Tested Bousfield–Friedlander Localization

• The paper demonstrates that a tested Quillen idempotent monad Q localizes functor categories by replacing abstract pullback criteria with explicit cotensor tests.
• The methodology uses evaluated cotensors and a prescribed set of test morphisms to characterize weak equivalences and fibrations in model structures.
• The approach unifies discrete, Goodwillie, and Weiss calculus by identifying the tested localization with an ordinary left Bousfield localization at explicit sets of maps.

Searching arXiv for papers on tested Bousfield–Friedlander localization and closely related localization frameworks. Tested Bousfield–Friedlander localization is a form of Bousfield–Friedlander localization on a simplicial functor category in which the defining homotopy-pullback condition for fibrations is detected by a prescribed set of test morphisms. In the framework of a tested Quillen idempotent monad $Q$ on $\mathrm{Fun}(\mathcal C,\mathcal D)$, the localization is controlled by cotensor tests in $\mathrm{Fun}(\mathcal C,s\mathbf{Sets})$, and—under proper cellular or combinatorial hypotheses on $\mathcal D$—it is identified with an ordinary left Bousfield localization at an explicit set of maps. This recasting yields a homogeneous model structure for any calculus arising from such a localization, shows that homogeneous functors in discrete calculus coincide up to homotopy with those in Goodwillie calculus, and realizes the polynomial model structure of Weiss calculus as an instance of tested Bousfield–Friedlander localization (Taggart, 23 Jul 2025).

1. Ambient framework and Bousfield–Friedlander data

The starting point is a right proper model category $\mathcal M$ together with a homotopical endofunctor

$Q:\mathcal M\to \mathcal M$

that is a Quillen idempotent monad, also called a Bousfield endofunctor. The required data are a natural transformation

$\eta:\mathrm{Id}\to Q$

such that $Q$ preserves homotopy pullback squares and both induced maps

$\eta_Q,\;Q(\eta):Q\to Q^2$

are levelwise weak equivalences. Under these hypotheses one obtains a Bousfield–Friedlander localization $\mathcal M_Q$ whose weak equivalences are the maps $\mathrm{Fun}(\mathcal C,\mathcal D)$0 such that $\mathrm{Fun}(\mathcal C,\mathcal D)$1 is a weak equivalence in $\mathrm{Fun}(\mathcal C,\mathcal D)$2, whose cofibrations are unchanged, and whose fibrations are those underlying fibrations for which

$\mathrm{Fun}(\mathcal C,\mathcal D)$3

is a homotopy pullback (Taggart, 23 Jul 2025).

The tested theory specializes this construction to functor categories

$\mathrm{Fun}(\mathcal C,\mathcal D)$4

where $\mathrm{Fun}(\mathcal C,\mathcal D)$5 is an essentially small simplicial category, $\mathrm{Fun}(\mathcal C,\mathcal D)$6 is a simplicial right proper model category, and the ambient model structure is the projective one. In that underlying projective model structure, cofibrations are projective cofibrations, fibrations are projective fibrations, and weak equivalences are objectwise weak equivalences. The tested condition does not alter the Bousfield–Friedlander definition of weak equivalence; it replaces the abstract homotopy-pullback criterion for fibrations by an explicit family of cotensor tests (Taggart, 23 Jul 2025).

2. Test morphisms and the evaluated cotensor

A technical feature of the framework is that $\mathrm{Fun}(\mathcal C,\mathcal D)$7 need not be cotensored over $\mathrm{Fun}(\mathcal C,\mathcal D)$8, even when $\mathrm{Fun}(\mathcal C,\mathcal D)$9 is cotensored over simplicial sets. The remedy is the evaluated cotensor. For $\mathrm{Fun}(\mathcal C,s\mathbf{Sets})$0 and $\mathrm{Fun}(\mathcal C,s\mathbf{Sets})$1, it is defined by

$\mathrm{Fun}(\mathcal C,s\mathbf{Sets})$2

This construction is central because the tests live in simplicial-set-valued functors, while the localization itself lives in the $\mathrm{Fun}(\mathcal C,s\mathbf{Sets})$3-valued functor category (Taggart, 23 Jul 2025).

A Quillen idempotent monad $\mathrm{Fun}(\mathcal C,s\mathbf{Sets})$4 on $\mathrm{Fun}(\mathcal C,s\mathbf{Sets})$5 is tested when there exists a collection $\mathrm{Fun}(\mathcal C,s\mathbf{Sets})$6 of morphisms in $\mathrm{Fun}(\mathcal C,s\mathbf{Sets})$7 such that, for each fibration $\mathrm{Fun}(\mathcal C,s\mathbf{Sets})$8, the square

$\mathrm{Fun}(\mathcal C,s\mathbf{Sets})$9

is a homotopy pullback in $\mathcal D$0 if and only if, for every test morphism $\mathcal D$1 in $\mathcal D$2, the square

$\mathcal D$3

is a homotopy pullback in $\mathcal D$4 (Taggart, 23 Jul 2025).

This immediately yields the fibrancy criterion. Applying the condition to $\mathcal D$5, one finds that an object $\mathcal D$6 is fibrant in the Bousfield–Friedlander localization if and only if $\mathcal D$7 is underlying fibrant and

$\mathcal D$8

is a weak equivalence for every $\mathcal D$9. Thus the local objects are detected by a fixed set of probes rather than by a global homotopy-pullback condition (Taggart, 23 Jul 2025).

3. Characterization as a left Bousfield localization

The main structural theorem states that a tested Bousfield–Friedlander localization is an ordinary left Bousfield localization at an explicit set of maps. If $\mathcal M$0 is equipped with a proper cellular, or proper combinatorial, simplicial model structure, and $\mathcal M$1 is a set of cofibrant homotopy generators for $\mathcal M$2, then one defines

$\mathcal M$3

Here $\mathcal M$4 is the object of $\mathcal M$5 given objectwise by

$\mathcal M$6

The theorem identifies the Bousfield–Friedlander localization $\mathcal M$7 with the left Bousfield localization of the projective model structure at the set $\mathcal M$8 (Taggart, 23 Jul 2025).

The proof proceeds by comparing fibrant objects. In the left Bousfield localization at $\mathcal M$9, a fibrant object is an underlying fibrant functor $Q:\mathcal M\to \mathcal M$0 such that for every map

$Q:\mathcal M\to \mathcal M$1

the induced map

$Q:\mathcal M\to \mathcal M$2

is a weak equivalence. By adjunction, this is equivalent to

$Q:\mathcal M\to \mathcal M$3

being a weak equivalence for all $Q:\mathcal M\to \mathcal M$4, and since $Q:\mathcal M\to \mathcal M$5 is a set of cofibrant homotopy generators, this is equivalent to

$Q:\mathcal M\to \mathcal M$6

being a weak equivalence for all $Q:\mathcal M\to \mathcal M$7. That is exactly the tested fibrancy condition for the Bousfield–Friedlander localization, so the two model structures have the same fibrant objects; since they also keep the same cofibrations, they are the same model structure (Taggart, 23 Jul 2025).

A direct corollary is that, under these hypotheses, the tested Bousfield–Friedlander localization is proper cellular, or proper combinatorial, simplicial, and in particular cofibrantly generated. The theorem is not an assertion about every Bousfield–Friedlander localization; it applies to those equipped with a suitable set of test morphisms (Taggart, 23 Jul 2025).

4. Homogeneous model structures

The second major application of the tested framework is the construction of homogeneous model structures for functor calculus. Suppose that for each $Q:\mathcal M\to \mathcal M$8 there is a degree $Q:\mathcal M\to \mathcal M$9 approximation functor $\eta:\mathrm{Id}\to Q$0, and that the degree $\eta:\mathrm{Id}\to Q$1 model structure on $\eta:\mathrm{Id}\to Q$2 is a tested Bousfield–Friedlander localization. An object $\eta:\mathrm{Id}\to Q$3 is degree $\eta:\mathrm{Id}\to Q$4 if $\eta:\mathrm{Id}\to Q$5 is an equivalence, $\eta:\mathrm{Id}\to Q$6-reduced if $\eta:\mathrm{Id}\to Q$7 is trivial, and $\eta:\mathrm{Id}\to Q$8-homogeneous if it is both degree $\eta:\mathrm{Id}\to Q$9 and $Q$0-reduced (Taggart, 23 Jul 2025).

Under these hypotheses there is a model structure in which the cofibrant objects are the underlying cofibrant $Q$1-reduced functors and the fibrations are the fibrations of the degree $Q$2 model structure. Its bifibrant objects are the bifibrant $Q$3-homogeneous functors. This is the $Q$4-homogeneous model structure (Taggart, 23 Jul 2025).

The construction uses the tested description in an essential way. Since the degree $Q$5 model structure is tested, it is a left Bousfield localization at

$Q$6

One then forms the set of homotopy cofibres

$Q$7

and

$Q$8

The $Q$9-homogeneous model structure is the right Bousfield localization of the degree $\eta_Q,\;Q(\eta):Q\to Q^2$0 model structure at $\eta_Q,\;Q(\eta):Q\to Q^2$1 (Taggart, 23 Jul 2025).

A key mechanism is the mapping-space fibre sequence

$\eta_Q,\;Q(\eta):Q\to Q^2$2

for $\eta_Q,\;Q(\eta):Q\to Q^2$3. If $\eta_Q,\;Q(\eta):Q\to Q^2$4 is degree $\eta_Q,\;Q(\eta):Q\to Q^2$5, then the rightmost map is a weak equivalence by the test condition, so the left term is contractible. Consequently degree $\eta_Q,\;Q(\eta):Q\to Q^2$6 functors become trivial in the $\eta_Q,\;Q(\eta):Q\to Q^2$7-homogeneous model structure. This isolates the $\eta_Q,\;Q(\eta):Q\to Q^2$8-homogeneous layer by right localization at homotopy cofibres of the degree $\eta_Q,\;Q(\eta):Q\to Q^2$9 tests (Taggart, 23 Jul 2025).

5. Discrete calculus, Goodwillie calculus, and Weiss calculus

The tested formalism is developed far enough to compare distinct calculi. Under the hypotheses guaranteeing tested degree $\mathcal M_Q$0 structures for both discrete calculus and Goodwillie calculus, the $\mathcal M_Q$1-homogeneous model structure for the discrete calculus is identical to the $\mathcal M_Q$2-homogeneous model structure for Goodwillie calculus. Equivalently, up to homotopy, a functor is $\mathcal M_Q$3-homogeneous in discrete calculus if and only if it is $\mathcal M_Q$4-homogeneous in Goodwillie calculus (Taggart, 23 Jul 2025).

For the degree $\mathcal M_Q$5 structure in discrete calculus, the test morphisms are

$\mathcal M_Q$6

where $\mathcal M_Q$7 denotes the representable functor on $\mathcal M_Q$8. Their homotopy cofibres are

$\mathcal M_Q$9

Right localization at this set produces the $\mathrm{Fun}(\mathcal C,\mathcal D)$00-homogeneous model structure for discrete calculus, and by the cited comparison with Biedermann–Röndigs, also the one for Goodwillie calculus (Taggart, 23 Jul 2025).

Weiss calculus fits the same pattern. Let $\mathrm{Fun}(\mathcal C,\mathcal D)$01 be the simplicial category of finite-dimensional inner product spaces over $\mathrm{Fun}(\mathcal C,\mathcal D)$02. A simplicial functor

$\mathrm{Fun}(\mathcal C,\mathcal D)$03

is $\mathrm{Fun}(\mathcal C,\mathcal D)$04-polynomial if the canonical map

$\mathrm{Fun}(\mathcal C,\mathcal D)$05

is a weak equivalence for all $\mathrm{Fun}(\mathcal C,\mathcal D)$06, and the universal $\mathrm{Fun}(\mathcal C,\mathcal D)$07-polynomial approximation is

$\mathrm{Fun}(\mathcal C,\mathcal D)$08

Using the evaluated cotensor form of Yoneda, the corresponding test morphisms are

$\mathrm{Fun}(\mathcal C,\mathcal D)$09

The $\mathrm{Fun}(\mathcal C,\mathcal D)$10-polynomial model structure is therefore a tested Bousfield–Friedlander localization with this explicit set of test morphisms, and in particular it is proper and cellular (Taggart, 23 Jul 2025).

6. Conceptual scope and related localization frameworks

Tested Bousfield–Friedlander localization does not replace the original Bousfield–Friedlander construction; it identifies a class of Bousfield–Friedlander localizations for which the defining homotopy-pullback condition is equivalent to locality with respect to a set of maps. The result is specific in two directions. First, the construction is formulated for the projective model structure on $\mathrm{Fun}(\mathcal C,\mathcal D)$11, though the proof makes clear that it extends to any proper cellular or combinatorial model structure on the functor category. Second, the identification with a left Bousfield localization requires additional hypotheses on $\mathrm{Fun}(\mathcal C,\mathcal D)$12: proper cellular or proper combinatorial simplicial structure, so that Hirschhorn’s existence theorem applies (Taggart, 23 Jul 2025).

A nearby generalization works in semimodel categories rather than full model categories. In that setting a BF-reflector is a homotopical endofunctor $\mathrm{Fun}(\mathcal C,\mathcal D)$13 with a natural transformation $\mathrm{Fun}(\mathcal C,\mathcal D)$14 such that $\mathrm{Fun}(\mathcal C,\mathcal D)$15 and $\mathrm{Fun}(\mathcal C,\mathcal D)$16 are natural equivalences, and localized equivalences can be tested by derived mapping objects against local objects; the resulting theory extends both the Bousfield–Friedlander theorem and Hirschhorn localization to semimodel settings (Carmona, 2022). This is adjacent to the tested theory rather than identical with it: the semimodel paper does not use a set of test morphisms in the sense of $\mathrm{Fun}(\mathcal C,\mathcal D)$17, but it shares the same emphasis on reducing localization to explicit test conditions.

Another nearby line studies left Bousfield localization through mapping-space tests against a set $\mathrm{Fun}(\mathcal C,\mathcal D)$18 of morphisms and then asks when the resulting localization is monoidal or preserves operadic algebra. In that framework an object $\mathrm{Fun}(\mathcal C,\mathcal D)$19 is $\mathrm{Fun}(\mathcal C,\mathcal D)$20-local when

$\mathrm{Fun}(\mathcal C,\mathcal D)$21

is a weak equivalence for every $\mathrm{Fun}(\mathcal C,\mathcal D)$22 in $\mathrm{Fun}(\mathcal C,\mathcal D)$23, and a monoidal Bousfield localization is characterized by the condition that every map $\mathrm{Fun}(\mathcal C,\mathcal D)$24, with $\mathrm{Fun}(\mathcal C,\mathcal D)$25 and $\mathrm{Fun}(\mathcal C,\mathcal D)$26 cofibrant, is a $\mathrm{Fun}(\mathcal C,\mathcal D)$27-local equivalence (White, 2014). This is not tested Bousfield–Friedlander localization, but it exhibits the same general model-categorical pattern: localization data become manageable when they are controlled by an explicit family of tests.

In that sense, tested Bousfield–Friedlander localization occupies a precise intermediate position. It remains genuinely Bousfield–Friedlander in its definition of weak equivalences and fibrations, but it is “tested” because its fibrancy condition can be expressed by a fixed set of cotensor probes, and this tested form is strong enough to identify the localization with a left Bousfield localization at an explicit set of maps.