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Notes on categories
Jonas Karlsson
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History and terminology
Historically, Eilenberg and MacLane wanted to define natural transformations, which go between functors, which go between morphisms, which
go between objects. This would have been fine, except that we nowadays
want to consider maps between maps between . . . between objects. A uniform terminology is to let objects be 0-morphisms, and let a map between
two n-morphisms be an (n + 1)-morphism. As for categories themselves,
they were at some point called galaxies; functors would then be intergalactic travel agencies. It is amusing that Kant, no stranger to categories, talked
about galaxies, which he called “island universes”.
Examples of categories
Madamina, il catalogo è questo
Delle belle che amò il padron mio
un catalogo egli è che ho fatt’io
Osservate, leggete con me.
from Mozart’s Don Giovanni
Everybody’s first example of a category is sets with functions between
them. You can have a lot of fun with sets, but they form a rather special
category (it’s a topos). To build intuition, it is necessary to look at lots of
examples, the more varied the better.
• The category Set of sets and arbitrary functions.
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• The category FinSet of finite sets and arbitrary functions.
• More generally, the category of sets of cardinality < κ for κ a strong
limit cardinal; FinSet corresponds to κ = ℵ0 .
• The category FinBij of finite sets and bijections between them (we’ll
see combinatorial applications of this).
• The category Set◦ of sets, where a morphism from X to Y is a function from Y to X; i.e. the opposite of Set.
• The category Set2 of pairs of sets and pairs of functions.
• More generally, the category Setn of n-tuples of sets and n-tuples of
functions.
• The category Set∗ of pointed sets and functions preserving the distinguished point.
• The category Set of sets with an endomap.
• For a fixed group G, the category G-Set of sets with an action of G.
• The category Pos of partially ordered sets and order-preserving maps.
• The category FinPos of finite partially ordered sets and order-preserving
maps.
• The category ∆ of finite ordinals. Non-obviously, ubiquitous in algebraic topology.
• While we’re at it, the category Λ of cyclically ordered sets and degree
one maps.
• Every set is itself a category, where the only morphisms are the identities.
• Similarly, every poset is a category with morphisms encoding the order relation.
• A special case of the previous construction is the poset of open subsets of a topological space, the morphisms being inclusion maps. This
gets used to define sheaves.
• The category Top of topological spaces and continuous maps.
• The category hTop of topological spaces and homotopy classes of
continuous maps.
• The category Top∗ of pointed topological spaces and continuous maps
preserving the distinguished point.
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• The category Haus of Hausdorff topological spaces and continuous
maps.
• The category Met of metric spaces and non-distance-increasing maps.
• More generally the category λ-Met of metric spaces and λ-Lipschitz
maps.
• The category Grp of groups and group homomorphisms.
• Every group is itself a category with one object, the group elements
being the self-maps of that object.
• The category Ring of unital rings and ring homomorphisms. It is
convenient to assume that our rings are unital, letting Rng be the category of not-necessarily-unital rings (“Rings without a multiplicative
Identity”). Also. . .
• . . . the category Rig of rigs, i.e. “Rings without Negatives” or rather
rings-not-necessarily-equipped-with additive inverses.
• The category Field of fields and field homomorphisms.
• The category Mon of monoids and homomorphisms thereof.
• For a fixed monoid M , the category M -Set of sets with an action of
M . The category Set of sets with an endomap is the special case
where M = (N, +), the monoid of non-negative numbers under addition.
• The category Vectk of vector spaces over the field k.
• More generally, the category ModR of modules over a fixed ring R.
• Another generalization of Vectk : the category Bunk (X) of k-vector
bundles over a topological space X, Vectk being the case where X is
a point.
• The category Bun(X) of fiber bundles over X.
• The category G-Bun of principal G-bundles.
• The category Aff of affine schemes and morphisms.
• The category Aff k of affine schemes over a field k, or rather over the
point Spec k.
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Some general constructions
• For C a category and c an object in it, there is a category (C ↓ c) whose
objects are objects of C together with a morphism to c, and whose
morphisms are morphisms from C commuting with the morphisms
to c. This is called a slice category, or the category of objects over c. Examples abound. Trivially, if C has a terminal object •, then (C ↓ •) is
just C itself. Less trivially, Aff k is (Aff ↓ Spec k). The collection of
covering spaces of a given topological space is another example; the
universal covering space is the initial object in this category.
• Dually, there is a notion of coslice categories (C ↑ c), or categories under a fixed object. Just take the previous example and turn all arrows
around. Pointed categories arise in this way; Set∗ is (Set ↑ •), &c.
Also Algk = (Ring ↑ k); the terminal object is any algebraic closure
of k. The analogy with covering spaces can be pushed quite far.
• To any category C we may associate its arrow category C→ , whose
objects are morphisms in C and whose morphisms are commutative
squares in C.
• If C and D are categories, there is a functor category DC of functors
F : C → D, where morphisms are natural transformations. Taking
C = G, a group considered as a category, and D = Vectk gives the
category of linear representations of G over the field k. Ditto for other
categories of representations.
Mnemonics
The image of a morphism f under a covariant functor tends to be called f∗ ,
while if the functor is contravariant, the notation is f ∗ . Dually, covariant
functors like homology and Tor have lower indices (Hn and Torn ), while
contravariant functors like cohomology and Ext have the index upstairs
(H n and Extn ). After a while this becomes obvious. My initial mnemonic
was that contrarians raise their fists.
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This convention extends pretty far: in a chain complex with lower indices,
the boundary maps lower the degree:
· · · → An+1 → An → An−1 → · · · ,
while in a cochain complex, the degree is raised:
· · · → An−1 → An → An+1 → · · · .
I also have trouble distinguishing ∧ and ∨ in a poset. One way to think
about them is that ∧ is like the logical “and”, hence taking x ∧ y gives you
something smaller (because there are more constraints to satisfy), hence
this should be lower in the poset. Alternatively, it’s an arrow pointing up,
meaning that you start at the bottom and go as far up as you can without
losing sight of x and y. And indeed conversely, if not more so, just as I was
about to say myself, Piglet. For some reason I have no difficulty remembering that right adjoints are continuous and left adjoints cocontinuous. Make
up your own mnemonic, or remember that ⊗, as a functor on modules, is a
left adjoint and preserves right-exactness (obviously) but not left-exactness
(almost as obvious). Then realize that the kernel is a limit while the cokernel is a colimit. The Yoneda embedding sends an object A to a contravariant
functor hA ; this association is covariant in A, so it makes sense to have A
downstairs. Doing algebraic geometry seems to make it obvious that, despite having its adornment below, hA is in fact contravariant: sending a
probe into a space lets you pull functions back.
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Ways in which sets are special
Suppose A → C is a monomorphism in Set, i.e. an injective function. Then
there exists a set B, unique up to isomorphism, such that the coproduct
A+B is isomorphic to C. This B is of course the complement of A inside C.
Mulling over the definition of coproduct, we realize that this comes down
to the following fact: an arbitrary function on C “is” an arbitrary function
on A, and an arbitrary function on the complement C \A. Basically, sets lack
cohesion. There is no way for points to talk to each other. By contrast, if A
and C are topological spaces, picking a continuous function on A restricts
what you can do on C \ A. It can also happen that a complement exists, but
not uniquely. For example, in a poset considered as a category, a c does
not fix a b making a ∨ b = c. You could always take b = c (!) but in general
there will be many possibilities. Note that in a poset-as-a-category, every
arrow is both an epi and a mono, for not very interesting reasons. This also
shows that in a general category, a morphism may be both epi and mono
without being an isomorphism, another way in which Set is special.
Family-friendly category theory: combinatorial species
Recall the category Bij of finite sets and bijections. A species is a covariant
functor F : Bij → Bij. The intended interpretation is that F encodes some
fixed structure you can put on a finite set, and assigns to a set S the set
F (S) of all possible ways to give S that structure. Moreover, this “structure” must be independent of the identity of the elements of the set: an
isomorphism class of objects in Bij is a set up to isomorphism, so the elements are on equal footing. This being the case, the structure may be transported along bijections, which is precisely what the functoriality is saying.
Examples abound:
• The species of graphs: F (S) is the collection of all graphs with vertex
set S.
• The species of trees: F (S) is the collection of all trees with vertex setS.
• The species of subsets: F (S) is the collection of all subsets of S.
• The species of total orders: F (S) is the collection of all total orders on
S.
• The species of directed graphs: F (S) is the collection of all directed
graphs with vertex set S.
• The species of partitions: F (S) is the collection of all partitions of S.
• The species of sets: F (S) is the set {S}.
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• The species of elements: F (S) is S itself.
• The species of being a seven-element set: F (S) is a singleton if |S| = 7,
∅ otherwise. There is nothing special about seven.
• The species of perfect matchings: F (S) is the collection of all partitions of S into two-element sets.
This would all be pretty pointless unless we could operate on the functors. We can, because they take values in a very nice category: Bij has
products and coproducts. Thus, for species F and G, we may form F + G,
defined by (F + G)(S) = F (S) + G(S), where the second plus sign is the
coproduct, i.e. disjoint union. Similarly the product of F and G is defined
by
a
(F G)(S) =
F (S0 ) × G(S1 ),
S=S0 +S1
that is, we split S into disjoint subsets in all possible ways, equip those
with the respective structures, then take the disjoint union of all those assignments. This looks like a convolution, and indeed the theory of species
is supposed to categorify generating functions. Also note that we think
of the sets F (S) as trees, orders &c, but the functor takes values in Bij,
which consists of naked sets. Where, then, is the combinatorial information
stored? In the available operations on the functors! For example, a binary
tree equipped with an embedding in the plane (planar tree, for short), is
either a singleton or a pair of planar trees (expand on this). This is saying
that there is an isomorphism
T ∼
= 1 + T2
between the species T of planar trees and the species of a singleton or a
pair of planar trees. Thus, the sets are thrice removed from the interesting
action: there are sets, bijections between them, functors between such, and
finally natural isomorphisms between functors.
Properties versus structure
A delightful spin-off from category theory is that we can make precise the
distinction between properties and structure (at least within algebra). This
seems to be due to James Dolan; I got it from John Baez. Consider the
forgetful functor Ab → Grp which takes abelian groups and regards them
as not-necessarily-abelian groups. This functor is fully faithful, meaning that
it is bijective on hom sets. That is, the collection of homomorphisms from
one abelian group to another is exactly the same as the collection of group
homomorphisms, period. This is because being abelian is a property, not a
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structure. By contrast, the forgetful functor Grp → Set is injective but not
surjective on hom sets: dropping the requirement that maps preserve the
group structure (ding!) allows for more maps.
Things that are special cases of each other
The terminal object of a category is the empty product, hence a special case
of the product. The product of a collection of objects is a special case of
a limit, namely one where the diagram category only has identity morphisms. A limit is a universal cone, that is to say, a terminal object in the
category of cones over some diagram. Thus, all these things are special
cases of special cases of special cases of themselves. As far as I can tell,
there is no fact of the matter as to which notion is most fundamental (and
the question was pretty contrived to begin with).
Limits interacting with colimits
Why add prime numbers?
Prime numbers are made to be
multiplied, not added.
Lev Landau
This is hairy. You know from elementary number theory that the decimal representation of integers is good for addition but not multiplication;
to multiply, you’d rather have the prime factorization as your standard
form. Going between the representations is computationally costly. Goldbach’s conjecture is a typical example of the complexities you can get into
by asking for relations between limits (products) and colimits (sums). Finite fields exhibit the same phenomenon: the additive group is simply a
vector space over some Fp , and life is good. The multiplicative group is
cyclic, which is even easier. Taking logarithms is no fun at all, especially
since there is no nonvacuous topological constraint keeping the maps in
check. Looking at more serious number theory, there is the celebrated product formula for the zeta function,
ζ(s) =
∞
X
Y
1
1
=
,
ns
1 − p−s
n=1
p prime
which has an elementary proof but whose proper context is representation
theory and the Langlands program. (expand this handwaving: duality &c)
Returning to elementary examples, the identity
x3 + 4x2 + x − 6 = (x − 1)(x + 2)(x + 3)
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can be thought of as translating between a colimit of a limit (sum of products) and a limit of a colimit (product of sums). Likewise, in Boolean algebra a disjunctive normal form such as
xyz + xyz + xyz + xyz + xyz
also has a conjunctive normal form:
(x + y + z)(x + y + z)(x + y + z).
Here the duality comes to our aid: negation switches conjunctions and disjunctions, so rewriting is easy. Nothing of that sort is available for factoring
polynomials. (List some more examples: circuits )
Uniqueness versus existence
It is a commonplace fact in category theory that objects defined by universal properties may or may not exist, but if they do, they are unique up
to unique isomorphism. In fact, similar things happen in the decidedly
un-categorical context of analysis, namely if you define the derivative of a
function to be its best local approximation. Explicitly, let f : R → R be a
function, and let’s ask for a number α such that
f (x) = f (a) + α(x − a) + o(|x − a|)
as x → a.
Then if β is another such number,
(α − β)(x − a) = o(|x − a|),
whence α = β; if derivatives exist, they are unique. This argument is completely independent of any nastiness hidden inside the map f 7→ f 0 . Optimality forces uniqueness. This should probably not be taken as a law of
nature; after all, mathematical optimization teaches us that many problems
have non-unique optima. Still, some general principle is at work here.
The Koch curve, both nasty and antsy.
Similarly, the normalization of a singular algebraic curve is unique for general reasons.
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Specification versus construction
This got its own section just to keep thoughts from getting too entangled.
A useful piece of terminology used in computer science and who knows
where else, is specification of an object (program?) versus its actual construction. This is of course closely related to the existence/uniqueness business,
but we need more terminology, not less. (Linguists are puzzled by how
children growing up in linguistically deprived environments still manage
to learn their language well. They would be even more puzzled if they visited the math department.) Defining things by induction is somewhere in
between the two (just add water).
Maps in, maps out
In elementary group theory, you learn that groups have subgroups, and
some of them have the property of being normal. Likewise, you learn in
commutative algebra that rings have subrings, some of which are ideals.
A better way to think about it is this: every subgroup H ≤ G is a mono
H ,→ G, while a normal subgroup H C G “is” an epi G G/H. Ditto for
rings: a subring is a subobject, an ideal is a (something you can) quotient
(by). Seen in this light, the notions are revealed to be dual, rather than
specializations. This is utterly trivial, but I wish someone had told me long
ago. Sometimes trivial things must be said.
Properties of properties
At some point in your upbringing, you were told that
1. continuity means that a function doesn’t “tear a space apart”, and
2. a map is continuous if the preimage of an open set is open.
This is confusing, because statement 1 is about points of a space moving
forward (i.e. in the direction of the map), while the definition talks about
subsets moving the opposite way. We are used to maps being required to
preserve structure, but this sounds like something is being reflected. This
is nevertheless the right definition, because what is being preserved is not
a structure but a property, and properties are maps out of a space, hence
move backwards. More precisely, a map f : X → Y of sets induces a map
f ∗ : P(Y ) → P(X); open sets of Y are sent to sets in X, and we ask that the
images be open. You might object that there is a covariant powerset functor,
and you’d be right. A fuller explanation of what is going on requires (...)
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Points
The points of a set ought to mean its elements; the points of a topological
space ought to mean what it always does, and so on. Now notice that both
Set and Top have terminal objects, which are one-element sets and onepoint topological spaces, respectively; and notice also that elements of a set
X biject with maps 1 → X, and likewise for topological spaces. In view of
this, we make the following
Definition. Suppose that a category C has a terminal object 1. A (global)
point, or a (global) element of an object X in C is a morphism 1 → X.
Terminal objects may not exist, and if they do, they may have too few
maps to other objects for this to be useful. Thus:
Definition. If X and T are objects of a category C, we call a morphism
T → X a (generalized) point, or a (generalized) element of X; also a T point, if that needs to be emphasized.
The idea is that a T -point of X looks like a copy of T , so that we may
probe X using extended objects. You have seen this in singular homology,
and just like in that example, the map need not be well-behaved (beyond
being a morphism of the category, of course). Notice also that if you have
a map S → T , you can compose with it and turn T -points into S-points.
In particular, global points can be turned into T -points for any T , and this
pluripotency makes them a bit bland.
Pluripotent stem cells
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Behaviourism vindicated
Joke: two behaviorists have sex. Afterwards, one turns to the other and
says, “That was good for you; how was it for me?”. The following lemma
is indispensable.
Lemma (Yoneda). Behaviourism works, at least in category theory.
OK, what Yoneda actually said was the following:
Lemma (Yoneda, take 2). The assignment A 7→ hA embeds a category C in
PreSh(C), the category of set-valued presheaves on C. This embedding is
full and faithful. If F : C → Set is an arbitrary functor, natural transformations hA ⇒ F biject with F (A).
In other words, we may work with functors of points rather than objects, and we don’t lose any information that way. To know an object is to
know how it interacts with other objects.
This dog is determined by its functor of points.
But why bother? The answer is that PreSh(C) is often a nicer category than
C itself. Namely, a presheaf category is a topos, at least if C is small.
Representable functors
Analogies
Several elementary-but-subtle constructions are roughly analogous to the
Yoneda embedding. Consider, for example, the process whereby the ring
of integers Z is made into the field Q of rational numbers. We form pairs
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(a, b) ∈ Z × (Z \ {0}), then define addition and multiplication, and finally
notice that the map Z ,→ Q defined by n 7→ (n, 1) is an injective ring homomorphism. If we did not exclude 0 from the set of denominators, we would
still get a ring R and a ring homomorphism Z 7→ R, but R would be trivial:
every (n, 1) is equal to (0, 0), hence any two elements in the ring are equal.
Adding too many new objects makes the old ones collapse. It is also possible to
add too few new objects: if we compactify the affine plane A2 by adding a
single point at infinity, it would be possible for two distinct lines to intersect
in two points, one in the plane and the other at infinity. It would then no
longer be true that any two distinct points determine a unique line. Hence,
we must add an entire projective line at infinity to get A2 ,→ P2 , which
has a good intersection theory. This suggests that whenever we complete
some structure by adding “ideal objects”, we have to worry a bit about
how those new objects interact with the old ones. Will the putative inclusion C ,→ PreSh(C) really be an inclusion? Will there be strange new maps
between ideal objects that didn’t come from C? The Yoneda lemma says
that this is not the case: no collapsing occurs.
What are real numbers?
Along the same lines, recall the different completions of the rationals. If we
regard Q as being just a field, the natural thing to do is to pass to an algebraic closure: Q ,→ Q. On the other hand, Q has an absolute value, making
it into a metric space, and completing it produces the Cauchy reals, RC .
Also, Q is an ordered abelian group, and we may construct the Dedekind
reals, RD . We are brought up to think that it’s obvious that RC ∼
= RD , but
it can’t be, because Q has many valuations in addition to the archimedean
one; this produces the p-adics. Penguins probably think it’s obvious that
frozen water has lower density than the liquid.
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Yoneda
First, let’s consider “little Yoneda”. This is the statement that the Yoneda
embedding is surjective on morphisms, i.e.
homCb (hA , hB ) ∼
= homC (A, B).
A priori, the left hand side is bigger: a natural transformation η has a component ηA at every object A of C. However, naturality constrains the data.
Namely, for an arbitrary morphism g : D → C in C, the following diagram
commutes:
ηC
/ hB (C)
hA (C)
◦g
hA (D)
ηD
◦g
/ hB (D)
If m ∈ hA (C), this says that ηC (m) ◦ g = ηD (m ◦ g). Now the trick is to take
C = A and m = 1A . Then ηD (g) = ηA (1A ) ◦ g, that is, all of η is determined
by the map ηA (1A ) : A → B. In particular, η arises from a map in C, as was
to be shown. Now let’s rewrite the bijection as follows:
homCb (hA , hB ) ∼
= hB (A).
This suggests that maybe
homCb (hA , F ) ∼
= F (A)
b and this is the general Yoneda lemma. The proof is
for an arbitrary F ∈ C,
essentially the same, and is left as an exercise in accordance with tradition.
Equalizers and coequalizers
The equalizer of two parallel morphisms is what is sounds like: the inclusion of the subobject of the common domain on which the morphisms
agree. Namely, given f and g from A to B, the equalizer is an object E together with a map e, universal with respect to the following commutative
diagram:
e
E → A ⇒ B.
In other words, f ◦ e = g ◦ e, and any other object/arrow pair satisfying
this factors uniquely through e. It is an instructive exercise to prove that e
is necessarily a monomorphism, and that in the category of sets,
E = {a ∈ A|f (a) = g(a)}.
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The coequalizer of a pair of parallel arrows is the dual notion; it is an epimorphism, and you get the proof for free by dualizing the previous exercise. As an amusing application, you can check that the equalizer of the
two maps k[t] → k given by evaluation at t = 1 and t = −1 (assuming
that 1 6= −1) is k[t2 , t3 − t] ,→ k[t] and that precomposing with the map
k[x, y] → k[t2 , t3 − t] given by x 7→ t2 and y 7→ t3 − t produces the following
picture:
An affine line with two points glued together
Non-locality
If the state spaces of two classical systems are S1 and S2 , the joint system
will have state space S1 × S2 or some subset thereof, and this product has
projections onto its factors. Thus, in classical physics, one can meaningfully
ask about the state of a subsystem, even if there are important interactions
with other parts of the whole. In quantum mechanics, by contrast, the joint
system will be the tensor product of the Hilbert spaces of the subsystems,
and this is not the categorical product. Thus, without additional structure,
no projection maps exist. This, in essence, is what makes quantum physics
non-local.
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Linear algebra
I used to find it hard to remember how the dependent and independent
variables of a linear transformation Rm → Rn (say) are to be matched up
with the rows and columns of its matrix. Here’s my current way of thinking: vectors, covectors and matrices are all just linear maps. A covector on
Rm is a map Rm → R1 , and a vector in Rm is a map R1 → Rm . Acting
with transformations on vectors and evaluating covectors on vectors are
both special cases of composing maps. Then I just have to remember that a
vector is a column vector, which is a m × 1 matrix. Clearly, then, columns
refer to the domain and rows to the codomain. This is a lot like identifying
a set X with its global points 1 → X. Somehow it also feels like having
vacuum states left and right of your operators in QFT:
h0|Ψ . . . Ψ† |0i.
Secretaries and censors
The product of a set of objects in a category is a single object in that same
category, maps into which biject with collections of maps into the whole
set of objects. This is like how emails about teaching positions reach us
graduate students. Rather than emailing us all directly, they email the department secretary, who passes on the email unchanged. She represents the
whole collection of graduate students. A representing object is not unique:
sometimes they email the director of graduate studies, but Dick and Bonnie
are isomorphic in this regard.
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