Functors & Limits

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.

The dual notions read off at once by reversing every arrow. The functor \(F\) 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, characterized by the requirement that its composite with each projection \(\lim (F \circ D) \to FD(I)\) be \(F p_I\), the corresponding projection of the image cone. The functor preserves the limit of \(D\) precisely when \(F \circ D\) also has a limit and 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.

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. The first example of the next section preserves every limit but does not reflect limits.

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. Across the categories of algebraic objects met so far, these forgetful functors behave consistently. They respect limits and fail to respect colimits, and that pattern sets the theme for the section. A topological example comes first, as a warning about reflection, and the algebraic categories follow, where limits are preserved and colimits are not.

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, a fact we state without proof. 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. The functor \(U\) 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. If \(\{x\}\) is not open in \(X\), then no basic open set \(V \times W\) equals \(\{(x, y)\}\), so that singleton is not open in the product. The projections of \(U(X) \times U(Y)\) lift to continuous maps out of \(Z\), and these maps form 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. If it were, the identity function \(Z \to X \times Y\), which is the unique map commuting with the projections, would be an isomorphism in \(\mathbf{Top}\) by uniqueness of limits. It is continuous, but its inverse is not, since \(Z\) carries the strictly finer topology. The image is a limit while the source is not, so 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 groups, abelian groups, rings (with the ring homomorphisms sending \(1\) to \(1\) as maps), and 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 the trivial group to a one-element set, not to the empty set, so it does not preserve initial objects. Likewise the two inclusions of \(X\) and \(Y\) into a direct sum \(X \oplus Y\) of vector spaces both send \(0\) to \(0\), so their image in \(\mathbf{Set}\) is not a coproduct cocone of the underlying sets, and 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 also 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 \(\mathbf{Grp}\) in full. For a limit of any shape in any of \(\mathbf{Grp}\), \(\mathbf{Ab}\), \(\mathbf{Vect}_k\), \(\mathbf{Ring}\), the same argument runs on the set of compatible families inside the product of the underlying sets, a set closed under the componentwise operations because the maps of the diagram are homomorphisms. We do not write out those cases.

Take groups \(X_1\) and \(X_2\), write \(U : \mathbf{Grp} \to \mathbf{Set}\) for the underlying-set functor of groups, reusing the letter \(U\), 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 \(p_2 : U(X_1) \times U(X_2) \to U(X_2)\).

The argument used no group theory beyond the componentwise checks. Its only inputs were 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 proposition treated the standard product set, but its argument applies verbatim to any product cone on \(\big(U(X_1), U(X_2)\big)\) in \(\mathbf{Set}\), since such a cone is isomorphic to the standard one through a bijection commuting with the projections, and the group structure transports along that bijection. Beyond preserving the product, the forgetful functor 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. Creation turns the underlying-set functor into a tool for building limits in algebra, not merely for respecting them.

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 creating functor into a category that already has limits lets the source category inherit them.

From the lemma follows a broad conclusion. Since \(\mathbf{Set}\) has all limits and the forgetful functors on these algebraic categories 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 same vocabulary, applied to epics instead of limits and to a hom-functor instead of a forgetful functor, 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 next definition singles out the objects \(P\) for which it does.

Unwound, since epics in \(\mathbf{Set}\) are exactly the surjections, \(P\) 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 \(S\). Indeed, under the adjunction the map \(\mathscr{B}(F(S), f)\) corresponds to \(\mathbf{Set}(S, Gf)\), and postcomposition with the surjection \(Gf\) is surjective on functions out of \(S\), a preimage being chosen for each value. 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. For instance \(\mathbb{Z}\) is projective but not injective, \(\mathbb{Q}\) is injective but not projective, and \(\mathbb{Z}/2\mathbb{Z}\) is neither. We record these facts about \(\mathbf{Vect}_k\) and \(\mathbf{Ab}\) without proof. Projective and injective objects are the building blocks of homological algebra, and the lifting condition defined here is where that subject begins.