Cofibration

In algebraic topology, a cofibration is a continuous map i: A \to X between topological spaces that satisfies the homotopy extension property (HEP): for any space Y, any continuous map g: X \to Y, and any homotopy H: A \times I \to Y (where I = [0,1]) such that H(a, 0) = g(i(a)) for all a \in A, there exists a homotopy \tilde{H}: X \times I \to Y such that \tilde{H}|_{A \times I} = H and \tilde{H}(x, 0) = g(x) for all x \in X.^[1] This property ensures that homotopies defined on the domain A can be extended over the codomain X while preserving initial conditions. Cofibrations are dual to fibrations and form a fundamental class of morphisms in homotopy theory.^[1] Cofibrations play a central role in model categories and homotopical algebra to facilitate the study of homotopy quotients and cofiber sequences, where the cofiber C_i = X \cup_{i} CA (the mapping cone on i) plays a central role in long exact sequences of homotopy groups.^[1] A key characterization is that a map is a cofibration if and only if it is a retract of the inclusion of a mapping cylinder, and pushouts preserve cofibrations, allowing constructions like cell attachments in CW complexes.^[1] For example, inclusions of closed subspaces often qualify as cofibrations when the pair (X, A) admits a neighborhood deformation retract, enabling homotopy equivalences between X/A and the cone Ci.^[1] In cohomology theories, cofibrations induce isomorphisms \tilde{E}^*(X/A) \cong E^*(X, A) for reduced cohomology E^*, which underpins exact sequences and cellular approximations in algebraic topology.^[1] They are essential in the study of Hopf invariants for maps between spheres, linking to deeper results like the Hopf invariant one problem resolved via K-theory.^[1] Overall, cofibrations provide a rigorous framework for handling quotients and extensions in homotopy, influencing modern areas such as stable homotopy theory and spectra.^[1]

Definitions

In model categories

A model category is a category equipped with three distinguished classes of morphisms: cofibrations, fibrations, and weak equivalences.^[2]^[3] These classes satisfy a set of axioms that enable the category to model homotopy theory in an abstract setting, allowing for the construction of derived functors and homotopy limits and colimits.^[2] The cofibrations form one of the key components, providing a notion of "good" inclusions that facilitate approximations and resolutions.^[3] In a model category, cofibrations are the morphisms that have the left lifting property with respect to acyclic fibrations. The class of cofibrations is closed under pushouts, transfinite compositions, and retracts by the model category axioms.^[3] This axiomatic characterization ensures that cofibrations behave well under the operations needed for homotopical constructions, such as building cell complexes or resolving objects.^[2] Specifically, the closure properties allow cofibrations to be generated from a small set of morphisms via iterative processes, while the lifting property guarantees compatibility with the homotopy structure.^[3] The small object argument provides a mechanism for generating the class of cofibrations from a set of cofibrant generators, typically denoted I, through a transfinite process of pushouts and compositions.^[3] In cofibrantly generated model categories, every cofibration is a retract of a relative I-cell complex, ensuring that the class is accessible and manageable for computational purposes.^[3] This generation technique is fundamental for establishing functorial factorizations in the model structure.^[2] Formally, the left lifting property for a cofibration i: A \to B with respect to an acyclic fibration p: E \to F states that for every commutative diagram

\begin{CD} A @>>> E \\ @ViVV @VVpV \\ B @>>> F \end{CD}

there exists a lift B \to E making both triangles commute.^[2]^[3] This property is one of the core axioms (M1 in Quillen's original formulation) and underpins the weak factorization system formed by cofibrations and acyclic fibrations.^[2] Cofibrations enable resolutions and approximations in homotopical algebra by allowing objects to be replaced with cofibrant ones via cofibrant resolutions, which preserve the homotopy type while facilitating computations in the homotopy category.^[3] This is particularly useful for deriving functors like homotopy groups or Ext groups in algebraic contexts.^[2] By duality, fibrations form the right orthogonal class to acyclic cofibrations.^[3]

In homotopy theory

In homotopy theory, particularly in the study of topological spaces and simplicial sets, a cofibration is a morphism A \hookrightarrow X characterized by the homotopy extension property (HEP). This property states that for any space Y, any map f: X \to Y, and any homotopy H: A \times I \to Y such that H_0 = f|_A, there exists a homotopy \tilde{H}: X \times I \to Y extending H (i.e., \tilde{H}|_{A \times I} = H) and satisfying \tilde{H}_0 = f.^[4] The HEP ensures that inclusions of subspaces behave well under homotopy, allowing extensions of homotopies from the subspace to the whole space without altering the initial map.^[4] This definition aligns with the structure of model categories tailored to homotopy theory. In the Quillen model structure on the category of topological spaces, cofibrations are precisely the maps with the left lifting property (LLP) with respect to acyclic fibrations (trivial fibrations), where weak equivalences are weak homotopy equivalences and fibrations are Serre fibrations.^[5] Similarly, in the Serre model structure on simplicial sets, cofibrations are the monomorphisms, which satisfy the LLP against acyclic Kan fibrations, with weak equivalences again being weak homotopy equivalences.^[6] These model structures formalize homotopy theory by designating cofibrations as the class of maps that "build" spaces in a controlled manner, dual to fibrations which handle path liftings.^[5] The concept of cofibrations plays a central role in constructing homotopy types through cell complexes, as discussed by Allen Hatcher in the context of CW-complexes. Here, cofibrations arise as relative cell attachments, where attaching an n-cell via a map from the boundary sphere S^{n-1} to the current space yields a cofibration, preserving the HEP for successive stages of construction.^[4] Such inclusions capture "good" subspaces that do not disrupt homotopy computations, enabling the inductive assembly of arbitrary homotopy types from elementary building blocks like disks and spheres while maintaining essential homotopy invariants.^[4]

Examples

Topological spaces

In the category of topological spaces, a fundamental characterization of cofibrations relies on the homotopy extension property (HEP), which requires that for any map f: X \to Y and homotopy H: A \times I \to Y extending f, there exists an extension to a homotopy on X \times I. A classic example is the inclusion of a closed subset A \hookrightarrow X, where A is closed in X. Such inclusions satisfy the HEP with respect to all topological spaces, rendering them Hurewicz cofibrations in the standard model structure.^[5] This property holds because homotopies defined on the closed subset can be extended across the entire space using the closedness to control neighborhoods and avoid boundary issues during extension.^[7] Cofibrations also arise prominently in the construction of CW-complexes through cell attachments. The boundary inclusion S^{n-1} \hookrightarrow D^n is a cofibration, as the sphere is closed in the disk, satisfying the HEP via radial extension of homotopies. More generally, when attaching an n-cell to a space via a map \phi: S^{n-1} \to X, the resulting inclusion of the mapping cylinder of \phi into the attached space preserves the cofibration property, ensuring skeletal inclusions in CW-complexes are cofibrations.^[8] This inductive construction allows homotopies on lower skeletons to extend cell by cell without obstruction. Specific verifications highlight these properties in simple cases. The inclusion of a single point into a Hausdorff space is a cofibration, since the singleton is closed and the HEP follows from the ability to extend constant homotopies trivially across the point while adjusting the rest of the space.^[5] Similarly, for a locally compact Hausdorff subspace A, its inclusion into the one-point compactification \alpha A is a cofibration; although the image of A is not closed, the local compactness ensures homotopies extend by compactifying supports near infinity, satisfying the lifting properties against acyclic fibrations. In contrast, open inclusions generally fail to be cofibrations, as they violate the left lifting property (LLP) with respect to certain acyclic fibrations; for instance, the inclusion (0,1) \hookrightarrow \mathbb{R} does not allow extension of homotopies relative to endpoints, leading to obstructions in the mapping cylinder construction. These examples are particularly relevant in Quillen's model structure on the category of compactly generated topological spaces, where cofibrations are defined as maps satisfying the LLP with respect to acyclic Serre fibrations and coincide with closed inclusions possessing the HEP. This structure ensures that cofibrations in compactly generated spaces behave well under colimits and provide a foundation for homotopy-theoretic computations in topology.^[5]

Simplicial sets

In the Kan-Quillen model structure on the category of simplicial sets, the cofibrations are precisely the monomorphisms, which are the maps that are injective on simplices in each dimension.^[9] These form a proper class of maps closed under pushouts, transfinite compositions, and retracts, and every simplicial set is cofibrant with respect to this structure.^[9] The class of cofibrations is generated by the boundary inclusions \partial \Delta^n \hookrightarrow \Delta^n for all n \geq 0, where \Delta^n denotes the standard n-simplex and \partial \Delta^n its boundary.^[9] A monomorphism f: A \to B is a cofibration if and only if it has the left lifting property (LLP) with respect to all acyclic Kan fibrations, which are the Kan fibrations that are also weak homotopy equivalences.^[9] The boundary inclusions \partial \Delta^n \hookrightarrow \Delta^n serve as basic cofibrations; for example, the map \partial \Delta^2 \hookrightarrow \Delta^2 injects the three edges of a 2-simplex into the full filled triangle, enabling the construction of more complex simplicial sets via pushouts and compositions.^[9] This combinatorial generation allows cofibrations to capture discrete analogues of cellular inclusions in topology. Cofibrations in simplicial sets facilitate skeletal filtrations: for any simplicial set X, the n-skeleton \mathrm{sk}_n X consists of all nondegenerate simplices of dimension at most n together with their degeneracies, and the natural inclusion \mathrm{sk}_{n-1} X \hookrightarrow \mathrm{sk}_n X is a cofibration.^[10] Under geometric realization |\cdot|: \mathrm{sSet} \to \mathrm{Top}, which sends simplicial sets to CW-complexes, these skeletal inclusions map to cofibrations in the category of topological spaces, preserving the cellular structure and establishing a Quillen equivalence between the two model categories.^[9] Cofibrations play a key role in computing homotopy groups through the singular simplicial set functor \mathrm{Sing}: \mathrm{Top} \to \mathrm{sSet}, which assigns to a topological space Y the Kan complex \mathrm{Sing} Y_n = \mathrm{Top}(\Delta^n, Y) and induces isomorphisms on homotopy groups \pi_k(\mathrm{Sing} Y) \cong \pi_k(Y) for all k.^[11] This allows homotopy groups of spaces to be computed combinatorially via the Kan-Quillen structure on \mathrm{Sing} Y, where cofibrations ensure proper control over extensions and liftings in the fibrant replacement.^[11]

Chain complexes

In the category of chain complexes of modules over a ring R, the projective model structure defines cofibrations as degreewise split monomorphisms whose cokernels are degreewise projective modules.^[12] This structure equips the category \mathrm{Ch}(R) with weak equivalences given by quasi-isomorphisms, enabling the homotopy category to model the derived category of R-modules.^[12] A representative example of a cofibration is the inclusion of the zero complex into a chain complex consisting of a projective module P concentrated in degree n, where the map is the zero map in all degrees except degree n, which is the zero map to P; this is a degreewise monomorphism (trivially split) with cokernel the projective complex. More generally, inclusions of subcomplexes into projective complexes where the degreewise cokernels are projective yield cofibrations.^[12] In this model structure, cofibrations satisfy the left lifting property with respect to acyclic fibrations, which are quasi-isomorphisms possessing the right lifting property against cofibrations.^[12] This lifting ensures compatibility with the axioms of a model category, facilitating resolutions and homotopy limits. Cofibrant objects in \mathrm{Ch}(R) are precisely the chain complexes that are degreewise projective, meaning each component module is projective; thus, projective resolutions serve as cofibrant replacements for arbitrary complexes.^[12] In applications to homological algebra, this identifies cofibrant chain complexes with projective resolutions, allowing derived functors such as \mathrm{Ext} and \mathrm{Tor} to be computed via these resolutions. For unbounded chain complexes, cofibrations play a key role in ensuring exactness when applying derived functors, as the projective model structure on \mathrm{Ch}(R) admits cofibrant replacements that preserve homology and enable precise computation of higher derived objects.^[12]

Properties

Lifting properties

In model categories, a morphism i: A \to B is said to have the strict left lifting property (LLP) with respect to a morphism p: X \to Y if, for every commutative square

\begin{CD} A @>>> X \\ @ViVV @VVpV \\ B @>>> Y \end{CD}

formed by morphisms f: A \to X and g: B \to Y such that p \circ f = g \circ i, there exists a lift h: B \to X making both triangles commute, i.e., h \circ i = f and p \circ h = g.^[13]^[14] This property ensures the existence of strict (non-homotopy) lifts and is fundamental to the definition of cofibrations, which are precisely the morphisms satisfying the LLP against all acyclic fibrations.^[13] The strict LLP extends to the homotopy left lifting property (HLLP) in enriched model categories, particularly simplicial or topological ones, where lifts are required only up to homotopy. Specifically, for a cofibration i: A \to B and fibration p: X \to Y, the HLLP demands that in any commutative square as above, there exists a lift h: B \to X up to homotopy (e.g., via simplicial homotopies or path objects), such that the triangles commute up to homotopy equivalence.^[14] This homotopy-coherent version accommodates the enriched structure, allowing for flexible lifts in homotopy-theoretic contexts while preserving the essential orthogonality of cofibrations to fibrations.^[14]^[13] In the acyclic case, a cofibration i has the LLP with respect to all trivial (acyclic) fibrations, which implies that i preserves weak equivalences in pushouts and other colimit constructions, ensuring homotopy invariance.^[13] A key characterization theorem states that in a model category, the class of cofibrations is exactly the class of morphisms with the LLP against all acyclic fibrations; this follows from the weak factorization system generated by the model structure.^[13] As a consequence, in settings where objects are cofibrant and fibrant, such lifts are unique up to homotopy, facilitating the construction of homotopy limits and colimits while maintaining coherence in derived functors.^[14] This uniqueness underpins the homotopy theory of model categories, where cofibrations enable precise control over homotopical data without strict equality.^[14]

Stability under colimits

In model categories, the class of cofibrations exhibits significant stability under colimits, which facilitates the inductive construction of cofibrant objects and enables the small object argument for factorization. This closure ensures that basic cofibrations can be extended systematically while preserving the left lifting property (LLP) with respect to fibrations. A fundamental aspect of this stability is closure under pushouts. Specifically, if i: A \to B is a cofibration and f: C \to X is any morphism, then the pushout inclusion A \sqcup_C X \to B \sqcup_C X is also a cofibration. This follows from the fact that pushouts preserve the LLP, as the lifting property transfers through the universal property of the pushout diagram.^[3] Cofibrations are likewise closed under coproducts (disjoint unions). If \{i_\alpha: A_\alpha \to B_\alpha\}_{\alpha \in \Lambda} is a family of cofibrations, then the induced map \coprod_\alpha A_\alpha \to \coprod_\alpha B_\alpha is a cofibration, provided the underlying category admits coproducts. This stability arises because coproducts preserve colimits and the LLP holds componentwise in the product diagram.^[3] Further stability holds under transfinite compositions, allowing for sequential or ordinal-indexed gluings. For any cardinal \lambda, if \{i_\beta: A_\beta \to B_\beta\}_{\beta < \lambda} is a \lambda-sequence of cofibrations, their transfinite composition is a cofibration. This property is crucial for attaching cells in transfinite stages, as it ensures the composite map satisfies the LLP at each successive step. The class of cofibrations is also closed under retracts. If i: A \to B is a cofibration and r: B' \to B is a retraction (with section s: B \to B'), then the composite A \to B' \to B is a cofibration. This closure, part of Quillen's axiom M5, guarantees that approximations by simpler cofibrations remain within the class. In cellular model categories, these stability properties imply that the cofibrations are precisely the retracts of relative \mathcal{I}-cell complexes, where \mathcal{I} is a small set of generating cofibrations, via the small object argument. This generative aspect underpins the existence of functorial cofibrant replacements and ensures the model structure is accessible for homotopy-theoretic computations.^[3]

Constructions

Cofiber

In homotopy theory, the cofiber of a cofibration i: A \to B is constructed as the pushout B \sqcup_A \Sigma A, where \Sigma A denotes the suspension of A.^[4] This construction, also known as the mapping cone in the pointed case, attaches the cone on A to B via i, forming C_i = B \cup_i CA with identifications (a,1) \sim i(a).^[4] The resulting object captures the "quotient" after attaching A to B and is homotopy invariant, meaning that if i and i' are homotopic, then \operatorname{cofib}(i) \simeq \operatorname{cofib}(i').^[4] The cofiber gives rise to a cofiber sequence A \xrightarrow{i} B \to \operatorname{cofib}(i) \to \Sigma A, which is exact in the homotopy category.^[4] This sequence extends via the Puppe construction to a long exact sequence \cdots \to \Sigma A \to \Sigma B \to \Sigma \operatorname{cofib}(i) \to \Sigma^2 A \to \cdots, natural with respect to maps between cofibrations.^[4] In particular, the map B \to \operatorname{cofib}(i) is a weak equivalence if i is a weak equivalence, ensuring that cofibers preserve homotopy types under localization to the homotopy category.^[4] In the category of topological spaces, when i: A \hookrightarrow B is a closed inclusion (a cofibration), the cofiber \operatorname{cofib}(i) is the quotient space B/A, obtained by collapsing A to a basepoint.^[4] This topological realization aligns with the abstract pushout, as the suspension \Sigma A corresponds to the unreduced cone modulo its base.^[4] Cofiber sequences connect homotopy and homology through long exact sequences. For the homotopy groups, the sequence induces

\cdots \to \pi_n(A) \to \pi_n(B) \to \pi_n(\operatorname{cofib}(i)) \to \pi_{n-1}(A) \to \cdots,

which is exact.^[4] Similarly, in homology with coefficients in an abelian group, applying a homology functor yields a long exact sequence

\cdots \to H_n(A) \to H_n(B) \to H_n(\operatorname{cofib}(i)) \to H_{n-1}(A) \to \cdots,

linking the relative homology H_n(B,A) to the absolute homology of the cofiber.^[4] These sequences are foundational for computing invariants and establishing duality in stable homotopy theory.^[4]

Cofibrant replacement

In model categories, a cofibrant replacement is a functor Q: \mathcal{C} \to \mathcal{C} that assigns to each object X a cofibrant object QX together with a natural weak equivalence q_X: QX \to X, often taken to be a trivial fibration, ensuring that the induced map on homotopy categories is an equivalence.^[15]^[3] This construction resolves non-cofibrant objects to cofibrant ones via cofibrations, enabling the performance of homotopy-invariant operations that require cofibrancy, such as forming correct homotopy colimits.^[3] The standard construction of a functorial cofibrant replacement relies on the small object argument, which provides functorial factorizations of morphisms into cofibrations followed by trivial fibrations.^[3] Specifically, for a set of generating acyclic cofibrations J, one iteratively builds QX as a transfinite colimit of pushouts along coproducts of maps from J, starting from the initial object and attaching cells to approximate X up to weak equivalence; this process terminates at a regular cardinal and yields QX \hookrightarrow X as a cofibration with QX cofibrant.^[15]^[3] Key properties include that QX is cofibrant by construction, the map q_X: QX \to X is a weak equivalence, and if X is already cofibrant, then QX \simeq X up to weak equivalence.^[15] Moreover, the replacement is unique up to weak equivalence in the homotopy category, and Q preserves weak equivalences between cofibrant objects, inducing an equivalence \operatorname{Ho}(\operatorname{cof}(\mathcal{C})) \simeq \operatorname{Ho}(\mathcal{C}).^[3] In the category of topological spaces with the Quillen model structure, the cofibrant replacement QX is a CW-complex weakly equivalent to X, obtained by singularization and geometric realization or direct cell attachment.^[3] For simplicial sets in the Kan-Quillen model structure, every object is already cofibrant, so the replacement functor is essentially the identity, yielding a cofibrant Kan complex upon further fibrant replacement if needed for homotopy computations.^[3] In the category of chain complexes of modules, QX is a bounded-below projective resolution weakly equivalent to X.^[3] Cofibrant replacements play a central role in defining derived functors, where the total left derived functor LF of a left Quillen functor F is computed as LF(X) = F(QX) for cofibrant X, ensuring that homotopy limits and colimits are correctly formed in the homotopy category.^[15]^[3] This facilitates Quillen adjunctions and equivalences by replacing objects with cofibrant approximations before applying functors.^[3]