Preservation and Reflection
A functor carries objects to objects and maps to maps, and so it carries a cone on a diagram to a
cone on the image diagram. The question that organizes this section is whether it carries the
best cone to the best cone: does a functor send a limit to a limit? The answer divides
functors into those that respect the universal constructions and those that do not, and the
vocabulary for the distinction is the subject of the following definitions.
Let \(F : \mathscr{A} \to \mathscr{B}\) be a functor and
\(D : \mathbf{I} \to \mathscr{A}\) a diagram. A
cone
\(\big(A \xrightarrow{p_I} D(I)\big)_{I \in \mathbf{I}}\) on \(D\) has an image
\(\big(F(A) \xrightarrow{F p_I} FD(I)\big)_{I \in \mathbf{I}}\), and this image is a cone on the
diagram \(F \circ D : \mathbf{I} \to \mathscr{B}\): functoriality turns the compatibility equations
\(p_J = Du \circ p_I\) into \(F p_J = F(Du) \circ F p_I\), which is exactly compatibility for
\(F \circ D\). The cone is always carried to a cone. Whether the universal property survives the
crossing is the content of the next definition.
Definition: Preservation of Limits
Let \(F : \mathscr{A} \to \mathscr{B}\) be a functor.
(a) For a small category \(\mathbf{I}\), the functor \(F\) preserves limits of
shape \(\mathbf{I}\) if for every diagram \(D : \mathbf{I} \to \mathscr{A}\) and every
cone \(\big(A \xrightarrow{p_I} D(I)\big)_{I \in \mathbf{I}}\) on \(D\),
\[
\big(A \xrightarrow{\;p_I\;} D(I)\big)_{I \in \mathbf{I}} \text{ is a
limit
cone on } D
\]
implies that
\[
\big(F(A) \xrightarrow{\;F p_I\;} FD(I)\big)_{I \in \mathbf{I}} \text{ is a limit cone on }
F \circ D .
\]
(b) The functor \(F\) preserves limits if it preserves limits of shape
\(\mathbf{I}\) for every small category \(\mathbf{I}\).
(c) The functor \(F\) reflects limits of shape \(\mathbf{I}\) if the implication
of (a) runs the other way: whenever the image cone
\(\big(F(A) \xrightarrow{F p_I} FD(I)\big)_{I \in \mathbf{I}}\) is a limit cone on \(F \circ D\),
the original cone \(\big(A \xrightarrow{p_I} D(I)\big)_{I \in \mathbf{I}}\) is already a limit
cone on \(D\).
The dual notions read off at once by reversing every arrow: \(F\) preserves colimits of
shape \(\mathbf{I}\) when it carries
colimit
cocones
to colimit cocones, preserves colimits when it does so for every shape, and reflects colimits when a
cocone whose image is a colimit was already one. Everything said below about limits has its mirror
for colimits, obtained by passing to the opposite categories, and is not restated.
There is a second way to state preservation that removes the quantifier over cones and makes the
structure of the definition visible. Suppose \(D\) has a limit \(\lim D\) in \(\mathscr{A}\), with
projections \(p_I : \lim D \to D(I)\). The image cone
\(\big(F(\lim D) \xrightarrow{F p_I} FD(I)\big)_{I \in \mathbf{I}}\) is a cone on \(F \circ D\). If
\(F \circ D\) also has a limit, then by the universal property of that limit there is a unique map
\[
F(\lim D) \longrightarrow \lim (F \circ D),
\]
the canonical comparison map, whose \(I\)-component is
\(F p_I : F(\lim D) \to FD(I)\), the projection of the image cone. The functor preserves the limit of
\(D\) precisely when, the composite \(F \circ D\) also having a limit, this comparison map is an
isomorphism.
The distinction worth keeping is that preservation asks for more than an abstract isomorphism between
\(F(\lim D)\) and \(\lim (F \circ D)\). The two objects might happen to be isomorphic for unrelated
reasons; preservation requires that they be isomorphic through the canonical map, the one
assembled from the projections. Establishing only the bare isomorphism
\(F(\lim D) \cong \lim (F \circ D)\), without checking that the comparison map is the isomorphism in
question, verifies something strictly weaker. It is this canonical map, not a coincidence of objects,
that the universal property controls, and it is the comparison-map test that will be used throughout.
Reflection, by contrast, is a faithfulness of detection: a functor that reflects limits cannot be
fooled into reporting a non-limit cone as a limit, for if the image is universal then the source
was. The two properties are independent, and a functor may have either without the other, as the
examples of the next part show.
How Forgetful Functors Behave
The definitions are best seen at work on the functors that strip an object of its structure and
remember only an underlying set. These forgetful functors behave consistently across the categories
of algebraic objects met so far, and the pattern they exhibit sets the theme for the section: they
respect limits and fail to respect colimits. Three examples make the pattern precise, the first a
warning about reflection and the others the positive statement for limits.
Topology: preservation without reflection
Write \(U : \mathbf{Top} \to \mathbf{Set}\) for the functor sending a topological space to its
underlying set and a continuous map to itself as a function. This functor preserves both limits and
colimits; the underlying set of a product space is the product of the underlying sets, the
underlying set of a quotient is the quotient of the underlying set, and so throughout. It does not,
however, reflect limits, and a single span shows why.
Choose spaces \(X\) and \(Y\) that are not discrete, and let \(Z\) be the set \(U(X) \times U(Y)\)
carried with the discrete topology. The discrete topology is strictly finer than the
product
topology
here, since a non-discrete factor forces the product topology to be coarser than discrete. The
projections of the underlying set lift to continuous maps out of \(Z\), giving a cone
\[
X \longleftarrow Z \longrightarrow Y
\]
in \(\mathbf{Top}\). Its image in \(\mathbf{Set}\) is
\(U(X) \longleftarrow U(X) \times U(Y) \longrightarrow U(Y)\), which is the product cone of the
underlying sets and so a limit cone in \(\mathbf{Set}\). Yet the cone in \(\mathbf{Top}\) is not a
product: the product of \(X\) and \(Y\) carries the product topology, and \(Z\) carries the strictly
finer discrete one, so \(Z\) is not the categorical product. The image is a limit while the source
is not; reflection fails. A functor may thus preserve every limit and still mistake a non-limit cone
for a limit one when read through its image: preservation carries universal structure forward, but
gives no guarantee that universal structure can be detected by passing to the image.
Algebra: limits preserved, colimits not
The forgetful functors out of the categories of algebraic objects — groups, abelian groups, rings,
vector spaces over a field — display the asymmetry between limits and colimits plainly.
On the colimit side they fail. The
initial
object
of \(\mathbf{Grp}\) is the trivial group, whose underlying set has one element, while the initial
object of \(\mathbf{Set}\) is the empty set; the forgetful functor sends a one-element set to a
one-element set, not to the empty set, so it does not preserve initial objects. Likewise the
underlying set of a direct sum \(X \oplus Y\) of vector spaces is not the disjoint union of the
underlying sets of \(X\) and \(Y\), so binary sums are not preserved either. Forgetful functors out
of categories of algebras very seldom preserve all colimits.
On the limit side they succeed, and the mechanism is worth seeing once in full. The underlying-set
functor not only preserves limits but exhibits them: a limit in the algebraic category is computed by
taking the limit of the underlying sets and equipping the result with the unique compatible
structure. We examine binary products in \(\mathbf{Grp}\); the same argument runs for any limit in
any of \(\mathbf{Grp}\), \(\mathbf{Ab}\), \(\mathbf{Vect}_k\), \(\mathbf{Ring}\).
Take groups \(X_1\) and \(X_2\) and form the
product
set
\(U(X_1) \times U(X_2)\), with its projection functions
\(p_1 : U(X_1) \times U(X_2) \to U(X_1)\) and \(p_2 : U(X_1) \times U(X_2) \to U(X_2)\).
Proposition: Products of Groups from Products of Sets
Let \(X_1\) and \(X_2\) be groups. There is exactly one group structure on the set
\(U(X_1) \times U(X_2)\) making the projections \(p_1\) and \(p_2\) homomorphisms, and with this
structure
\[
X_1 \xleftarrow{\;p_1\;} U(X_1) \times U(X_2) \xrightarrow{\;p_2\;} X_2
\]
is a
product
in \(\mathbf{Grp}\).
Proof
Uniqueness. Suppose a group structure on \(U(X_1) \times U(X_2)\) makes both
projections homomorphisms. Take elements \((x_1, x_2)\) and \((x'_1, x'_2)\) and write their
product as \((y_1, y_2)\). Since \(p_1\) is a homomorphism,
\[
y_1 = p_1(y_1, y_2) = p_1\big((x_1, x_2) \cdot (x'_1, x'_2)\big) = p_1(x_1, x_2) \cdot
p_1(x'_1, x'_2) = x_1 \cdot x'_1,
\]
and likewise \(y_2 = x_2 \cdot x'_2\) from \(p_2\). Hence the product is forced:
\[
(x_1, x_2) \cdot (x'_1, x'_2) = (x_1 x'_1, \, x_2 x'_2).
\]
The same reasoning applied to inverses and the identity forces
\((x_1, x_2)^{-1} = (x_1^{-1}, x_2^{-1})\) and the identity element to be \((1, 1)\). At most one
group structure makes the projections homomorphisms.
Existence. Define multiplication, inversion, and identity by the componentwise formulas
just forced. The group axioms hold componentwise because they hold in \(X_1\) and \(X_2\)
separately, and the projections are homomorphisms by construction, so the cone displayed above
lives in \(\mathbf{Grp}\). It is a product cone: given homomorphisms \(f_1 : A \to X_1\) and
\(f_2 : A \to X_2\), the function \(a \mapsto (f_1 a, f_2 a)\) is the unique homomorphism
\(A \to U(X_1) \times U(X_2)\) commuting with the projections, since it is forced
set-theoretically by the product property in \(\mathbf{Set}\) and is a homomorphism by the
componentwise structure.
The argument used no group theory beyond the componentwise checks; it used the product structure of
\(\mathbf{Set}\) and the demand that the projections be homomorphisms. Stripped of the language of
groups, what was shown is this. Given objects \(X_1\) and \(X_2\) of \(\mathbf{Grp}\), for any
product cone on \(\big(U(X_1), U(X_2)\big)\) in \(\mathbf{Set}\) there is a unique cone on
\((X_1, X_2)\) in \(\mathbf{Grp}\) whose image under \(U\) is the cone started with, and this cone on
\((X_1, X_2)\) is a product cone. The forgetful functor does not merely preserve the product; it
lifts the product of underlying sets back to a product of groups, uniquely. That stronger property is
the one named next.
Creation of Limits
Preservation says that a functor carries existing limits forward. The group computation did more: it
began with a limit downstairs in \(\mathbf{Set}\) and produced from it a limit upstairs in
\(\mathbf{Grp}\), uniquely determined. A functor with this lifting power is said to create
limits — the property that turns the underlying-set functor into a tool for building limits in
algebra, not merely respecting them.
Definition: Creation of Limits
A functor \(F : \mathscr{A} \to \mathscr{B}\) creates limits of shape
\(\mathbf{I}\) if for every diagram \(D : \mathbf{I} \to \mathscr{A}\) the following
holds: for any limit cone \(\big(B \xrightarrow{q_I} FD(I)\big)_{I \in \mathbf{I}}\) on the
diagram \(F \circ D\), there is a unique cone
\(\big(A \xrightarrow{p_I} D(I)\big)_{I \in \mathbf{I}}\) on \(D\) with \(F(A) = B\) and
\(F(p_I) = q_I\) for all \(I \in \mathbf{I}\), and moreover this cone
\(\big(A \xrightarrow{p_I} D(I)\big)_{I \in \mathbf{I}}\) is a limit cone on \(D\).
Read in words: every limit cone downstairs has a unique lift to a cone upstairs sitting exactly over
it, and that lift is automatically a limit. A limit found in \(\mathscr{B}\) on the image diagram is
thus carried back up into \(\mathscr{A}\), with both the lifting object and its projections pinned
down by the requirements \(F(A) = B\) and \(F(p_I) = q_I\). The forgetful functors out of
\(\mathbf{Grp}\), \(\mathbf{Ring}\), \(\mathbf{Ab}\), \(\mathbf{Vect}_k\) all create limits: in the
group computation, the limit downstairs was the product of sets, its unique lift was that set
carrying the forced group structure, and the lift was a product upstairs.
The reward for creation is that it manufactures completeness: a functor into a category that already
has limits lets the source category inherit them.
Lemma: Creation Yields Limits and Preserves Them
Let \(F : \mathscr{A} \to \mathscr{B}\) be a functor and \(\mathbf{I}\) a small category. Suppose
\(\mathscr{B}\) has, and \(F\) creates, limits of shape \(\mathbf{I}\). Then \(\mathscr{A}\) has,
and \(F\) preserves, limits of shape \(\mathbf{I}\).
Proof
Let \(D : \mathbf{I} \to \mathscr{A}\) be any diagram. The image diagram
\(F \circ D : \mathbf{I} \to \mathscr{B}\) has a limit, since \(\mathscr{B}\) has limits of shape
\(\mathbf{I}\); fix a limit cone \(\big(B \xrightarrow{q_I} FD(I)\big)_{I \in \mathbf{I}}\) on
\(F \circ D\). Because \(F\) creates limits of shape \(\mathbf{I}\), this cone lifts to a cone
\(\big(A \xrightarrow{p_I} D(I)\big)_{I \in \mathbf{I}}\) on \(D\) with \(F(A) = B\) and
\(F(p_I) = q_I\), and this lifted cone is a limit cone on \(D\). In particular \(D\) has a limit,
and since \(D\) was arbitrary, \(\mathscr{A}\) has all limits of shape \(\mathbf{I}\).
It remains to see that \(F\) preserves them. Let \(\big(A' \xrightarrow{p'_I} D(I)\big)_{I \in
\mathbf{I}}\) be any limit cone on \(D\); we must show its image is a limit cone on \(F \circ D\).
The cone just constructed, \(\big(A \xrightarrow{p_I} D(I)\big)\), is also a limit cone on \(D\),
so by uniqueness of limits there is an isomorphism \(\theta : A' \to A\) with
\(p_I \circ \theta = p'_I\) for all \(I\). Applying \(F\), the map \(F\theta : F(A') \to F(A) = B\)
is an isomorphism with \(q_I \circ F\theta = F(p_I) \circ F\theta = F(p_I \circ \theta) =
F(p'_I)\) for all \(I\). Thus the image cone
\(\big(F(A') \xrightarrow{F p'_I} FD(I)\big)\) is carried by the isomorphism \(F\theta\) onto the
limit cone \(\big(B \xrightarrow{q_I} FD(I)\big)\), commuting with the projections; a cone
isomorphic to a limit cone through a map commuting with the projections is itself a limit cone.
Hence \(F\) preserves the limit of \(D\).
From the lemma follows a broad conclusion. Since \(\mathbf{Set}\) has all limits and the forgetful
functors out of the categories of algebras create them, those categories have all limits too, and
the forgetful functors preserve them. Completeness of algebra is imported wholesale from
completeness of sets.
A caution on equality
The definition of creation contains a feature that should be viewed with suspicion. It refers to
equality of objects, in the clauses \(F(A) = B\) and \(F(p_I) = q_I\), and equality of
objects of a category is a relation usually too strict to be the right one; isomorphism is almost
always the healthier notion. Replacing equality by isomorphism throughout gives a more inclusive
property: one asks instead that if \(F \circ D\) has a limit then some cone on \(D\) has a limit cone
as its image, and that every such cone is itself a limit cone. The property defined above, resting on
strict equality, is properly called strict creation, and in much of the literature
the unqualified word "creates" denotes the more inclusive isomorphism-based version. The strict form
is used here because it is simpler to state and because the examples at hand — the forgetful functors
of algebra — satisfy it outright.
Projective and injective objects
The dual vocabulary, applied not to a forgetful functor but to a represented one, isolates a class of
objects defined by their interaction with
epics.
The covariant
hom-functor
\(\mathscr{B}(P, -) : \mathscr{B} \to \mathbf{Set}\) need not preserve epics; the objects \(P\) for
which it does are singled out.
Definition: Projective and Injective Object
An object \(P\) of a category \(\mathscr{B}\) is projective if the functor
\(\mathscr{B}(P, -) : \mathscr{B} \to \mathbf{Set}\) preserves epics: whenever \(f\) is epic, so
is \(\mathscr{B}(P, f)\). An object \(I\) is injective if it is projective in
\(\mathscr{B}^{\mathrm{op}}\), equivalently if the contravariant functor
\(\mathscr{B}(-, I) : \mathscr{B}^{\mathrm{op}} \to \mathbf{Set}\) preserves epics.
Unwound, \(P\) is projective when every map out of \(P\) into the target of an epic lifts along that
epic. Projectivity is inherited along adjunctions in the following sense: if
\(F \dashv G\) is an
adjunction
between \(\mathbf{Set}\) and \(\mathscr{B}\) whose right adjoint \(G\) preserves epics, then
\(F(S)\) is projective for every set \(S\). In \(\mathbf{Vect}_k\) every object is injective; in
\(\mathbf{Ab}\), by contrast, neither projectivity nor injectivity is automatic, and objects of each
kind sit alongside objects of neither. Projective and injective objects are the building blocks of
homological algebra, and the lifting condition defined here is where that subject begins.