Dwyer–Kan Localisation in Homotopy Theory

• Dwyer–Kan localisation is a refinement of ordinary localization, retaining higher mapping-space data to encode complete homotopical structures.
• It employs simplicial and hammock localization methods to compute homotopy function complexes and derived mapping spaces effectively.
• DK equivalences and tailored model structures ensure the preservation of weak equivalences on mapping objects and essential surjectivity on homotopy categories.

Dwyer–Kan localisation is the simplicial or $\infty$-categorical refinement of ordinary localization of a category with weak equivalences: instead of retaining only the homotopy category $C[W^{-1}]$, it retains higher mapping-space data and thereby encodes a homotopy theory as a simplicially enriched category or as an $\infty$-category. In the modern formulation used for model categories, one writes $N(M)[W^{-1}]$ for the Dwyer–Kan localization of a model category $M$ at its weak equivalences $W$; its homotopy category is the ordinary homotopy category $\mathsf{Ho}(M)$, and it is modeled by the hammock localization after passage to the homotopy coherent nerve (Péroux, 2020). In Hinich’s $\infty$-categorical formulation, for a marked $\infty$-category $(C,W)$, the localization $C[W^{-1}]$0 is characterized by the universal property

$C[W^{-1}]$1

where $C[W^{-1}]$2 is the maximal Kan subcomplex of $C[W^{-1}]$3 (Hinich, 2013).

1. Classical simplicial localization and its purpose

The starting point of the classical theory is a category $C[W^{-1}]$4 together with a subcategory $C[W^{-1}]$5 of weak equivalences. Ordinary Gabriel–Zisman localization $C[W^{-1}]$6 only remembers the homotopy category, whereas Dwyer–Kan localization is designed to retain the simplicial hom-spaces that record higher homotopies (Low, 2014).

A standard construction recalled in the literature is the simplicial localization $C[W^{-1}]$7. If $C[W^{-1}]$8 denotes the standard resolution of a category $C[W^{-1}]$9, then the simplicial localization of $\infty$0 is the simplicially enriched category corresponding to

$\infty$1

obtained by inverting $\infty$2 in $\infty$3 levelwise (Low, 2014). Its objects are those of $\infty$4, and for each pair $\infty$5 it provides a simplicial set of morphisms refining the hom-set in $\infty$6 (Low, 2014).

The hammock localization $\infty$7 is the other classical model. It is built from reduced hammocks, and for model categories it supplies mapping spaces equivalent to homotopy function complexes. Hinich shows that for a fibrant simplicial category $\infty$8 and a fibrant simplicial subcategory $\infty$9 with the same objects, the map $N(M)[W^{-1}]$0 induces a weak equivalence of marked simplicial sets

$N(M)[W^{-1}]$1

so the homotopy coherent nerve of the hammock localization presents the $N(M)[W^{-1}]$2-categorical localization (Hinich, 2013).

This simplicial enrichment is the essential distinction between Dwyer–Kan localization and ordinary localization. A plausible implication is that any account of “Dwyer–Kan localisation” restricted to $N(M)[W^{-1}]$3-level data is describing only a shadow of the full construction.

2. Homotopy function complexes and derived mapping spaces

One of the central classical results is that Dwyer–Kan localization computes the correct homotopy function complexes. Low reformulates the Dwyer–Kan homotopy function complexes as total right derived functors and proves that they functorially compute the homotopy type of the hom-spaces in simplicial localization (Low, 2014).

For a model category $N(M)[W^{-1}]$4, if $N(M)[W^{-1}]$5 is a cosimplicial resolution of $N(M)[W^{-1}]$6 and $N(M)[W^{-1}]$7 is a simplicial resolution of $N(M)[W^{-1}]$8, then the total hom-complex

$N(M)[W^{-1}]$9

models the homotopy function complex from $M$0 to $M$1 (Low, 2014). The derived bifunctor $M$2 is then identified with the hammock-localization hom-space by the theorem

$M$3

as functors

$M$4

(Low, 2014).

The same paper gives a presheaf-theoretic computation. For a category with weak equivalences $M$5, one localizes the projective model structure on simplicial presheaves $M$6 at the Yoneda images of weak equivalences,

$M$7

and fibrant replacement $M$8 in that left Bousfield localization satisfies

$M$9

(Low, 2014). This identifies simplicial-localization hom-spaces with derived representables.

For model categories, the same perspective is reiterated in later work: $W$0 is “what one should call the Dwyer–Kan localization because it is modeled by the hammock localization $W$1 after passing through the homotopy coherent nerve” (Péroux, 2020). This places classical function complexes, hammock localization, and $W$2-categorical localization in a single framework.

3. Universal properties and the $W$3-categorical formulation

Hinich’s reformulation places Dwyer–Kan localization directly inside $W$4. For a saturated marking $W$5, localization is expressed as the pushout

$W$6

and is characterized by the mapping-space formula

$W$7

(Hinich, 2013). This states that maps out of the localization are exactly maps out of $W$8 that send the marked arrows to equivalences.

The same universal-property language appears in the quasi-categorical treatment of model categories. For an $W$9-category $\mathsf{Ho}(M)$0 and a class of morphisms $\mathsf{Ho}(M)$1, the localization $\mathsf{Ho}(M)$2 is equipped with a functor $\mathsf{Ho}(M)$3 such that for every $\mathsf{Ho}(M)$4-category $\mathsf{Ho}(M)$5,

$\mathsf{Ho}(M)$6

where the right-hand side is the full subcategory of functors sending $\mathsf{Ho}(M)$7 to equivalences (Péroux, 2020). In the model-categorical case this is realized as $\mathsf{Ho}(M)$8 (Péroux, 2020).

An important family-level extension is that localization behaves well with cocartesian fibrations. If

$\mathsf{Ho}(M)$9

is a marked cocartesian fibration in the sense of Hinich, then

$\infty$0

is again a cocartesian fibration, and the induced map from a fiber $\infty$1 to the homotopy fiber of $\infty$2 is an $\infty$3-localization (Hinich, 2013). This suggests that DK localization is not only an objectwise inversion procedure but also a parameterized one.

The operator-algebraic theorem “Bicategories of $\infty$4-correspondences as Dwyer–Kan localisations” formulates the same principle in a bicategorical/quasi-categorical setting. The canonical map

$\infty$5

exhibits the bicategory of proper correspondences as the Dwyer–Kan localisation of $\infty$6-algebras at corner embeddings, in the sense that for every quasi-category $\infty$7,

$\infty$8

is a homotopy equivalence (Meyer, 29 Aug 2025). The paper emphasizes that this localization is already $\infty$9-truncated: the target happens to be represented by a bicategory with only invertible $\infty$0-morphisms (Meyer, 29 Aug 2025).

4. Dwyer–Kan equivalences and model structures

In practice, Dwyer–Kan localization is often accessed through model structures whose weak equivalences are DK-equivalences. The common pattern is “local weak equivalence plus essential surjectivity on $\infty$1.”

For $\infty$2-enriched categories, Muro defines a $\infty$3-functor $\infty$4 to be homotopically fully faithful if

$\infty$5

is a weak equivalence in $\infty$6 for all objects $\infty$7, and homotopically essentially surjective if

$\infty$8

is essentially surjective. A Dwyer–Kan equivalence is one satisfying both conditions (Muro, 2012). Under the hypotheses that $\infty$9 is combinatorial, closed symmetric monoidal, and satisfies the monoid axiom, $(C,W)$0 admits the Dwyer–Kan model structure (Muro, 2012).

The same template appears for topological categories. A $(C,W)$1-enriched functor $(C,W)$2 is a Dwyer–Kan equivalence if

$(C,W)$3

is a weak equivalence for every pair of objects, and

$(C,W)$4

is an equivalence of ordinary categories (Körschgen, 2017). Gepner–Henriques’ result, elaborated in that paper, is that such an $(C,W)$5 induces a Quillen equivalence on topologically enriched presheaf categories (Körschgen, 2017).

For more elaborate algebraic gadgets, the Dwyer–Kan recipe is transferred through an “underlying categorical part.” In algebras over augmented operadic collections, a map $(C,W)$6 is a weak equivalence iff it is entrywise a weak equivalence in $(C,W)$7 and

$(C,W)$8

is essentially surjective (Yau, 2016). The resulting Dwyer–Kan model structures apply to enriched wheeled props, wheeled properads, and wheeled operads (Yau, 2016). For colored cyclic operads enriched in simplicial sets, the paper on cyclic operads defines a Dwyer–Kan equivalence by local weak equivalence together with equivalence of the induced categories $(C,W)$9, and proves a positive Dwyer–Kan type model structure (Drummond-Cole et al., 2018). For enriched colored PROPs, Yalin constructs a Dwyer–Kan model structure whose weak equivalences are local weak equivalences together with essential surjectivity on the underlying enriched category $C[W^{-1}]$00 (Caviglia, 2015).

A plausible implication is that, across enriched categories, operads, cyclic operads, and PROPs, “Dwyer–Kan localisation” is operationally realized by model structures whose weak equivalences are calibrated to preserve both mapping objects and object/color data up to homotopy.

5. Model categories, monoidal refinements, and concrete applications

For an ordinary model category $C[W^{-1}]$01, Dwyer–Kan localization is written as

$C[W^{-1}]$02

and one has

$C[W^{-1}]$03

assuming functorial fibrant and cofibrant replacement (Péroux, 2020). This is the basic passage from model-categorical presentations to underlying $C[W^{-1}]$04-categories.

The monoidal refinement is one of the most technically significant modern developments. For a symmetric monoidal model category $C[W^{-1}]$05 with cofibrant unit, there is a symmetric monoidal Dwyer–Kan localization

$C[W^{-1}]$06

characterized by the universal property

$C[W^{-1}]$07

(Péroux, 2020). The paper “Coalgebras in the Dwyer–Kan localization of a model category” proves that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $C[W^{-1}]$08-categories after DK localization, and uses this to derive equivalences of $C[W^{-1}]$09-categories of $C[W^{-1}]$10-coalgebras (Péroux, 2020). One main application is the derived Dold–Kan correspondence: $C[W^{-1}]$11 as symmetric monoidal $C[W^{-1}]$12-categories (Péroux, 2020).

A second application is the operadic Dold–Kan correspondence for enriched operads. The normalization functor induces an equivalence between the relative categories of simplicial $C[W^{-1}]$13-linear operads and connective dg $C[W^{-1}]$14-linear operads with varying color sets, after inverting Dwyer–Kan equivalences: $C[W^{-1}]$15 (Truong, 2023). When the DK model structure exists on $C[W^{-1}]$16, this upgrades to a Quillen equivalence (Truong, 2023).

A third application is derived localisation of dg algebras. Braun–Chuang–Lazarev compare derived localization of the chain dg algebra $C[W^{-1}]$17 of a simplicial monoid $C[W^{-1}]$18 with Dwyer–Kan localization of $C[W^{-1}]$19, proving

$C[W^{-1}]$20

and using Dwyer–Kan’s theorem

$C[W^{-1}]$21

to obtain

$C[W^{-1}]$22

(Braun et al., 2015). This is a precise bridge between simplicial localization and derived algebraic localization.

More recent work uses DK localization to compare genuinely different algebraic models. The homotopical algebra of Lie–Rinehart pairs proves that the Dwyer–Kan localization of suitably cofibrant dg Lie–Rinehart pairs at pairs of quasi-isomorphisms is equivalent to the corresponding localization of strong homotopy Lie–Rinehart pairs, first with linear SH morphisms and then with general SH morphisms (Pištalo, 6 Jan 2026). This shows that DK localization can function as the comparison mechanism between strict and strong-homotopy presentations even when a full model structure is not the primary tool (Pištalo, 6 Jan 2026).

6. Scope, related constructions, and common confusions

A recurring source of confusion is that “Dwyer–Kan” names several related but distinct notions: simplicial localization, hammock localization, homotopy function complexes, DK-equivalences, DK model structures, and Dwyer–Kan–Smith cohomology. The literature in the data block makes these distinctions explicit.

Some papers are directly about localization, while others are only adjacent. “Comonad Cohomology of Track Categories” does not construct simplicial localizations $C[W^{-1}]$23 or hammock localizations directly, but works inside the homotopy theory of simplicial categories via Dwyer–Kan–Smith cohomology (Blanc et al., 2018). “Comparing cohomology obstructions” is likewise about $C[W^{-1}]$24-cohomology, Dwyer–Kan–Smith rectification, and Dwyer–Kan–Stover realization, not about localization as such (Baues et al., 2010). “A variant of a Dwyer–Kan theorem for model categories” reformulates a classical Dwyer–Kan theorem about categories of homotopy functors in model-categorical terms, using $C[W^{-1}]$25-equivalences on bifibrant objects as the analogue of equivalence after simplicial localization (Chorny et al., 2018).

Another confusion concerns 1-categorical substitutes. Thomas’s “On the 3-arrow calculus for homotopy categories” develops a localization theory yielding a 3-arrow calculus for ordinary localizations and homotopy categories, but it is explicitly “not a paper on Dwyer–Kan simplicial or hammock localisation in the strict sense”; it captures only the ordinary localization category, not higher mapping-space data (Thomas, 2010). This suggests a useful contrast: DK localization addresses the higher homotopical content that 1-categorical calculi intentionally omit.

Finally, not every paper carrying the name “Dwyer–Kan” concerns localization. “The Dold–Kan theorem for paracyclic modules” studies the Dwyer–Kan operator in cyclic/paracyclic homological algebra and explicitly states that it “does not discuss simplicial localization, hammock localization, simplicial categories, derived mapping spaces, or Dwyer–Kan equivalences of simplicial categories/model categories” (Getzler, 8 Feb 2026). For the topic of Dwyer–Kan localisation, that paper is therefore a namesake rather than a direct source.

Taken together, the cited literature presents Dwyer–Kan localisation as a unifying mechanism for encoding homotopy theories beyond their homotopy categories: classically via simplicial and hammock localization, abstractly via universal properties in $C[W^{-1}]$26, and practically via model structures whose weak equivalences are DK-equivalences. Across enriched categories, operads, PROPs, coalgebras, $C[W^{-1}]$27-correspondences, and Lie–Rinehart pairs, the recurring pattern is the same: local weak equivalence on mapping objects or operation objects, combined with essential surjectivity on the homotopy category, is the datum that the localization inverts (Hinich, 2013).