Representable Functors

Representable Functors

The previous page left a promise unredeemed. Each object \(A\) of a locally small category was to be probed by the functor \(\mathscr{A}(A, -)\) that records all the maps out of it, turning the object into the data of its relationships. It is time to make the construction precise and to ask the question it forces: which set-valued functors arise this way? The functors naturally isomorphic to an object's web of outgoing maps are the representable ones, and they are the second route, after adjunctions, to the idea of a universal property.

The functor of maps out of an object

Fix an object \(A\) of a category \(\mathscr{A}\). To each object \(B\) assign the set \(\mathscr{A}(A, B)\) of maps from \(A\) to \(B\). This assignment is functorial in \(B\). A map \(g : B \to B'\) acts on a map \(p : A \to B\) without any choice, since there is exactly one thing to do with \(p\), namely follow it by \(g\). Post-composition by \(g\) thus carries \(\mathscr{A}(A, B)\) into \(\mathscr{A}(A, B')\).

That \(H^A\) is a functor is a one-line check. Identities are preserved because \(H^A(1_B)\) sends \(p\) to \(1_B \circ p = p\), so \(H^A(1_B) = 1_{\mathscr{A}(A,B)}\). Composition is preserved because for \(g : B \to B'\) and \(h : B' \to B''\), applying \(H^A(h) \circ H^A(g)\) to \(p\) yields \(h \circ (g \circ p)\), which by associativity equals \((h \circ g) \circ p = H^A(h \circ g)(p)\). The single fact that composition is associative is what makes post-composition functorial. Nothing about the particular category \(\mathscr{A}\) is used.

Representability

One should not expect a functor picked at random to be representable. The interest of the notion lies in how often the functors that arise in practice turn out to be representable after all. The forgetful functors, which strip structure from an object and return its underlying set, are the leading examples.

The smallest examples

Take \(\mathscr{A} = \mathbf{Set}\) itself and let \(A = 1\) be a one-element set. A map \(1 \to B\) is precisely a choice of one element of \(B\), so \(H^1(B) = \mathbf{Set}(1, B)\) is in bijection with \(B\). In this sense the terminal set sees each set as its own collection of points. The bijection is natural in \(B\), because a map \(g : B \to B'\) acts on both sides by applying \(g\) to the chosen element, so \(H^1\) is naturally isomorphic to the identity functor \(1_{\mathbf{Set}}\). The identity functor on \(\mathbf{Set}\) is therefore representable, represented by the one-element set. The observation that an element of a set is the same thing as a map out of \(1\) was made on the previous page, and here it is promoted to a natural isomorphism of functors.

The pattern extends to the forgetful functors. The functor \(\mathbf{Top} \to \mathbf{Set}\) that forgets a topology is naturally isomorphic to \(\mathbf{Top}(1, -)\), where \(1\) is the one-point space, because a continuous map from a one-point space into \(X\) is just a point of \(X\). The functor \(\mathbf{Grp} \to \mathbf{Set}\) forgetting the group structure is \(\mathbf{Grp}(\mathbb{Z}, -)\), because a homomorphism out of the infinite cyclic group is determined freely by where the generator goes, hence by an arbitrary element of the target group. In each case a single chosen object, the one-point space or the integers, represents the operation of reading off the underlying set.

Representables among structured categories

Beyond the underlying-set functors, representables turn up wherever a construction is governed by a universal property. Fixing two finite-dimensional real vector spaces \(U\) and \(V\), consider the functor \(\mathbf{Vect}_{\mathbb{R}} \to \mathbf{Set}\) whose value at \(W\) is the set of bilinear maps \(U \times V \to W\). This functor is representable. There is a space \(T\) and a natural isomorphism between the bilinear maps out of \(U \times V\) and the linear maps out of \(T\), \[ \{\text{bilinear } U \times V \to W\} \cong \mathbf{Vect}_{\mathbb{R}}(T, W), \quad \text{natural in } W . \]

The representing object \(T\) is the tensor product \(U \otimes V\), and the natural isomorphism above is precisely its universal property, met earlier as the statement that bilinear maps out of a product factor uniquely through the tensor product. Naturality in \(W\) is the uniqueness clause of that property. For a linear map \(\ell : W \to W'\) and a bilinear map \(\beta\) with linear factorization \(\widetilde{\beta}\), both \(\ell \circ \widetilde{\beta}\) and the factorization of \(\ell \circ \beta\) are linear maps out of \(U \otimes V\) that agree with \(\ell \circ \beta\) on the image of \(U \times V\), so they coincide. What was then phrased as a universal mapping property is, in the present language, the single assertion that a certain functor is representable, with \(U \otimes V\) the representing object.

The basic spaces met across these pages furnish further instances. In the category of based spaces, the based maps from the circle into a space \(X\) form the underlying set of its loop space \(\Omega X\), so the functor sending \(X\) to its set of loops is represented by the circle. In the based homotopy category, where maps are taken up to based homotopy, this same circle represents the fundamental group \(\pi_1(X)\), whose elements are the homotopy classes of those loops. These examples share one shape. In each, a functor that looked like a free-standing construction is unmasked as the maps out of one well-chosen object.

Seeing and Being Seen

The examples just gathered were not produced one at a time by luck. The forgetful functors share a feature that forces representability on them, and naming that feature converts a list of instances into a theorem. The feature is the possession of a left adjoint.

Adjoints produce representables

Recall that an adjunction \(F \dashv G\) supplies a bijection \(\mathscr{B}(F A, B) \cong \mathscr{A}(A, G B)\) natural in both arguments. Reading it with \(A\) fixed and \(B\) varying exhibits the composite \(\mathscr{A}(A, G(-))\) as a hom-functor in disguise.

The lemma specializes at once to set-valued functors. A forgetful functor lands in \(\mathbf{Set}\) and, in the algebraic cases, carries a left adjoint in the form of the free construction. Having such an adjoint is enough to force representability.

The proposition explains the earlier list at a stroke, and it produces new instances. The forgetful functor \(\mathbf{Vect}_k \to \mathbf{Set}\) has the free vector space functor as its left adjoint. The value of that adjoint at the one-element set is the one-dimensional space \(k\), so the forgetful functor is \(H^k\). Directly, a linear map out of \(k\) is fixed by the image of \(1\), which may be any vector of the target. Likewise, the forgetful functor on commutative rings with identity, with ring homomorphisms required to send \(1\) to \(1\), is represented by the polynomial ring \(\mathbb{Z}[x]\), because such a homomorphism out of \(\mathbb{Z}[x]\) is determined by the image of \(x\), which may be any element of the target. Across algebra, the free-forgetful pattern is in every case a certificate of representability.

Putting the views together

We have, for each object \(A\), a functor \(H^A\) describing how \(A\) sees the rest of the category. As \(A\) varies the view varies, yet it is always the same category being viewed, so the views are related. A map \(A' \to A\) converts maps out of \(A\) into maps out of \(A'\) by pre-composition. Note the reversal. The assignment \(A \mapsto H^A\) is itself contravariant.

Both claims in this definition are instances of associativity. Each \(H^f\) is natural because, for a map \(g : B \to B'\), both routes around the naturality square send \(p \in \mathscr{A}(A, B)\) to \(g \circ p \circ f\). The assignment \(f \mapsto H^f\) preserves identities because \(p \circ 1_A = p\), and it reverses composition because for \(f' : A'' \to A'\) one has \(H^{f'} \circ H^f : p \mapsto (p \circ f) \circ f' = p \circ (f \circ f')\), which is \(H^{f \circ f'}\). The reversed order is the contravariance of \(H^\bullet\).

Dualizing: how objects are seen

Every definition so far can be dualized by reversing the arrows. At the formal level the move is the trivial one of replacing \(\mathscr{A}\) by \(\mathscr{A}^{\mathrm{op}}\), but the flavor changes. We stop asking what an object sees and start asking how it is seen. Fixing a target \(A\) and letting the source vary gives the maps into \(A\), and a map \(g : B' \to B\) now acts by pre-composition. It sends a map \(B \to A\) to a map \(B' \to A\), in the opposite direction to \(g\).

Representability for presheaves is defined by the mirror condition. Strictly the notion is already available, since a contravariant functor on \(\mathscr{A}\) is a covariant functor on \(\mathscr{A}^{\mathrm{op}}\), to which the earlier definition applies. A direct statement is nonetheless convenient.

The contravariant examples are as natural as the covariant ones. The power-set construction is a presheaf \(\mathbf{Set}^{\mathrm{op}} \to \mathbf{Set}\). It sends a set \(B\) to its power set \(\mathcal{P}(B)\) and a map \(g : B' \to B\) to the operation of taking preimages, \(U \mapsto g^{-1}U\). Since a subset of \(B\) is the same as a map \(B \to 2\) into the two-element set, this presheaf is represented by \(2\), that is, \(\mathcal{P} \cong H_2\).

In the same way, the operation of sending a topological space to its set of open subsets is a representable presheaf on the category of spaces. It is represented by the Sierpiński space, the two-point space in which exactly one singleton is open. The assignment to each space of its ring of continuous real-valued functions, post-composed with the forgetful functor to sets, is represented by the real line. In each case the object being mapped into, whether the classifier \(2\), the Sierpiński space, or the line, is the universal receptacle through which the construction is read.

The Yoneda Embedding

The covariant representables were bundled, in the previous section, into a single contravariant functor \(H^\bullet\). The contravariant representables can be bundled in the same way, and it is this second bundling that the theory takes as its main object. As the target \(A\) varies, each map \(f : A \to A'\) gives a natural transformation \(H_A \to H_{A'}\) between the presheaves. This time the direction is preserved, so the assignment \(A \mapsto H_A\) is covariant.

Naturality of each \(H_f\) and functoriality of \(H_\bullet\) follow from associativity exactly as for \(H^\bullet\), with the roles of the two slots exchanged, and this time the order of composition is preserved. The choice to work with the presheaf bundling rather than its covariant mirror is a matter of convenience, not of substance, since any theorem about one dualizes to a theorem about the other. The advantage is that, when \(\mathscr{A}\) is small, the target \([\mathscr{A}^{\mathrm{op}}, \mathbf{Set}]\) is a richly structured category, with limits and colimits and exponentials that \(\mathscr{A}\) itself may lack. Embedding \(\mathscr{A}\) into it therefore places a category inside a well-behaved one. The word embedding anticipates a fact proved in the pages ahead: \(H_\bullet\) is full and faithful, and consequently injective on isomorphism classes of objects, so that no information about \(\mathscr{A}\) is lost in the passage to its presheaves. Non-isomorphic objects give non-isomorphic presheaves. An object is determined, up to isomorphism, by the bare pattern of maps into it.

A summary of the four functors

Four functors have now been built from the single operation of forming hom-sets, and it is worth setting them side by side. For each object \(A\) there is a covariant functor \(H^A : \mathscr{A} \to \mathbf{Set}\) of maps out of \(A\), and collecting these over all \(A\) gives the contravariant \(H^\bullet : \mathscr{A}^{\mathrm{op}} \to [\mathscr{A}, \mathbf{Set}]\). Dually, for each object \(A\) there is a presheaf \(H_A : \mathscr{A}^{\mathrm{op}} \to \mathbf{Set}\) of maps into \(A\), and collecting these gives the covariant Yoneda embedding \(H_\bullet : \mathscr{A} \to [\mathscr{A}^{\mathrm{op}}, \mathbf{Set}]\). The second pair is the dual of the first. Both involve a contravariant step that cannot be avoided. Whether one fixes the source and varies the target or the reverse, exactly one of the two slots reverses arrows.

The two slots at once

One further functor unifies all four. Rather than fix either argument of the hom-set, we let both vary. Since the first slot reverses arrows and the second preserves them, the home of that functor is \(\mathscr{A}^{\mathrm{op}} \times \mathscr{A}\).

The form \(p \mapsto g \circ p \circ f\) is forced. A morphism in \(\mathscr{A}^{\mathrm{op}} \times \mathscr{A}\) from \((A,B)\) to \((A',B')\) is a pair \(A' \xrightarrow{f} A\) and \(B \xrightarrow{g} B'\). The only way to turn a map \(p : A \to B\) into a map \(A' \to B'\) using \(f\) and \(g\) is to precede \(p\) by \(f\) and follow it by \(g\). Functoriality follows from associativity.

Packaging the hom-sets into a bifunctor also clarifies what the naturality clauses in the definition of an adjunction were asserting. An adjunction \(F \dashv G\) between \(F : \mathscr{A} \to \mathscr{B}\) and \(G : \mathscr{B} \to \mathscr{A}\) gives, for each \(A\) and \(B\), a bijection \(\mathscr{B}(F A, B) \cong \mathscr{A}(A, G B)\). Both sides are values of functors \(\mathscr{A}^{\mathrm{op}} \times \mathscr{B} \to \mathbf{Set}\), namely \(\mathscr{B}(F-, -)\) and \(\mathscr{A}(-, G-)\), each obtained from a hom-bifunctor. The requirement that the bijection be natural in both arguments is precisely the requirement that these two functors be naturally isomorphic, because a morphism \((p, q)\) of the product category factors as \((1, q) \circ (p, 1)\), so naturality in each variable separately amounts to naturality in the pair. What once looked like a pair of commuting-square conditions is the single statement that an adjunction is a natural isomorphism \(\mathscr{B}(F-, -) \cong \mathscr{A}(-, G-)\).

Objects Probed by Elements

This page has turned on one reversal: an object is known not by what it contains but by how it relates to others. The most concrete form of that reversal comes now. Objects of an arbitrary category have no elements in any obvious sense, since there is nothing inside an abstract object to point at. But sets do have elements, and at the start of this page an element of a set \(A\) was seen to be the same thing as a map out of the terminal object, a map \(1 \to A\). That identification seeds a definition restoring a notion of element to every object of every category.

The term is a synonym for map. Its value lies in the change of attitude it encourages. Generalized elements recover the ordinary ones and reach beyond them. When \(A\) is a set, a generalized element of shape \(1\) is an ordinary element, while a generalized element of shape \(\mathbb{N}\) is a sequence in \(A\). In the category of spaces the generalized elements of shape \(1\), the one-point space, are the points, and the generalized elements of shape the circle are, by definition, the loops. In categories of geometric objects one may therefore equally well speak of figures of a given shape. Among based spaces, the based loops of \(X\), that is, its generalized elements of shape the based circle, form the underlying set of its loop space, and their based homotopy classes form the fundamental group \(\pi_1(X)\). The first section of this page found both of these, as functors of \(X\), to be represented by the circle.

Algebra furnishes the same pattern. To study the solutions of an equation such as \(x^2 + y^2 = 1\) over a commutative ring \(A\) is to study the pairs \((a, b) \in A \times A\) with \(a^2 + b^2 = 1\). Each such pair corresponds to exactly one ring homomorphism out of \(\mathbb{Z}[x, y]/(x^2 + y^2 - 1)\) sending \(1\) to \(1\), namely the one whose values on \(x\) and \(y\) form the chosen pair. A solution in the ring \(A\) is thus a generalized element of \(A\) whose shape is the coordinate ring of the equation. The fixed object \(\mathbb{Z}[x, y]/(x^2 + y^2 - 1)\) is the shape, and the maps out of it into the varying ring \(A\) are exactly the solutions there. A geometric figure, a sequence, a point, and a solution of an equation are each a map into an object, and the totality of maps of a given shape is a representable functor evaluated at that object.

Probing is functorial

Fixing the shape and letting the probed object vary gives, for each object \(S\), the covariant functor \(H^S = \mathscr{A}(S, -)\) sending an object to its set of generalized elements of shape \(S\). The functoriality of \(H^S\) carries a plain meaning: any map \(A \to B\) transforms \(S\)-shaped elements of \(A\) into \(S\)-shaped elements of \(B\), simply by composition. A continuous map of spaces transforms loops into loops, a ring homomorphism transforms solutions of an equation into solutions, and a linear map transforms the chosen probes of one space into those of another. The probe is held fixed and the map pushes its elements forward.

Functoriality of this kind has already appeared, in concrete form, in the manifold series. To each smooth map of manifolds \(F : M \to N\) and each point \(p\) there is the differential \(dF_p : T_pM \to T_{F(p)}N\), and the properties of the differential established there include the chain rule \(d(G \circ F)_p = dG_{F(p)} \circ dF_p\) and the identity \(d(\mathrm{Id})_p = \mathrm{Id}\). Read in the present language these two equations are preservation of composition and preservation of identities. It follows that the operation of taking the tangent space at a chosen point is a functor on manifolds equipped with a point, and the differential is what it does to maps. The pushforward notation \(F_* := dF_p\), with its rule \((G \circ F)_* = G_* \circ F_*\), was the functoriality asserting itself before the word was available.