Memo in a reduction strategy of a map

This is my random thought on how we should investigate the properties of a map in general.

The mapping path space construction provides a canonical homotopy-theoretic replacement of a map $f:A\to B$ by a fibration

$A \xrightarrow{\sim} E_f \to B,$

functorial in $f$ and canonical up to homotopy equivalence over $B$ . It may be viewed as a homotopy-theoretic “normal form” of $f$ , not in the strict algebraic sense of a matrix normal form, but as a distinguished fibration model of the same homotopy-theoretic information.

This construction is especially natural in $\mathbf{Top}$ because the category has a canonical interval object $I$ and well-behaved path objects $B^I$ , so homotopies into $B$ are encoded by maps into $B^I$ . This makes fibration replacement functorial and uniform.

More generally, a standard homotopy-theoretic strategy is to replace a map by a model carrying additional structure—typically a fibration or cofibration—so that exact sequences and other derived tools become available. In $\mathbf{Top}$ , the mapping path space gives the fiber/fibration side of this story, while the mapping cone gives the cofiber/cofibration (or pair/relative) side.

In a practical or application-oriented setting, the direction in which one “replacing” a map is rarely controllable a priori. Standard constructions—such as replacing a map $f:A \to B$ by a fibration via the mapping path space as described earlier—provide a canonical way to analyze $f$ up to weak homotopy equivalence. However, this form of replacement is inherently rigid: it forces the analysis into a framework governed by weak homotopy types of fibrant objects, which may be too coarse to retain finer structural or application-relevant features of the original map.

From this perspective, the issue is not merely the existence of a canonical replacement, but rather the lack of flexibility in what aspects of the map are preserved or emphasized by that replacement. Different contexts may require retaining different layers of information—geometric, combinatorial, or partially homotopical—none of which are universally captured by a single replacement procedure.

This motivates the following question: given a map $f$ , is there a systematic way to parameterize the direction of replacement, so that one can control which features are preserved and which are discarded? In other words, can one construct a family of “structured replacements” of $f$ , interpolating between different regimes of equivalence, rather than committing to a single, fixed notion such as weak homotopy equivalence?

In one direction, there is such example of parametrizing the weak equivalence (a class of morphisms) in a model category, called Bousfield localization in that as a standard example the (Quillen) rationalization replaces a space $X$ with the torsion free version $X_\mathbb{Q}$ , up on the choice of generalized homotopy theory $H_\ast(-;\mathbb{Q})$ .

I set a deep research with ChatGTP 5.4 for the latest threads of further extensions, variants or generalization of such parameterization. These details have been summarized in the table below for ease of reference.

Approach family	Primary parameter(s)	What the replacement tends to preserve	Typical contexts / applications	Key sources
Model-localization (Bousfield / reflective localization)	Set S of maps; theory E; locality notion (“S-local”)	Exactly the invariants detected by the localized regime (e.g. E-homology); often preserves certain limits/colimits depending on left/right exactness	Homological/chromatic regimes; stable $\infty$ -categories; forcing specific descent/local properties	Hirschhorn; Bousfield (spaces, spectra); Mantovani (stable $\infty$ -cats)
Map-class choice in Top (q/h/mixed)	“Good maps” (Serre vs Hurewicz), plus weak equivalences	Geometric lifting (Hurewicz) vs CW/weak equivalence sensitivity (q); some versions make all objects fibrant/cofibrant	Geometric topology vs homotopy theory; comparing strict vs weak regimes	Strøm; Cole; Ronan notes
AWFS / algebraic model structures	Generators; choice of AWFS (double categorical / finitary); level of algebraic structure	Canonical chosen fillers; coherent functorial factorizations; better interaction with computation/formalization	Constructive homotopy theory; semantics of type theory; structured fibrations	Garner; Riehl; Seip; van den Berg–Bourke–Seip; Cavallo–Sattler; Hilhorst et al.
Towers (Postnikov / Moore–Postnikov / Goodwillie)	Integer stage n; excisiveness degree n; convergence hypotheses	Partial information up to degree n (homotopy groups; obstruction layers); polynomial/excisive approximation	Obstruction theory; classification problems; functor calculus; “interpolation” between unstable and stable	May; Rezk (truncation/OFS); ABFJ (Goodwillie in ∞-topoi); recent Goodwillie work
$\infty$ -modalities / $\infty$ -topoi	Modality/localization functor; truncation level n; generating maps $\Sigma$	Left-exact structure (if lex); canonical factorization systems; internal language compatibility	Higher topos theory; descent; synthetic homotopy theory	HTT; Hinich; Rezk; ABFJ I/II; HoTT modalities