The divisors of 12 under divisibility, $a ∣ b,$ form such a category (Figure 5.1.1). So do the real numbers under $\leq .$ The arrows are not functions of any kind: an arrow is nothing more than the fact “ $a \leq b ”.$

Commutative diagrams, and one word for sameness

Reading a diagram

Mathematicians draw categories the way you would draw a map of roads: dots for objects, arrows for arrows. Such a picture is a diagram. A diagram commutes if any two routes along arrows with the same starting point and the same end point give the same composite. For a square

A h ↓ ⏐ C f k B ↓ ⏐ g D

commuting means one single equation: $g \circ f = k \circ h .$ “Across, then down” equals “down, then across”. For a triangle it means that the long side is the composite of the two short ones. This is the same idea that commutative diagrams had when they first appeared in §1.1 Sets and Functions; now we will use them constantly.

All roads agree Set the starting temperature with the slider · press “Follow both roads”

gold road: convert, then warm by 18°F → 86°F teal road: warm by 10°C, then convert → 86°F The square commutes: every starting temperature gives the same answer both ways.

Starting temperature (°C) 20°C

Figure 5.1.2 A commutative square from everyday life. Converting Celsius to Fahrenheit and then warming by

1 8^{\circ} F

gives the same reading as warming by

1 0^{\circ} C

and then converting, for every starting temperature: the square commutes. Change the right-hand arrow to “

+ 1 0^{\circ} F ”

and it no longer does — a diagram is a claim, and it can be false.

Why draw pictures instead of writing equations? Because in a large diagram there are many routes, and “the diagram commutes” packs all their equations into one glance. You have already met several commutative squares in disguise. The definition of a chain map $f$ between chain complexes (§3.5 Maps, Invariance, and First Triumphs) is the statement that every square

C_{n} f_{n} ↓ ⏐ D_{n} \partial \partial C_{n - 1} ↓ ⏐ f_{n - 1} D_{n - 1} commutes: f_{n - 1} \circ \partial = \partial \circ f_{n} .

A diagram chase is an argument that follows a single element around a diagram: start with $c \in C_{n},$ push it across, push it down, compare with the other route. Proving that chain maps send cycles to cycles is a two-step chase: if $\partial c = 0,$ then $\partial (f_{n} c) = f_{n - 1} (\partial c) = f_{n - 1} (0) = 0 .$ In §5.2 Homological Algebra you will watch a longer chase, the snake lemma, animated step by step.

Isomorphism: one word for every kind of sameness

Throughout the book we needed different words for “the same”: bijection, group isomorphism, linear isomorphism, homeomorphism, homotopy equivalence. In category language they are all one word.

The inverse is unique: if $g$ and $g^{'}$ both work, then $g = g \circ 1_{B} = g \circ (f \circ g^{'}) = (g \circ f) \circ g^{'} = 1_{A} \circ g^{'} = g^{'} .$ That little proof used nothing but the axioms, so it holds in every category at once: it proves that inverse functions, inverse matrices and inverse homeomorphisms are unique, and that a homotopy inverse is unique up to homotopy, all in one line. This is Poincaré’s promise kept — the demonstration made for a known object applies immediately to the new ones. Now read the definition in each of our categories:

Category	An isomorphism is…
$Set$	a bijection
$Grp,$ $Ab$	a group isomorphism (§1.3 Groups)
$Vect$	an invertible linear map (an invertible matrix)
$Top$	a homeomorphism (§2.1 Spaces and Continuity)
$hTop$	a homotopy equivalence (§2.3 Homotopy)
a poset	only the identity arrows (by antisymmetry)
a group, as a one-object category	every arrow

The line for $hTop$ is worth pausing on. A homotopy equivalence was defined in §2.3 Homotopy as a map $f : X \to Y$ with a map $g : Y \to X$ such that $g \circ f ≃ id_{X}$ and $f \circ g ≃ id_{Y} .$ In the homotopy category, where homotopic maps are equal arrows, that is literally the definition of an isomorphism. The “weaker” notion of sameness is the ordinary notion in a different category.

Functors: maps between categories

You have already met the most important example. Homology takes a space and returns a group; it takes a map of spaces and returns a homomorphism; and it sends a composite of maps to the composite of the homomorphisms. It translates one world of objects and arrows into another, and the arrows go on talking to each other in the same way. Such a translation is a functor.

In words: a functor sends dots to dots and arrows to arrows, and it sends every commutative diagram to a commutative diagram. (Check this: if $g \circ f = k \circ h,$ apply $F$ to both sides and use the first rule.) That last sentence is the whole reason functors are useful, as we will see shortly. Tom Leinster describes the strategy of algebraic topology in exactly these terms:

“…the strategy is to learn about a space by extracting data from it in some clever way, assembling that data into an algebraic structure, then studying the algebraic structure instead of the original space.” — Tom Leinster, Basic Category Theory (Leinster 2014, Examples 1.2.5)

Here are functors you already know, some of them old friends wearing a new badge.

Homology, $H_{n} : Top \to Ab .$ A space $X$ goes to the group $H_{n} (X),$ a map $f$ goes to $f_{*} = H_{n} (f) .$ The two functor laws are the equations from §3.5 Maps, Invariance, and First Triumphs quoted above. Because homotopic maps induce the same homomorphism, $H_{n}$ is even a functor $hTop \to Ab$ (Hatcher 2002, §2.3).
Forgetful functors, such as $U : Grp \to Set,$ which sends a group to its underlying set and a homomorphism to the same function, forgetting that there was an operation. Likewise $Ab \to Grp,$ $Top \to Set,$ $Vect \to Ab .$ They sound trivial, and they are; but they let us say precisely “this is the same map, viewed with less structure”.
The free functor $Z [-] : Set \to Ab .$ A set $S$ goes to the free abelian group $Z [S]$ of formal sums $n_{1} s_{1} + \dots + n_{k} s_{k}$ (§1.4 Abelian Groups and Formal Sums); a function $S \to T$ goes to the homomorphism that applies it to each symbol and keeps the coefficients. The chain group $C_{k} (K)$ is $Z [the k -simplices of K] .$
Path components, $π_{0} : Top \to Set .$ A space goes to the set of its path components; a continuous map sends paths to paths, so it sends each component into a component.
The fundamental group, $π_{1} : Top_{*} \to Grp$ (§2.3 Homotopy). Here $Top_{*}$ is the category of based spaces: spaces with a chosen point, and continuous maps that send chosen point to chosen point. The basepoint is needed because a loop has to start somewhere.
Composites of functors. Simplicial homology was built in two stages: first the chain complex $C_{∙} (K),$ then its homology. Each stage is a functor — from simplicial complexes to $Ch,$ and from $Ch$ to $Ab$ — and $H_{n}$ is their composite. Functors compose like arrows, so categories and functors themselves form a category (as long as we stick to small categories, whose objects form a set).

The next figure shows one functor at work on a small diagram. On top are three spaces and two continuous maps. The map $f : S^{1} \to T^{2}$ wraps the circle $p$ times around one direction of the torus and $q$ times around the other: if we think of points of the circle as complex numbers $z$ of size 1, and points of the torus as pairs of them, $f (z) = (z^{p}, z^{q}) .$ The map $g : T^{2} \to S^{1}$ sends $(x, y)$ to $x^{r} y^{s} :$ it multiplies angles by $r$ and $s$ and adds them. On the bottom is what $H_{1}$ makes of all this.

A functor turns spaces into groups Change p, q (the loop) and r, s (the map)

H_{1} (g \circ f) = (3) = (21) (11) = H_{1} (g) \circ H_{1} (f)

The loop goes 1 time around one way and 1 around the other; the composite winds 3 times around the last circle.

loop

f

p 1

q 1

map

g

r 2

s 1

Figure 5.1.3

H_{1}

sends each circle to

Z,

the torus to

Z^{2},

and each map to a matrix:

f_{*}

sends the generator

1

to the class

(p, q)

of the gold loop, and

g_{*} (a, b) = r a + s b .

The composite

g \circ f

winds the circle

r p + s q

times around the circle on the right, and

H_{1}

turns it into multiplication by

r p + s q

— which is exactly the matrix product

(r s) (p q) .

Functoriality,

H_{1} (g \circ f) = H_{1} (g) \circ H_{1} (f),

holds for every choice you make.

When arrows turn around

One of the book’s four recurring ideas is the reversal of arrows. Preimages ran backwards along functions (§1.1 Sets and Functions); the transpose of a matrix ran backwards along a linear map (§1.5 Linear Algebra); differential forms were pulled back along smooth maps (§4.3 Differential Forms); and cochains, the measurements of cohomology, travelled backwards along continuous maps (§4.2 Cohomology Groups). Category theory has a home for all of these.

Here is the reversed-arrow thread of the book, now seen as a list of contravariant functors.

Preimage. Send a set $S$ to the set $P (S)$ of its subsets, and a function $f : S \to T$ to $f^{- 1} : P (T) \to P (S),$ $B \mapsto f^{- 1} (B) .$ Then $(g \circ f)^{- 1} = f^{- 1} \circ g^{- 1} :$ to find what lands in $B$ after doing $f$ then $g,$ first pull $B$ back along $g,$ then along $f .$
Duals and transposes. Send a vector space $V$ to its dual $V^{*} = Hom (V, R),$ the space of linear “measurements” $φ : V \to R,$ and a linear map $T : V \to W$ to $T^{*} : W^{*} \to V^{*},$ $φ \mapsto φ \circ T :$ to measure a vector of $V,$ push it to $W$ and measure there. In matrices, $T^{*}$ is the transpose, and the rule $(g \circ f)^{*} = f^{*} \circ g^{*}$ is the old identity $(B A)^{T} = A^{T} B^{T} .$
Measuring with values in a group. More generally, for a fixed abelian group $G,$ sending $A \mapsto Hom (A, G)$ is a contravariant functor $Ab^{op} \to Ab .$ Cochains are exactly this applied to chains: $C^{n} (X; G) = Hom (C_{n} (X), G) .$
Cohomology, $H^{n} (-; G) : Top^{op} \to Ab .$ A map $f : X \to Y$ induces $f^{*} : H^{n} (Y; G) \to H^{n} (X; G),$ going backwards, with $(g \circ f)^{*} = f^{*} \circ g^{*} .$ This is what we meant, back in §4.2 Cohomology Groups, by calling cohomology contravariant.

The same diagram, seen by cohomology Switch between H₁ and H¹ · change p, q, r, s

H^{1} (g \circ f) = (3) = (11) (21) = H^{1} (f) \circ H^{1} (g)

Arrows turned around, matrices transposed, and the order of composition reversed — yet the composite still matches.

loop

f

p 1

q 1

map

g

r 2

s 1

Figure 5.1.4 Switch from

H_{1}

H^{1}

and every algebraic arrow turns around:

f^{*} : Z^{2} \to Z

and

g^{*} : Z \to Z^{2} .

The matrices are transposed —

f^{*} = (p q),

g^{*} = (r s)

— and the composite is computed in the reverse order,

(g \circ f)^{*} = f^{*} \circ g^{*} .

Follow the gold labels: the basic measurement on the right-hand circle pulls back to the measurement

(r, s)

on the torus, and then to

p r + q s

on the left-hand circle.

Why should anyone care which way the arrows point? Two answers from earlier chapters. First, contravariance is what made the cup product possible (§4.5 The Cup Product). Two measurements on $X$ combine into one measurement on $X \times X;$ every space has the diagonal map $Δ : X \to X \times X,$ $x \mapsto (x, x);$ and a contravariant functor carries $Δ$ backwards, turning that measurement on $X \times X$ into a product measurement on $X .$ Homology, which can only push forward along $Δ,$ from $X$ into $X \times X,$ gets no product this way. Second, measurements can be restricted to smaller pieces of a space, which is what made sheaves work (§4.7 Sheaves and Čech Cohomology). Direction is not bookkeeping; it is structure.

Functoriality proves theorems

We can now state, in one line, why functors are so effective at proving that things are impossible or different.

Read the contrapositive (§0.2 How to Read Mathematics): if $F (A) \neq ≅ F (B),$ then $A \neq ≅ B$ . That is the logic of every invariant in this book. The circle and the point are not homotopy equivalent, because $H_{1} : hTop \to Ab$ sends them to $Z$ and $0 .$ The torus and the sphere are not homeomorphic, because $H_{1}$ gives $Z^{2}$ and $0 .$ One small proposition, used a hundred times.

Functors do more than tell objects apart. They also show that certain arrows cannot exist, because a functor would carry an impossible diagram of spaces to an impossible diagram of groups. Start with a warm-up in one dimension.

Example Warm-up: you cannot fold an interval onto its two ends

Is there a continuous map $r : [0, 1] \to {0, 1}$ that fixes both endpoints, $r (0) = 0$ and $r (1) = 1 ?$ Write $i : {0, 1} \to [0, 1]$ for the inclusion of the endpoints. The condition says $r \circ i = id_{{0, 1}} :$ the triangle ${0, 1} \to [0, 1] \to {0, 1}$ commutes. Apply the functor $π_{0} :$ the two-point space has two path components and the interval has one, so we would get functions ${2 elements} π_{0} (i) {1 element} π_{0} (r) {2 elements}$ whose composite is the identity. Impossible: the first function squashes both elements to one, and nothing afterwards can pull them apart again. So no such $r$ exists.

Now be suspicious. Why does ${0, 1}$ have two path components? Because no path in it runs from 0 to 1 — that is, no continuous map $[0, 1] \to {0, 1}$ sends 0 to 0 and 1 to 1. That is exactly the statement we set out to prove! Here the functor did all of the bookkeeping and none of the work: the input we fed it was the conclusion in disguise. (The real content is the connectedness of the interval, the fact behind the intermediate value theorem.) Keep this in mind for the next argument, where the input will be a hard-won computation and the output a genuinely new theorem.

Now the real thing. A retraction of the disk $D^{2}$ onto its rim $S^{1}$ would be a continuous map $r : D^{2} \to S^{1}$ that keeps every point of the rim where it is. The figure walks through the argument that none exists. Read the top half as geometry and the bottom half as algebra; the functor $H_{1}$ is the bridge.

Brouwer, by functoriality Step through with ‹ ›, or press play

i : S^{1} ↪ D^{2}

The inclusion i puts the circle into the disk as its rim.

The rim sits inside the disk

Figure 5.1.5 If a retraction

r

existed, the triangle of spaces

S^{1} i D^{2} r S^{1}

would commute with the identity. Applying

H_{1}

gives a commuting triangle

Z \to 0 \to Z

whose long side is the identity of

Z .

But anything that passes through the zero group comes out as

0 :

the gold

1

is crushed in the middle while the identity delivers

1 .

Contradiction — so

r

does not exist.

From here, Brouwer’s fixed-point theorem is the short geometric step you saw in §3.5 Maps, Invariance, and First Triumphs: if a map $f : D^{2} \to D^{2}$ had no fixed point, then for each $x$ the ray from $f (x)$ through $x$ would hit the rim at a point $r (x),$ and $r$ would be a retraction. In every dimension the same diagram works with $H_{n - 1},$ since $H_{n - 1} (S^{n - 1}) ≅ Z$ and $H_{n - 1} (D^{n}) = 0$ for $n \geq 2$ (Hatcher 2002, Cor. 2.15). Brouwer himself proved the theorem around 1911, with the degree of a map, a generation before anyone spoke of functors (Brouwer 1911).

Natural transformations: no arbitrary choices

Some maps feel more honest than others, and this book has used both kinds. The Hurewicz map of §3.5 Maps, Invariance, and First Triumphs sends a loop to the same loop, read as a cycle: nobody has to choose anything. The Universal Coefficient Theorem of §4.2 Cohomology Groups splits $H^{n} (X; G)$ into two pieces, but to write the splitting down you must make choices, and different choices give different isomorphisms. For decades mathematicians called the first kind of map “natural” or “canonical”, as a matter of taste. Eilenberg and Mac Lane turned the taste into a definition — while making sense of the Universal Coefficient Theorem, as the History box below tells. The cleanest place to watch the definition at work is linear algebra.

Two isomorphisms, one natural

Let $V$ be a finite-dimensional real vector space. Its dual $V^{*}$ is the space of linear measurements $φ : V \to R,$ and it has the same dimension as $V$ (§1.5 Linear Algebra). Two vector spaces of the same dimension are isomorphic, so $V ≅ V^{*} .$ But try to write down an isomorphism. The usual recipe is: choose a basis $e_{1}, \dots, e_{n}$ of $V,$ let $e_{1}^{*}, \dots, e_{n}^{*}$ be the dual basis ( $e_{i}^{*}$ measures the $i$ -th coordinate), and send $e_{i} \mapsto e_{i}^{*} .$ Call this map $β .$ It works — but it depends on the basis you chose. Choose a different basis and you get a different map.

Now take one more step and consider the double dual $V^{**} = (V^{*})^{*} :$ its elements are measurements of measurements, machines that eat a $φ \in V^{*}$ and return a number. Here there is an isomorphism you can write down with no choices at all. A vector $v$ gives the machine “evaluate at $v ”:$

ev_{V} : V \to V^{**}, ev_{V} (v) = (φ \mapsto φ (v)) .

It is linear, and for finite-dimensional $V$ it is an isomorphism (it is injective, and both sides have the same dimension). No basis was mentioned (Leinster 2014, Example 1.3.14). How do we turn the feeling “ $ev$ involves no choices but $β$ does” into mathematics?

The answer is to ask how each map behaves when we change perspective — when we move $V$ by a linear map $T : V \to W .$ If a construction involves no arbitrary choices, it cannot tell the difference between doing it before or after the move. For $ev,$ that is the claim that the square

V T ↓ ⏐ W ev_{V} ev_{W} V^{**} ↓ ⏐ T^{**} W^{**}

commutes for every linear map $T .$ Here $T^{**} = (T^{*})^{*}$ is the double-dual map, $T^{**} (ξ) = ξ \circ T^{*} .$ Check it on a vector $v :$ going across then down gives $ev_{V} (v) \circ T^{*},$ which eats $ψ \in W^{*}$ and returns $ev_{V} (v) (T^{*} ψ) = (T^{*} ψ) (v) = ψ (T v) .$ Going down then across gives $ev_{W} (T v),$ which eats $ψ$ and returns $ψ (T v) .$ The same. That calculation is the entire proof.

The double arrow $\Rightarrow$ is a reminder that $η$ is an arrow between functors, one level up from arrows between objects. Evaluation is a natural isomorphism from the identity functor of finite-dimensional vector spaces to the double-dual functor $V \mapsto V^{**}$ (which is covariant: two reversals cancel).

What about $β : V \to V^{*} ?$ Here the double-dual trick is not available: $V \mapsto V^{*}$ is contravariant, so there is not even a square of the right shape for a general $T .$ We can still test $β$ against changes of perspective that can be undone. If $T : V \to V$ is invertible, then $(T^{- 1})^{*} : V^{*} \to V^{*}$ points the right way, and a choice-free $β$ would make the square with $T$ on the left and $(T^{- 1})^{*}$ on the right commute. The figure lets you test that.

Natural versus chosen Pick a change of perspective T · drag the vector v

β (T v)

(T^{- 1})^{*} β (v)

Evaluation

V \to V^{**}

— no choices

→ then ↓

T^{**} (ev_{v}) \leftrightarrow (2, 1)

↓ then →

ev_{T v} \leftrightarrow (2, 1)

commutes

Dual basis

β : V \to V^{*}

— uses a basis

→ then ↓

(T^{- 1})^{*} β (v) \leftrightarrow (1, 0)

↓ then →

β (T v) \leftrightarrow (2, 1)

the two routes disagree

change of perspective

T

(1011)

Figure 5.1.6 Left: a vector

v

(gold), its image

T v

(blue), and two measurements drawn as stacks of level lines, as in §4.3 Differential Forms. Right: the evaluation square commutes for every

T,

because in coordinates

T^{**}

is just

T .

The square for the dual-basis map

β

commutes only when

T

is a rotation or a reflection: for “Double”, one route gives

2 v

and the other

\frac{1}{2} v .

On the left the two stacks then fail to coincide. The reason:

β (v)

is “take the dot product with

v ”,

a measurement built from lengths and angles, and a general linear map does not preserve lengths and angles.

So $V$ and $V^{*}$ are isomorphic, but not naturally isomorphic, while $V$ and $V^{**}$ are naturally isomorphic. The precise statement is this: there is no way to choose isomorphisms $V \to V^{*},$ one for every finite-dimensional space, that all commute with all invertible linear maps. (Already for $V = R$ and $T$ = doubling, the square forces $β = 4 β,$ so $β = 0 .)$ Mac Lane, describing a similar double-dual map for abelian groups, put it in words: naturality “is just a precise expression for the elementary observation that the definition … depends on no artificial choices of bases, generators, or the like” (Mac Lane 1971, §I.4).

Naturality in the wild

Two maps you have used in this book are natural transformations, and their naturality was doing work behind the scenes.

The Hurewicz map. For a space $X$ with basepoint $x_{0},$ the map $h_{X} : π_{1} (X, x_{0}) \to H_{1} (X)$ sends the class of a loop to the class of the loop viewed as a 1-cycle (§3.5 Maps, Invariance, and First Triumphs). For any based map $f : X \to Y,$ $f_{*} \circ h_{X} = h_{Y} \circ f_{*} :$ map a loop and then view it as a cycle, or view it as a cycle and then map it — the same. So $h$ is a natural transformation between the functors $π_{1}$ and $H_{1}$ (both regarded as functors from based spaces to groups). This is why the Hurewicz isomorphism $π_{1} (X)^{ab} ≅ H_{1} (X)$ for path-connected $X$ is compatible with every map you apply (Hatcher 2002, Thm 2A.1).
The connecting homomorphism. In the long exact sequence of a pair (§3.6 Exact Sequences and Mayer–Vietoris), the connecting map $\partial : H_{n} (X, A) \to H_{n - 1} (A)$ is natural: a map of pairs $f : (X, A) \to (Y, B)$ gives a commuting square $H_{n} (X, A) f_{*} ↓ ⏐ H_{n} (Y, B) \partial \partial H_{n - 1} (A) ↓ ⏐ f_{*} H_{n - 1} (B)$ So a map of pairs induces a whole ladder of commuting squares between two long exact sequences. Comparing such ladders is how many computations in §3.6 Exact Sequences and Mayer–Vietoris really work.

Now the Universal Coefficient Theorem from the start of this section can be stated precisely. It says $H^{n} (X; G) ≅ Hom (H_{n} (X), G) \oplus Ext (H_{n - 1} (X), G);$ the isomorphism exists, but the splitting into two pieces cannot be chosen naturally. For the homology version of the theorem, the next chapter exhibits a single map of spaces that no natural splitting could survive.

A little history Born from homology

Samuel Eilenberg 1913 – 1998

Saunders Mac Lane 1909 – 2005

In 1941 Saunders Mac Lane, an algebraist who had finished his doctorate in Göttingen in 1933 as the Nazis emptied its mathematics department, lectured at the University of Michigan on group extensions. Samuel Eilenberg, a young topologist recently arrived from Poland — and later the owner of one of the world’s great collections of Southeast Asian art — missed the last lecture and asked for a private repeat (O’Connor & Robertson 2015; O’Connor & Robertson 2000). He noticed at once that Mac Lane’s algebraic answer was a group he had seen before: Steenrod had found it as a homology group of the complement of a solenoid, a strange coiled space Eilenberg had been studying. After an all-night session and months of puzzling, they explained the coincidence with what we now call the universal coefficient theorem (Eilenberg & Mac Lane 1942a). To say precisely that $Hom (A, B)$ “varies naturally” with $A$ and $B,$ they invented functors and natural isomorphisms that same year (Eilenberg & Mac Lane 1942b); in 1945 they added categories and natural transformations (Weibel 1999).

Universal properties: the best solution to a problem

So far we have described objects by what they are made of: a product of sets is a set of pairs, a quotient group is a set of cosets. But look at how this book used them. To get a map out of $H_{n} = Z_{n} / B_{n},$ we defined it on cycles and checked that it killed boundaries. To define the boundary operator on chains, we said what it did to each simplex and let linearity do the rest. In those arguments we never once looked inside a coset. What mattered was how the object relates to every other object. Category theory turns that into a way of defining things: “among all solutions to a certain problem, this one is the best, in the sense that every other solution factors through it in exactly one way.” Such a description is called a universal property.

Products

Start with sets $A$ and $B .$ The Cartesian product $A \times B$ (§1.1 Sets and Functions) comes with two projections, $p_{1} (a, b) = a$ and $p_{2} (a, b) = b .$ Now suppose some other set $X$ has a function to each factor, $f : X \to A$ and $g : X \to B .$ Then there is a function $X \to A \times B$ that “remembers both”, namely $x \mapsto (f (x), g (x)),$ usually written $⟨ f, g ⟩ .$ It makes both triangles in the diagram commute,

p_{1} \circ ⟨ f, g ⟩ = f, p_{2} \circ ⟨ f, g ⟩ = g,

and it is the only function that does: if $h : X \to A \times B$ satisfies $p_{1} \circ h = f$ and $p_{2} \circ h = g,$ then the two coordinates of $h (x)$ are forced to be $f (x)$ and $g (x) .$ Turn this observation into a definition that never mentions elements:

Read it as a problem and its best solution. The problem: “an object with an arrow to $A$ and an arrow to $B ”.$ Every $X$ with such a pair of arrows is a solution. The product is the solution through which all others pass, uniquely. Now look at products in our categories:

In $Set :$ the Cartesian product, as we just saw.
In $Ab :$ the direct sum $A \oplus B$ of §1.4 Abelian Groups and Formal Sums, with the two coordinate projections.
In $Top :$ the product space with the product topology (§2.1 Spaces and Continuity). In fact the product topology is designed to make this work: it is the coarsest topology for which both projections are continuous.
In a poset: a product of $a$ and $b$ is an element $p \leq a,$ $p \leq b$ such that every $x$ with $x \leq a$ and $x \leq b$ satisfies $x \leq p$ — the greatest lower bound. For the real numbers under $\leq$ it is $min (a, b);$ for positive integers under divisibility it is the greatest common divisor.

Universal properties Product: click an element of X, then a pair · gcd: click two numbers · Kernel: drag g(1)

Both triangles commute:

p_{1} \circ h = f, p_{2} \circ h = g

. Try to find a different

h

that also works — click an element of

X

, then a pair.

u

f (u) =

g (u) =

v

f (v) =

g (v) =

w

f (w) =

g (w) =

Figure 5.1.7 Three “best solutions”. Product of sets: the purple arrow

h

is forced — try to send an element of

X

to any other pair and a triangle stops commuting. gcd in a poset: among all common divisors of two numbers (violet), the gcd (gold) is the one every other divides, along a unique dashed arrow; switch to the coproduct and the lcm plays the same role with arrows reversed. Kernel: a map

g : R \to R^{2}

with

f \circ g = 0

factors through the kernel line in exactly one way.

This is why we may say “the product”: any two are isomorphic in one, and only one, compatible way. The same argument works for every universal property below. (Products, kernels and their reversed-arrow twins are all special cases of limits and colimits, the subject of Chapter 5 of Leinster’s book (Leinster 2014, ch. 5).)

Coproducts: the same, with arrows reversed

Turn every arrow around and you get the dual notion. A coproduct of $A$ and $B$ is an object $Q$ with arrows $i_{1} : A \to Q$ and $i_{2} : B \to Q$ such that any pair $f : A \to X,$ $g : B \to X$ factors as $f = u \circ i_{1},$ $g = u \circ i_{2}$ for exactly one $u : Q \to X .$ In $Set$ and $Top$ the coproduct is the disjoint union $A ⊔ B :$ a map out of a disjoint union is the same as a map out of each piece. In a poset it is the least upper bound (the lcm for divisibility). And in $Ab$ it is — again — the direct sum $A \oplus B,$ with the inclusions $a \mapsto (a, 0)$ and $b \mapsto (0, b) :$ a homomorphism out of $A \oplus B$ is a pair of homomorphisms, added together, $u (a, b) = f (a) + g (b) .$ That one object is both the product and the coproduct is a special feature of abelian groups; it will reappear in the last section.

Kernels, cokernels, quotients and free groups

Four constructions you have used since Part I are universal.

Kernel. For a homomorphism $f : A \to B$ of abelian groups, the inclusion $ι : ker f \to A$ satisfies $f \circ ι = 0,$ and it is the best such map: every $g : X \to A$ with $f \circ g = 0$ lands inside $ker f,$ so it factors as $g = ι \circ g^$ for exactly one $g^: X \to ker f .$ (The third tab of Figure 5.1.7 shows this for a linear map $R^{2} \to R .)$
Cokernel. Reverse the arrows. The cokernel of $f$ is the quotient $coker f = B / im f,$ with the projection $π : B \to coker f .$ It satisfies $π \circ f = 0,$ and every $g : B \to Y$ with $g \circ f = 0$ factors as $g = g ˉ \circ π$ for exactly one $g ˉ .$ The cokernel is the dual twin of the kernel — “what is not reached”, where the kernel was “what is crushed”.
Quotients. A special case you used constantly: a homomorphism out of a quotient group $G / H$ is the same thing as a homomorphism out of $G$ that sends $H$ to zero (§1.4 Abelian Groups and Formal Sums). Homology $H_{n} = Z_{n} / B_{n}$ is a quotient, and this universal property is exactly why a chain map induces a well-defined map on homology: it sends cycles to cycles and boundaries to boundaries, so it kills what needs killing.
Free abelian groups. A homomorphism out of $Z [S]$ is the same thing as an arbitrary function out of $S :$ you may send the generators anywhere, and linearity does the rest, uniquely. That is why the boundary operator could be defined just by saying what it does to each simplex.

Equivalences, and where homological algebra lives

When are two categories “the same”?

Categories and functors form a world of their own, so we can ask our usual question one level up: when are two categories the same? An isomorphism of categories — functors $F : C \to D$ and $G : D \to C$ with $G \circ F$ and $F \circ G$ exactly the identity functors — is far too strict: as the example below shows, it would separate two categories that everyone who has used a matrix regards as the same. The right notion lets the round trips be the identity only up to natural isomorphism.

The motivating example is linear algebra itself. Compare the category of finite-dimensional real vector spaces with the matrix category from the gallery (objects $0, 1, 2, \dots,$ arrows matrices). Send $n$ to $R^{n}$ and a matrix to the linear map it defines: every linear map between $R^{n}$ and $R^{m}$ comes from exactly one matrix, and every finite-dimensional space is isomorphic to some $R^{n} .$ Choosing a basis for every space gives a functor back, and the round trips are naturally isomorphic to the identities. So the two categories are equivalent, even though one has a single object for each dimension and the other has vast numbers of different-looking spaces of each dimension. This is the precise sense in which “doing linear algebra with matrices” loses nothing. In Mac Lane’s words, equivalence “allows us to compare categories which are ‘alike’ but of very different ‘sizes’” (Mac Lane 1971, §I.4).

Additive and abelian categories

Look again at the abelian-group world $Ab .$ It has features that $Set,$ $Top$ and even $Grp$ lack:

You can add arrows. Two homomorphisms $f, g : A \to B$ have a sum, $(f + g) (a) = f (a) + g (a),$ and composition distributes over this addition. Each hom-set $Ab (A, B)$ is itself an abelian group. (In $Grp$ this fails: the pointwise product of two homomorphisms of a non-abelian group need not be a homomorphism.)
There is a zero object, the trivial group $0,$ with exactly one arrow to and from every object; composing through it gives the zero maps.
Products and coproducts agree: $A \oplus B$ is both, as we saw.
Every arrow has a kernel and a cokernel, and the first isomorphism theorem holds: the natural map from $A / ker f$ to $im f$ is an isomorphism (§1.4 Abelian Groups and Formal Sums).

A category with the first three features (stated with care) is called additive; one with all four is called abelian. Abelian categories are the natural home of the words “exact sequence”, “kernel”, “image” and “homology”: in any of them, every theorem of the next chapter — the snake lemma, long exact sequences, derived functors — is proved once and for all. Examples include abelian groups, vector spaces, modules over a ring, chain complexes, and sheaves of abelian groups on a space (§4.7 Sheaves and Čech Cohomology). It was the last example that drove the definition. David Buchsbaum had axiomatised “exact categories” in his 1955 thesis; Alexander Grothendieck’s 1957 paper “Sur quelques points d’algèbre homologique”, known as the Tôhoku paper, fixed the name abelian category and its axioms, together with a hierarchy of optional extra ones, so that sheaf cohomology could be built with the same tools as the cohomology of spaces (Grothendieck 1957; Weibel 1999).

A little history A seminar under the bombs

Alexander Grothendieck 1928 – 2014

His childhood was spent with foster parents in Hamburg and then, with his mother, in internment camps in the south of France; at a school in Le Chambon-sur-Lignon he hid in the woods whenever the authorities came looking for Jews. As a student at Montpellier, unhappy that nobody had explained what volume is, he rediscovered a general form of the Lebesgue integral on his own. The Tôhoku paper of the paragraph above came about a decade later. In November and December 1967 he visited North Vietnam, then under American bombing, and spent a week and a half at Hanoi University, evacuated to the countryside about 100 km from the capital, giving what he called “a more specialized seminar on categories and homological algebra, with thirty to forty listeners” (O’Connor & Robertson 2014).

We end where Mac Lane’s epigraph began. Categories were defined so that functors could be defined, and functors so that natural transformations could be defined. Go back to the box at the top of this chapter: each of its loose sentences now has a precise meaning, and one of them, applied to a diagram that cannot exist, gave the no-retraction theorem, and with it Brouwer’s, in a few lines. The next chapter moves into one abelian category, chain complexes, and finds that the tools you built by hand for spaces — exact sequences, connecting maps, the Universal Coefficient Theorem — are really facts about chain complexes, whether or not a space is in sight.

Exercises

Exercise

π_{0}

is a functor

Explain why a continuous map $f : X \to Y$ sends each path component of $X$ into a single path component of $Y,$ and deduce that $π_{0}$ is a functor $Top \to Set .$ Then use it, as in the text, to show that there is no continuous map from the circle $S^{1}$ onto the two-point space ${0, 1} \subset R .$ Which fact about ${0, 1}$ does your argument take for granted?

Hint

The image of a path is a path. For the second part, think about $π_{0} (S^{1}) .$

Solution

If $x$ and $x^{'}$ are joined by a path $γ$ in $X,$ then $f \circ γ$ is a path from $f (x)$ to $f (x^{'})$ in $Y .$ So $f$ induces a function $π_{0} (f) : π_{0} (X) \to π_{0} (Y)$ sending the component of $x$ to the component of $f (x);$ this is well defined by what we just said. Composites and identities are respected because $π_{0} (g \circ f)$ and $π_{0} (g) \circ π_{0} (f)$ both send the component of $x$ to the component of $g (f (x)) .$ For the second part: a surjective continuous $f : S^{1} \to {0, 1}$ would give a surjective function $π_{0} (f)$ from the one-element set $π_{0} (S^{1})$ onto the two-element set $π_{0} ({0, 1})$ — impossible. (Surjectivity of $f$ passes to $π_{0} (f),$ since every component of ${0, 1}$ contains a point $f (x) .)$ The argument takes for granted that ${0, 1}$ has two path components, that is, that no path in it runs from 0 to 1: the connectedness of the interval. Unlike the warm-up in the text, this is not circular — the input is a fact about paths, the output a fact about circles — but it shows again where the real content lies.

Exercise Natural maps of sets

(a) For each set $X$ let $Δ_{X} : X \to X \times X$ be the diagonal map, $x \mapsto (x, x),$ the map behind the cup product. Show that the maps $Δ_{X}$ form a natural transformation from the identity functor of $Set$ to the functor $X \mapsto X \times X,$ which sends a function $f$ to $f \times f : (x, x^{'}) \mapsto (f (x), f (x^{'})) .$ (b) Now show that naturality is a strong condition: the only natural transformation from the identity functor of $Set$ to itself is the identity, $η_{X} = id_{X}$ for every $X .$

Hint

For (b), test the naturality square on the function $p_{x} : {*} \to X$ from a one-point set that picks out $x .$

Solution

(a) For $f : X \to Y$ we need $(f \times f) \circ Δ_{X} = Δ_{Y} \circ f .$ Both sides send $x$ to $(f (x), f (x)) .$

(b) Let $η$ be natural and let $x \in X .$ The naturality square for $p_{x}$ says $η_{X} \circ p_{x} = p_{x} \circ η_{{*}} .$ A one-point set has only one function to itself, so $η_{{*}}$ is the identity, and evaluating at $*$ gives $η_{X} (x) = p_{x} (*) = x .$ So $η_{X}$ fixes every element. A choice-free way of turning every set into itself can only be “do nothing”: shuffling the elements of a set would need a choice of which element goes where.

Exercise Groups as categories

Regard groups $G$ and $H$ as one-object categories. (a) Show that a functor $G \to H$ is the same thing as a group homomorphism. (b) Show that a natural transformation between two such functors $F, F^{'} : G \to H$ is a single element $h \in H$ with $F^{'} (g) = h F (g) h^{- 1}$ for every $g \in G .$

Solution

(a) There is only one object on each side, so the functor must send $★$ to $★ .$ On arrows it is a function $F : G \to H,$ and the functor laws say $F (g g^{'}) = F (g) F (g^{'})$ and $F (e) = e :$ exactly a homomorphism. (b) A natural transformation has one component, an arrow $η_{★} : ★ \to ★$ of $H,$ that is, an element $h \in H .$ The naturality square for the arrow $g$ says $F^{'} (g) \circ h = h \circ F (g),$ i.e. $F^{'} (g) h = h F (g),$ i.e. $F^{'} (g) = h F (g) h^{- 1} .$ So two homomorphisms are naturally isomorphic exactly when they differ by conjugation by a single element (every natural transformation here is an isomorphism, since $h$ is invertible).

Summary

Recap

A category has objects, arrows between them, identities and an associative composition. Sets, groups, vector spaces, spaces, chain complexes, posets, a single group and matrices all form categories.
A diagram commutes when all routes between two points agree; diagram chasing is following elements around it. An isomorphism is an arrow with a two-sided inverse; bijections, homeomorphisms and homotopy equivalences are all isomorphisms in suitable categories.
A functor sends objects to objects and arrows to arrows, preserving composition and identities. $H_{n}$ is a functor $Top \to Ab;$ so are $π_{0},$ $π_{1}$ (on based spaces), forgetful and free functors.
A contravariant functor reverses arrows: preimage, dual space (transpose), $Hom (-, G)$ and cohomology $H^{n} : Top^{op} \to Ab .$
Functors preserve isomorphisms and commutative diagrams, which is why invariants tell objects apart and why “no retraction $D^{2} \to S^{1} ”$ follows from $Z \to 0 \to Z .$
A natural transformation is a family of arrows commuting with every map. $V \to V^{**}$ is natural; any isomorphism $V \to V^{*}$ needs a choice and is not. The Hurewicz map and connecting homomorphisms are natural.
Universal properties describe objects as best solutions: products, coproducts, kernels, cokernels, quotients and free groups; they are unique up to unique isomorphism.
Equivalence is the right sameness for categories. Abelian categories — abelian groups, vector spaces, modules, chain complexes, sheaves — are where homological algebra lives.

Categories and Functors

You have been doing category theory all along

Categories: dots and arrows

A gallery of categories

Commutative diagrams, and one word for sameness

Reading a diagram

Isomorphism: one word for every kind of sameness

Functors: maps between categories

When arrows turn around

Functoriality proves theorems

Natural transformations: no arbitrary choices

Two isomorphisms, one natural

Naturality in the wild

Universal properties: the best solution to a problem

Products

Coproducts: the same, with arrows reversed

Kernels, cokernels, quotients and free groups

Equivalences, and where homological algebra lives

When are two categories “the same”?

Additive and abelian categories

Exercises

Summary

Further reading

References