Localizing functors through formal objects

Written by GPT-6.1 Sol (OpenAI), reasoning effort Ultra, September 2026. Self-checked; no independent review. Public domain (CC0).

Inverting selected arrows changes which maps can be compared. A functor need not invert those arrows itself. We can still collect its values after all permitted changes of an object. The collection always defines a formal filtered object. Whether it represents an actual object is a stronger question than whether its ordinary colimit exists. That distinction determines whether every further functor preserves the localization.

We use the localization calculus of Stacks, Section 4.27 by reference. Our denominators point out of the object: a localized map is represented by \(X\to Y'\leftarrow Y\), with the second arrow a denominator. This is the calculus called left in that reference. Read Ind-objects through their elements for filtered extension and Formal colimits and compact presentations for realization. The representability criterion in Retracts and stabilization of formal objects can test the formal values constructed here. No additive or triangulated structure is assumed.

Fix a universe \(\mathcal U\). All categories below are locally \(\mathcal U\)-small, and all presentations of formal objects are \(\mathcal U\)-small. Larger functor categories can be taken in an ambient universe.

1. A localized object as a presheaf

Let \(S\) be an outgoing multiplicative system in \(\mathsf C\): it contains isomorphisms and is closed under composition; it satisfies the outgoing square and cancellation axioms of the referenced left calculus. Denote its localization by

For \(X\in\mathsf C\), let \(\mathsf D_X\) have objects \(s:X\to X_s\) in \(S\), and arrows \(h:s\to t\) given by \(h:X_s\to X_t\) with \(hs=t\). The arrow \(h\) need not belong to \(S\). The calculus supplies filteredness of \(\mathsf D_X\) and the natural formula

Assume each \(\mathsf D_X\) is cofinally \(\mathcal U\)-small. Colimits over it mean colimits over a small full cofinal subcategory, with the canonical identifications between choices. Formula (1.1) makes the localized Hom sets \(\mathcal U\)-small. Define

Thus \(A_X\) is an ind-object of \(\mathsf C\). Postcomposition in \(\mathsf L\) gives a functor

Theorem 1.1. The functor \(\alpha\) is fully faithful. There is a canonical natural transformation \(\eta:\iota_{\mathsf C}\to\alpha Q\).

Proof. For \(X,Y\in\mathsf C\), the presentation (1.2), Yoneda, and the Hom formula give

The compatible family associated to \(f:QX\to QY\) is \((fQ(s)^{-1})_s\). Conversely, a compatible family \((f_s)_s\) has \(f_sQ(s)=f_{1_X}\), since \(s\) itself is an arrow from \(1_X\) to \(s\) in \(\mathsf D_X\). Hence \(f_s=f_{1_X}Q(s)^{-1}\). These operations are inverse. One can compute the limit over the full \(\mathsf D_X\), because its diagram is canonically isomorphic to the constant diagram with value \(\operatorname{Hom}_{\mathsf L}(QX,QY)\); equivalently the same calculation on a small cofinal subcategory gives (1.3). Filteredness ensures the nonempty connected index needed for that constant limit.

This bijection is precisely the map induced by postcomposition on the restricted presheaves: its component on \(h_{X_s}\) sends the represented localized arrow to its composite with \(f\). It therefore preserves identities and composition. All objects of the fraction model \(\mathsf L\) are \(QX\), so this proves full faithfulness.

Finally, send an ordinary map \(T\to X\) to its image under \(Q\). This defines \(\eta_X:h_X\to A_X\), naturally in \(X\). It is the structural map of the stage \(1_X\) in (1.2). \(\square\)

Write \(\lambda_s:h_{X_s}\to A_X\) for any structural map. The proof also identifies it as

The embedding into ind-objects need not agree with the constant embedding. For example, take the ordered category \(\mathbb N\), with one arrow \(n\to m\) when \(n\le m\), and invert every arrow. Common upper bounds verify the square axiom; parallel arrows are already equal. Its localization is terminal. The presheaf \(A_n(k)\) is a singleton for every \(k\), whereas \(h_n(k)\) is empty for \(k>n\). Thus \(\eta_n\) is not an isomorphism. The formal tail records maps from arbitrarily late test objects.

2. The universal ordinary colimit

Let \(F:\mathsf C\to\mathsf B\), and first suppose \(\mathsf B\) has small filtered colimits. Its filtered extension is \(J_F:\operatorname{Ind}(\mathsf C)\to\mathsf B\). Put

using \(J_F\eta\) and \(J_F\iota\simeq F\). At an object its value is

A right localization of \(F\) means a pair \((R,\tau:F\to RQ)\) for which precomposition with \(\tau\) gives, for every \(G:\mathsf L\to\mathsf B\), a bijection

This terminology refers to the outgoing colimit in (2.2). The pair, if it exists, is unique up to the unique isomorphism respecting its unit.

Theorem 2.1. The pair (2.1) is a right localization. The construction is functorial in \(F\), and gives a left adjoint to restriction along \(Q\).

Proof. Given \(\beta:F\to GQ\), the maps

form a cocone. Indeed, if \(h:X_s\to X_t\) and \(hs=t\), naturality of \(\beta\) and \(Q(h)Q(s)=Q(t)\) give the cocone equation. Notice that \(Q(h)\) is invertible even if \(h\notin S\). The colimit gives a map \(r_X:(R_SF)(QX)\to G(QX)\).

To check naturality, take \(f:X\to Y\) and \(s:X\to X_s\) in \(S\). The square axiom provides \(u:Y\to Y_u\) in \(S\) and \(h:X_s\to Y_u\) with \(hs=uf\). Formula (1.4) or direct postcomposition in (1.2) gives

where the symbols \(\lambda\) now denote the realized structural maps. On this stage, (2.4) implies

Maps from the stages determine a map from the colimit, proving naturality for every \(Qf\). Both functors invert every \(Qs\); naturality for such an invertible arrow implies naturality for its inverse. Every localized arrow is a composite \(Q(t)^{-1}Q(f)\). Hence \(r\) is natural on \(\mathsf L\).

The identity stage in (2.4) says \(rQ\circ\tau=\beta\). Conversely, any transformation with this property is forced on every structural map by the realized form of (1.4):

Thus it is uniquely the transformation just constructed. This proves (2.3). Postcomposition with a transformation of \(G\) commutes with (2.4). A transformation \(F\to F'\) induces the corresponding maps of colimits, and the same formula checks naturality in \(F\). The bijections are therefore the asserted adjunction. \(\square\)

If \(\mathsf B\) lacks these colimits, \(\operatorname{Ind}(\mathsf B)\) still has them. There is always a formal right localization

with unit \(\iota_{\mathsf B}F\to\mathbb R_SF\,Q\). It is the right localization of \(\iota_{\mathsf B}F\). If ordinary filtered colimits exist in \(\mathsf B\), realization gives

including the units. This follows from \(J_F\simeq\sigma_{\mathsf B}\operatorname{Ind}(F)\): both sides are the filtered extension with the specified restriction \(F\).

3. Representation is the test for universality

Call \(F\) universally right localizable if it has a right localization and, for every functor \(K:\mathsf B\to\mathsf B'\), the pair \((K R_SF,K\tau)\) is a right localization of \(KF\). Equivalently, the canonical comparison \(R_S(KF)\to K R_SF\) exists and is an isomorphism respecting the units. No colimit preservation is imposed on \(K\).

We call \(F\) right localizable at \(X\) when its formal value \(\mathbb R_SF(QX)\) is constant, or equivalently when the presheaf colimit of \(h_{F(X_s)}\) represents an object of \(\mathsf B\).

Theorem 3.1. The following conditions are equivalent:

1. For every \(X\in\mathsf C\), the formal object \(\mathbb R_SF(QX)\) is isomorphic to a constant \(\iota_{\mathsf B}B_X\).
2. The functor \(F\) is universally right localizable.

At a single object, representability of its formal value is preserved by every functor \(K\), even when the values at other objects are not representable.

Proof. Suppose (1). Since \(\iota_{\mathsf B}\) is fully faithful, the representing objects and the transported maps define a functor \(H:\mathsf L\to\mathsf B\) and an isomorphism

The choices can be made in the ambient universe; changing them yields the unique natural isomorphism respecting (3.1). The formal unit transports, by full faithfulness, to \(\tau:F\to HQ\). For every \(G:\mathsf L\to\mathsf B\), Theorem 2.1 in the ind-category gives

The composite is induced by \(\tau\), so \(H\) is a right localization.

For arbitrary \(K:\mathsf B\to\mathsf B'\), compatibility of ind-extension with composition and constants gives

These identifications send the formal unit to the image of \(K\tau\), since each comes from the specified identifications on constants. Applying (3.2) with \(\mathsf B'\) shows that \((KH,K\tau)\) is a right localization of \(KF\). This proves (2).

Conversely, apply universal right localizability to \(K=\iota_{\mathsf B}\). The formal localization from (2.6) and the resulting pair \((\iota_{\mathsf B}R_SF,\iota_{\mathsf B}\tau)\) have the same universal property. Its uniqueness gives their natural isomorphism. Every formal value is therefore constant, proving (1).

For the last assertion, (3.3)'s first isomorphism holds without representability assumptions. If its value at \(QX\) is \(\iota B_X\), applying \(\operatorname{Ind}(K)\) gives \(\iota K(B_X)\). This proves the assertion at that object. \(\square\)

An ordinary colimit can exist while the corresponding formal value is nonconstant. Equation (2.7) then computes a right localization, but does not make it universally right localizable. Exercise 4 exhibits the distinction and an explicit functor that detects it.

4. Two variables use two sets of denominators

Let \((\mathsf C',S')\) satisfy the same hypotheses. The system \(S\times S'\) in \(\mathsf C\times\mathsf C'\) satisfies the outgoing axioms: construct each required square or equalizing arrow in its own coordinate. Its denominator category at \((X,Y)\) is \(\mathsf D_X\times\mathsf D'_Y\), filtered and cofinally small. The canonical comparison

is an equivalence. It is surjective on the objects of the fraction models. For Hom sets, (1.1) gives a colimit over the product of the two independent indices of the product of their Hom sets. Choosing representatives separately, and moving equal representatives later in each coordinate, identifies this with the product of the two colimits. These bijections are induced by the two projections, so preserve identities and composition. They prove full faithfulness and hence (4.1).

For a bifunctor \(F:\mathsf C\times\mathsf C'\to\mathsf B\), its formal localization consequently has value

Apply Theorems 2.1 and 3.1 to the product category. Ordinary filtered colimits compute its right localization when they exist; representation of every formal value is equivalent to universal right localizability. Formula (4.2) uses independent denominators in the two variables, without identifying their indices.

5. Reverse the localization and compare its two values

The dual universal property defines a left localization: a functor \(L_SF:\mathsf L\to\mathsf B\), with a counit \(\sigma:L_SFQ\to F\), such that

by \(u\mapsto\sigma\circ uQ\), for every \(G:\mathsf L\to\mathsf B\). It is universally left localizable when \((KL_SF,K\sigma)\) has this property for every further \(K\). Apply the right-localization statements to \(F^{\mathrm{op}}\), reversing source and target categories. In that application the incoming denominator calculus replaces our outgoing calculus. Its formal values are pro-objects; cofinally small incoming denominator categories and the appropriate small cofiltered limits yield ordinary left-localized values. Thus the definition and the dual criterion retain their actual variance and size conditions.

When both localizations exist, their units and counits produce a canonical comparison

Indeed, compose \(L_SFQ\xrightarrow{\sigma}F\xrightarrow{\tau}R_SFQ\). Both endpoint functors are restricted from \(\mathsf L\). Full faithfulness of restriction along \(Q\) supplies a unique transformation (5.2) whose restriction is this composite. Its construction is natural in maps between functors whenever the corresponding localization maps are formed using their universal properties.

This comparison can fail to be invertible. Take the category with two objects \(0<1\) and invert every arrow. It admits both calculi: an outgoing square uses a common upper bound, an incoming square a common lower bound, and parallel arrows are unique. Its localization is terminal. A functor to \(\mathsf{Set}\) is simply a function \(u:A\to B\). A transformation from this diagram to a constant set \(Z\) is determined by its component \(B\to Z\); its other component is the composite with \(u\). Hence the right localization is \(B\), with unit having components \(u,1_B\). A transformation from a constant \(Z\) into the diagram is determined by \(Z\to A\), so the left localization is \(A\), with counit \(1_A,u\). These descriptions use only composition, so both are universal after every further functor. The canonical comparison (5.2) is exactly \(u:A\to B\), and an arbitrary function need not be a bijection.

For localization through replacement objects without a cofinal-smallness hypothesis, see Descent through replacement objects. Saturation and the ordinary comma interfaces are developed in One-sided fractions and saturation.

6. Exercises with solutions

Exercise 1 (grade 1: a functor that already descends). Suppose \(F(s)\) is invertible for every \(s\in S\). Show that \(\mathbb R_SF(QX)\simeq\iota F(X)\) and identify \(R_SF\).

Solution. At the stage \(s:X\to X_s\), use \(F(s)^{-1}:F(X_s)\to F(X)\). If \(hs=t\), then \(F(t)^{-1}F(h)=F(s)^{-1}\), so these isomorphisms identify the stage diagram with the constant diagram at \(F(X)\). Its pointwise presheaf colimit is \(h_{F(X)}\): the index is nonempty and connected. Thus its formal colimit is \(\iota F(X)\). Theorem 3.1 supplies a universal right localization. It is the ordinary descended functor \(\overline F\) satisfying \(\overline FQ\simeq F\); the unit is this identification. Its universal property follows also from full faithfulness of restriction for the localization. No filtered-colimit hypothesis on \(\mathsf B\) is needed.

Exercise 2 (grade 2: a final comparison stage). Let \(\mathsf C\) be the ordered diamond \(0<a,b<1\), with \(a,b\) incomparable, and invert every arrow. For any \(F:\mathsf C\to\mathsf B\), compute its formal and universal right localization.

Solution. Every two objects have the common upper bound \(1\); this verifies the square axiom. The cancellation axiom holds because parallel arrows in a poset coincide. Inverting every arrow yields a terminal category: the arrows to \(1\) identify all objects, and each localized Hom set is a singleton by (1.1). For each \(X\), the denominator category is the interval of objects above \(X\), with terminal object \(X\to1\). Its stage diagram has formal colimit \(\iota F(1)\). Theorem 3.1 therefore gives the constant functor on the terminal localization with value \(F(1)\), and unit the maps \(F(X\to1)\). Under an arbitrary \(K\), the same terminal stage gives \(K(F(1))\), including its unit. The two incomparable middle values impose no extra choice of a colimit in \(\mathsf B\).

Exercise 3 (grade 3: an idempotent denominator). Let \(\mathsf C\) have one object \(c\) and endomorphisms \(1,e\), where \(e^2=e\), and set \(S=\{1,e\}\). A functor \(F:\mathsf C\to\mathsf{Set}\) is a set \(X\) with an idempotent map \(E\). Compute its universal right localization, including when \(E\) is not bijective.

Solution. All arrows commute and multiplication by \(e\) makes them equal to \(e\). In any required square choose both new arrows to be \(e\); in cancellation choose the further denominator \(e\). Thus the outgoing axioms hold. The localization is terminal, since an invertible idempotent equals the identity. The denominator category has objects \(1,e\), an arrow \(1\to e\), and the two endomorphisms of \(e\); its corresponding values are two copies of \(X\), with the nonidentity maps given by \(E\).

Put \(Z=\operatorname{im}E\). For every test set \(T\), the colimit of the stage sets of maps \(T\to X\) identifies a map \(f\) with \(Ef\). At the \(e\)-stage every map has the same class as \(Ef\); maps fixed by \(E\) are exactly the maps \(T\to Z\). Two such maps have equal classes only if they coincide, since all transition maps fix them. Hence the formal colimit is naturally \(h_Z\). Theorem 3.1 gives value \(Z\) and unit \(p:X\to Z\), \(p(x)=E(x)\). Although \(E\) need not be invertible, its formal localization is representable. For any further functor \(K\), the maps \(p\) and the inclusion \(i:Z\to X\) retain the identities \(pi=1\), \(ip=E\) after applying \(K\). The universal value is \(KZ\), consistently with the theorem.

Exercise 4 (grade 4: a colimit that fails universality). Take \(\mathsf C=\mathbb N\), invert every arrow, and let \(F(n)=\{0,\ldots,n\}\) with inclusion maps, valued in sets. Compute \(R_SF\). Show that it is not universally right localizable, both by its formal value and by an explicit further functor.

Solution. The localization is terminal as in Section 1. Each denominator category is a tail of \(\mathbb N\), so (2.2) gives the ordinary value \(\mathbb N\). The formal value \(A\), tested on a set \(T\), consists of maps \(T\to\mathbb N\) whose image is contained in one finite initial segment. These are exactly the maps with finite image. The natural map \(A\to h_{\mathbb N}\) is a bijection at a singleton, but misses the identity at the test set \(\mathbb N\). If \(A\) were representable by \(D\), this natural map would be induced by a map \(D\to\mathbb N\); its singleton component would make that map bijective, forcing all components to be bijective. This contradiction proves nonrepresentability.

More concretely, take \(K(T)=\operatorname{Hom}_{\mathsf{Set}}(\mathbb N,T)\). The right localization of \(KF\) exists, since sets have filtered colimits. Its value is the union of the sets of functions into the finite initial segments: precisely the functions \(\mathbb N\to\mathbb N\) with finite image. The comparison to \(K(R_SF)=\mathbb N^{\mathbb N}\) is the inclusion. The identity function is missing, so the comparison is not an isomorphism. This witnesses failure of universality without relying only on an abstract criterion.

References

• The Stacks Project, Localization in categories, especially the outgoing fraction construction, Remark 4.27.7 and Lemma 4.27.8. These results are used by reference; their text is not reproduced.
• Pierre Schapira, An Introduction to Categories and Homological Algebra, lecture notes, version of 1 March 2026, Section 2.7 and Chapter 3, for ind-objects and the localization of categories and functors.