03.12.54 · modern-geometry / homotopy

Filtered space

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Anchor (Master): Brown-Higgins-Sivera *Nonabelian Algebraic Topology* §B.7 and Part III (connected filtered spaces, the connectivity condition $\phi$); Brown-Higgins on crossed complexes

Intuition Beginner

A filtered space is a space together with a record of how it was built up in stages. You name a growing chain of subspaces $X_{0} \subseteq X_{1} \subseteq X_{2} \subseteq \dots$ that eventually fills out the whole space $X$ . The index $n$ is a measure of "how far along the build" you are: $X_{0}$ is the starting material, $X_{1}$ is what you have after the first round of gluing, and so on.

The running example to keep in mind is a CW complex. There you build a shape one dimension at a time: dots, then sticks glued to dots, then patches glued onto the stick-figure, then solid blobs. The $n$ -skeleton $X_{n}$ is "everything of dimension $n$ or less." That nested sequence of skeleta is the filtration. Nothing new is added to the space; the filtration just remembers the staircase you climbed to reach it.

Why bother recording the staircase? Because many hard questions about a space become answerable one floor at a time. If you understand how floor $n$ sits inside floor $n + 1$ , you can often assemble an answer for the whole building from the answers on each landing. The filtration is exactly that floor-by-floor scaffold.

Visual Beginner

A schematic of nested subspaces: a small inner blob labelled $X_{0}$ , a larger region around it labelled $X_{1}$ , a still larger one $X_{2}$ , and so on, with the outermost boundary labelled $X$ . Arrows point outward to show the chain of inclusions growing to fill the whole space.

For a CW complex the rings are the skeleta: the innermost dots are $X_{0}$ , adding arcs gives $X_{1}$ , adding patches gives $X_{2}$ , and the union of all of them is the space.

Worked example Beginner

Filter the circle $S^{1}$ using its smallest CW build: one dot and one arc.

Step 1. Let $X_{0}$ be the single point $p$ . This is the 0-skeleton: one 0-cell, nothing else. Step 2. Glue an arc (a 1-cell) by attaching both of its endpoints to $p$ . The result is a loop. Call this loop $X_{1}$ . Step 3. There are no cells of dimension 2 or higher, so $X_{2} = X_{1}$ , $X_{3} = X_{1}$ , and the chain stops growing. Step 4. The filtration is $X_{0} = {p} \subseteq X_{1} = S^{1} = X_{2} = X_{3} = \dots$ , and the union of all the pieces is $S^{1}$ itself.

What this tells us: the filtration on $S^{1}$ has real content only in degrees 0 and 1. Floor 0 is a point; floor 1 is the whole circle; every later floor repeats floor 1. The "interesting" information — a loop appearing — happens exactly at the step from $X_{0}$ to $X_{1}$ , which matches the fact that the circle's first loop lives in dimension 1.

Check your understanding Beginner

Formal definition Intermediate+

A filtered space is a topological space $X$ together with a sequence of subspaces $$ X_0 \subseteq X_1 \subseteq X_2 \subseteq \cdots \subseteq X_n \subseteq \cdots \subseteq X $$ indexed by $n \in N$ , satisfying the exhaustion condition $X = ⋃_{n \geq 0} X_{n}$ . The subspace $X_{n}$ is the $n$ -th filtration stage. We write $X_{*}$ for the filtered space, reserving $X$ for the underlying total space ^{[Brown-Higgins-Sivera §B.7]}. By convention $X_{- 1} = \emptyset$ .

The exhaustion condition fixes the underlying set of $X$ but not yet its topology relative to the stages. Two regimes occur. In the subspace-filtration regime, $X$ carries a given topology and each $X_{n}$ has the subspace topology; this is the default in Hatcher's treatment of skeleta ^{[Hatcher §0]}. In the colimit regime, $X$ is recovered as $X = colim_{n} X_{n}$ , with the weak topology in which $A \subseteq X$ is closed iff $A \cap X_{n}$ is closed in $X_{n}$ for every $n$ . For a CW complex the two regimes agree, since the weak topology on the skeleta is the space's topology.

A filtered map $f : X_{*} \to Y_{*}$ is a continuous map $f : X \to Y$ that preserves the filtration degree by degree: $f (X_{n}) \subseteq Y_{n}$ for all $n$ . Filtered maps compose, and the identity is filtered, so filtered spaces and filtered maps form a category, denoted $FTop$ .

Three structural instances recur. The skeletal filtration of a CW complex sets $X_{n} = X^{(n)}$ , the $n$ -skeleton. The constant filtration on a space $Y$ sets $Y_{n} = Y$ for every $n$ ; the inclusions are identities. A based space $(Y, y_{0})$ becomes a filtered space concentrated in degree $0$ by setting $X_{0} = {y_{0}}$ and $X_{n} = Y$ for $n \geq 1$ — only the basepoint occupies the bottom stage.

Counterexamples to common slips

A decreasing chain $X \supseteq X_{1} \supseteq X_{2} \supseteq \dots$ (as appears in the spectral sequence of a filtered complex) is a descending filtration; it is a different object and does not satisfy the ascending-exhaustion convention used here. The two are related by reindexing only when the chain is finite.
Exhaustion is about the set $X = ⋃_{n} X_{n}$ . It does not by itself force the topology of $X$ to be the colimit topology. The Hawaiian earring filtered by finite sub-wedges exhausts as a set but is not the colimit of those sub-wedges as a space.
A continuous map with $f (X_{n}) \subseteq Y_{n + 1}$ but not $f (X_{n}) \subseteq Y_{n}$ is not a filtered map; the degree must be preserved on the nose, not merely up to a shift.

Key theorem with proof Intermediate+

Theorem. The skeletal filtration makes every CW complex $X$ into a filtered space $X_ $w i t h$ X_n = X^{(n)} $, an d a ce l l u l a r ma p$ f \colon X \to Y $b e tw ee n C W co m pl e x es i s a f i l t er e d ma p$ X_* \to Y_* $. M or eo v er$ \mathsf{FTop} $ha s a l l s ma l l co l imi t s, co m p u t e d o n u n d er l y in g s p a ces, an d t h e a ss i g nm e n t$ X \mapsto X_* $f r o m ce l l u l a r ma p so f C W co m pl e x es in t o$ \mathsf{FTop}$ is a functor.*

Proof. Let $X$ be a CW complex with skeleta $X^{(- 1)} = \emptyset \subseteq X^{(0)} \subseteq X^{(1)} \subseteq \dots$ . The skeleta are nested closed subspaces by construction, and $X = ⋃_{n} X^{(n)}$ is the defining union of a CW complex ^{[Hatcher §0]}. So $X_{n} := X^{(n)}$ exhibits $X$ as a filtered space. A cellular map is one with $f (X^{(n)}) \subseteq Y^{(n)}$ for all $n$ , which is exactly the filtered-map condition; identities are cellular, and a composite of cellular maps is cellular, so this assignment respects identities and composition and is a functor into $FTop$ .

For colimits, let $D : J \to FTop$ be a small diagram with $D (j) = (X^{j})_{*}$ . Form the colimit $X = colim_{j} X^{j}$ in $Top$ , with structure maps $ι_{j} : X^{j} \to X$ . Define the filtration on $X$ by $$ X_n = \bigcup_{j \in J} \iota_j\big( X^j_n \big) \subseteq X . $$ Each $X_{n}$ is a subspace of $X$ and $X_{n} \subseteq X_{n + 1}$ because every $X_{n}^{j} \subseteq X_{n + 1}^{j}$ . Exhaustion holds: $⋃_{n} X_{n} = ⋃_{n} ⋃_{j} ι_{j} (X_{n}^{j}) = ⋃_{j} ι_{j} (⋃_{n} X_{n}^{j}) = ⋃_{j} ι_{j} (X^{j}) = X$ , using exhaustion of each $X_{*}^{j}$ and surjectivity of the colimit cocone onto $X$ . This makes $X_{*}$ a filtered space, and the $ι_{j}$ are filtered maps since $ι_{j} (X_{n}^{j}) \subseteq X_{n}$ by definition.

It remains to check the universal property. Suppose $g_{j} : (X^{j})_{*} \to Z_{*}$ is a cocone of filtered maps. The underlying continuous maps factor through a unique continuous $g : X \to Z$ with $g \circ ι_{j} = g_{j}$ , by the colimit in $Top$ . The map $g$ is filtered: for $x \in X_{n}$ there is some $j$ and $y \in X_{n}^{j}$ with $ι_{j} (y) = x$ , whence $g (x) = g_{j} (y) \in Z_{n}$ because $g_{j}$ is filtered, so $g (X_{n}) \subseteq Z_{n}$ . Uniqueness of $g$ as a continuous map gives uniqueness as a filtered map. Thus $X_{*}$ with the cocone $(ι_{j})$ is the colimit in $FTop$ , computed on underlying spaces. $□$

Bridge. This builds toward the higher homotopy van Kampen theorem of nonabelian algebraic topology, and the colimit computation above appears again in 03.12.55, where the fundamental crossed complex functor is shown to preserve exactly these filtered colimits. The foundational reason filtered spaces are the right input is already visible here: the skeletal filtration is defined as a colimit of its skeleta, so a structure that turns filtrations into algebra must respect colimits, and this is exactly the functoriality just proved. The construction generalises the cellular-pushout assembly of a CW complex from its cells 03.12.10; putting these together, the bridge is the recognition that "build $X$ in stages, then read off invariants stage by stage" only works because both the building and the reading are colimit-compatible.

Exercises Intermediate+

Exercise 3 (medium, short-answer).

Show that the constant filtration defines a functor $Top \to FTop$ , and that it is left adjoint to the functor $FTop \to Top$ sending $X_{*}$ to its total space $X$ .

Hint

A filtered map out of the constant filtration on $Y$ is just a continuous map $Y \to Z$ landing in $Z_{0}$ ? Check the degree condition carefully.

Answer

Let $c (Y)$ denote the constant filtration $Y_{n} = Y$ . A continuous $g : Y \to Z$ induces $c (g) : c (Y) \to c (Z)$ , which is filtered since $g (Y) = Y \mapsto Z = Z_{n}$ ; this respects identities and composition, giving a functor. For the adjunction, a filtered map $c (Y) \to Z_{*}$ is a continuous $g : Y \to Z$ with $g (Y_{n}) \subseteq Z_{n}$ for all $n$ ; since $Y_{n} = Y$ , the binding constraint is $g (Y) \subseteq Z_{0}$ , i.e. $g$ factors through $Z_{0} \subseteq Z$ . The forgetful functor sends $Z_{*}$ to its total space $Z$ , not $Z_{0}$ , so the clean adjunction is between the constant filtration and the bottom-stage functor $Z_{*} \mapsto Z_{0}$ . Rubric: full credit for the functor check and identifying $Z_{0}$ (not $Z$ ) as the correct right adjoint target.

Exercise 4 (medium, short-answer).

Let $X_{*}$ and $Y_{*}$ be filtered spaces. Define a filtration on the product $X \times Y$ making it the product in $FTop$ , and verify the universal property.

Hint

A point reaches filtration level $n$ in the product when both coordinates are at level $\leq n$ . Set $(X \times Y)_{n} = ⋃_{p + q \leq n}$ ? No — try $X_{n} \times Y_{n}$ first and check projections.

Answer

Set $(X \times Y)_{n} = X_{n} \times Y_{n}$ . These are nested ( $X_{n} \times Y_{n} \subseteq X_{n + 1} \times Y_{n + 1}$ ) and exhaust $X \times Y$ since $⋃_{n} X_{n} \times Y_{n} = X \times Y$ . The projections $pr_{X}, pr_{Y}$ are filtered: $pr_{X} (X_{n} \times Y_{n}) = X_{n}$ . Given filtered maps $f : Z_{*} \to X_{*}$ and $g : Z_{*} \to Y_{*}$ , the pairing $(f, g) : Z \to X \times Y$ is continuous and filtered since $z \in Z_{n}$ gives $f (z) \in X_{n}$ , $g (z) \in Y_{n}$ , hence $(f, g) (z) \in X_{n} \times Y_{n}$ . Uniqueness follows from uniqueness in $Top$ . Rubric: full credit for the filtration $X_{n} \times Y_{n}$ and the universal-property verification.

Exercise 5 (medium, short-answer).

Give a filtered space whose underlying space is path-connected but whose bottom stage $X_{0}$ is empty, and explain why such a filtration cannot satisfy the connectivity condition that nonabelian algebraic topology requires of a connected filtered space.

Hint

The connectivity condition demands, among other things, that $X_{0}$ meet every path component of every later stage.

Answer

Take $X = S^{1}$ with $X_{0} = \emptyset$ , $X_{1} = X_{2} = \dots = S^{1}$ . This is a valid filtered space (nested, exhausting). But the connectivity condition requires the map $π_{0} (X_{0}) \to π_{0} (X_{n})$ to be surjective for each $n$ ; with $X_{0} = \emptyset$ , $π_{0} (X_{0}) = \emptyset$ cannot surject onto $π_{0} (S^{1})$ , which has one element. So the filtration fails to be connected in the NAT sense even though the total space is path-connected. Rubric: full credit for the empty- $X_{0}$ example and the failure of $π_{0}$ -surjectivity.

Exercise 6 (hard, short-answer).

Prove that the forgetful functor $U : FTop \to Top$ , $X_{*} \mapsto X$ , creates colimits: a diagram in $FTop$ has a colimit iff its image under $U$ does, and $U$ carries the colimit to the colimit.

Hint

The Key theorem computed the filtered colimit on underlying spaces. Run the universal property in both directions.

Answer

Let $D : J \to FTop$ with $U D$ having colimit $X = colim_{j} X^{j}$ in $Top$ . As in the Key theorem, define $X_{n} = ⋃_{j} ι_{j} (X_{n}^{j})$ ; this is a filtered space and the cocone $ι_{j}$ is filtered. Its universal property in $FTop$ was verified there, so the colimit exists and $U$ sends it to $X$ . Conversely, if $D$ has a colimit $C_{*}$ in $FTop$ , then $U (C_{*})$ is a cocone in $Top$ ; comparing with the constructed colimit $X_{*}$ over the same diagram, the universal properties give mutually inverse filtered maps $C_{*} \to X_{*}$ and $X_{*} \to C_{*}$ , so $U (C_{*}) ≅ X$ . Hence $U$ creates colimits. Rubric: full credit for both directions and the comparison isomorphism.

Exercise 7 (hard, short-answer).

Exhibit a filtered space whose total topology is not the colimit topology of its stages, and a second filtered space where it is, to show the two regimes of the formal definition genuinely differ.

Hint

The Hawaiian earring versus the infinite wedge of circles distinguishes the subspace and colimit regimes.

Answer

Let $H \subseteq R^{2}$ be the Hawaiian earring, filtered by $H_{n} =$ the union of the $n$ largest circles. As a subspace of $R^{2}$ , the wedge point has neighbourhoods meeting infinitely many circles, so the subspace topology on $H$ is strictly coarser than the colimit topology $colim_{n} H_{n}$ (in the colimit, a set hitting each $H_{n}$ in a closed set is closed, which fails for $H$ 's metric topology). By contrast, take the infinite wedge $W = ⋁_{n} S_{n}^{1}$ with the CW topology, filtered by partial wedges $W_{n} = ⋁_{k \leq n} S_{k}^{1}$ . There $W = colim_{n} W_{n}$ by definition of the CW topology, so the colimit regime holds on the nose. Rubric: full credit for the Hawaiian earring as the subspace-only example and the CW wedge as the colimit example.

Advanced results Master

The bare exhaustion axiom is too weak to support algebra. Nonabelian algebraic topology selects a subclass by imposing a connectivity condition, written $ϕ$ , on a filtered space $X_{*}$ ^{[Brown-Higgins-Sivera §B.7]}. The condition has two parts. First, $ϕ_{0}$ : each inclusion induces a surjection $π_{0} (X_{0}) \to π_{0} (X_{r})$ for every $r \geq 0$ — the bottom stage already meets every path component of every higher stage. Second, $ϕ_{n}$ for $n \geq 1$ : for every basepoint $v \in X_{0}$ and every pair $r > n \geq 1$ , the relative homotopy group vanishes, $$ \pi_n(X_r, X_n, v) = 0 . $$ A filtered space satisfying $ϕ$ for all degrees is a connected filtered space. The skeletal filtration of a connected CW complex satisfies $ϕ$ : cellular approximation lets one push any map of a disk of dimension $n$ off cells of dimension $> n$ , which is precisely the vanishing $π_{n} (X^{(r)}, X^{(n)}, v) = 0$ for $r > n$ .

The connectivity condition is the hinge on which the main theorems of the subject turn. To a connected filtered space $X_{*}$ one associates its fundamental crossed complex $Π X_{*}$ , whose degree- $n$ part packages the relative homotopy groups $π_{n} (X_{n}, X_{n - 1}, v)$ together with the boundary maps and the action of the fundamental groupoid $π_{1} (X_{1}, X_{0})$ . The condition $ϕ$ is exactly what makes this assignment lose no information at the level of cells: the higher van Kampen theorem of Brown-Higgins states that $Π$ carries certain colimits of connected filtered spaces (gluings along subcomplexes) to colimits of crossed complexes, and the proof requires $ϕ$ to subdivide maps and trade a continuous gluing for an algebraic one ^{[Brown-Higgins 1981]}.

The relationship between filtered spaces and crossed complexes is an equivalence after restriction. On the topological side, the connected filtered spaces with the homotopy relation; on the algebraic side, crossed complexes; the functor $Π$ and a geometric realisation functor pass between them and become mutually inverse on homotopy categories after the appropriate localisation. The cubical reformulation replaces $Π X_{*}$ by a cubical $ω$ -groupoid built from filtered maps of cubes $I_{*}^{n} \to X_{*}$ , and the two models — crossed complexes and cubical $ω$ -groupoids — are equivalent. The filtered space is the common geometric input feeding both.

Synthesis. The filtered space is the right category of input precisely because every higher-homotopical invariant in this circle of ideas is a staged invariant, and the staging is the foundational reason the theory computes. The skeletal filtration generalises the dimension grading of a CW complex into a free-standing structure, and the connectivity condition $ϕ$ is what makes that structure rigid enough to carry algebra: this is exactly the demand that the relative homotopy groups below the diagonal vanish, so that $Π X_{*}$ sees each cell once and only once. The higher van Kampen theorem is dual to the colimit computation in $FTop$ proved at Intermediate tier — gluing spaces along subcomplexes on the topological side becomes a colimit of crossed complexes on the algebraic side, and $ϕ$ is the bridge that licenses the trade. Putting these together, the central insight is that the move from spaces to filtered spaces is not a loss of generality but a gain of computability: one pays the price of remembering a filtration and is repaid with a functor $Π$ that turns local-to-global gluings into algebra. This appears again in 03.12.57, where the cubical $ω$ -groupoid model and the multiple compositions of filtered cubes make the same point in a form built for explicit calculation.

Full proof set Master

Proposition. The skeletal filtration of a connected CW complex $X$ is a connected filtered space: it satisfies the connectivity condition $ϕ$ in every degree.

Proof. Write $X_{n} = X^{(n)}$ . We verify $ϕ_{0}$ and $ϕ_{n}$ for $n \geq 1$ .

Condition $ϕ_{0}$ . We must show $π_{0} (X^{(0)}) \to π_{0} (X^{(r)})$ is surjective for all $r$ . Since $X$ is connected and CW, it is path-connected, and so is each skeleton $X^{(r)}$ for $r \geq 1$ once $X$ is connected: any two 0-cells are joined by an edge-path because the 1-skeleton of a connected CW complex is a connected graph. Every path component of $X^{(r)}$ therefore contains a 0-cell, i.e. a point of $X^{(0)}$ , so the map on $π_{0}$ is surjective. (For $r = 0$ the map is the identity.)

Condition $ϕ_{n}$ , $n \geq 1$ . Fix $v \in X^{(0)}$ and $r > n$ . We show $π_{n} (X^{(r)}, X^{(n)}, v) = 0$ , i.e. every map of pairs $f : (D^{n}, S^{n - 1}, s_{0}) \to (X^{(r)}, X^{(n)}, v)$ is homotopic rel $S^{n - 1}$ to a map into $X^{(n)}$ . The image $f (D^{n})$ is compact, so it meets only finitely many open cells of $X^{(r)}$ , hence lies in a finite subcomplex. Apply cellular approximation to the map $f : D^{n} \to X^{(r)}$ relative to the subspace $S^{n - 1}$ , on which $f$ already lands in $X^{(n)}$ (indeed in $X^{(n - 1)} \subseteq X^{(n)}$ after a preliminary approximation): a map of an $n$ -dimensional disk can be homotoped off every cell of dimension $> n$ because a compact set of covering dimension $n$ cannot surject onto an open cell of dimension $> n$ , and the deformation is supported away from $S^{n - 1}$ . The homotopy is rel $S^{n - 1}$ and ends in $X^{(n)}$ . Thus the class of $f$ in $π_{n} (X^{(r)}, X^{(n)}, v)$ is the class of a map into $X^{(n)}$ , which is the zero of the relative group. Both conditions hold for all degrees, so $X_{*}$ is a connected filtered space. $□$

Proposition. Every filtered map $f \colon X_ \to Y_* $b e tw ee n co nn ec t e df i l t er e d s p a ces in d u ces am or p hi s m$ \Pi(f) \colon \Pi X_* \to \Pi Y_* $o f cr osse d co m pl e x es, an d$ \Pi$ is functorial.*

Proof. A filtered map satisfies $f (X_{n}) \subseteq Y_{n}$ and $f (X_{n - 1}) \subseteq Y_{n - 1}$ , so $f$ restricts to a map of pairs $(X_{n}, X_{n - 1}) \to (Y_{n}, Y_{n - 1})$ for every $n$ , and to $(X_{1}, X_{0}) \to (Y_{1}, Y_{0})$ in degree one. Relative homotopy groups are functorial for maps of pairs, so $f$ induces homomorphisms $π_{n} (X_{n}, X_{n - 1}, v) \to π_{n} (Y_{n}, Y_{n - 1}, f (v))$ commuting with the boundary maps $\partial$ and compatible with the fundamental-groupoid action, since that action is natural for maps of pairs preserving basepoints in $X_{0}$ . These assembled maps are exactly the data of a crossed-complex morphism $Π (f)$ . Functoriality $Π (g \circ f) = Π (g) \circ Π (f)$ and $Π (id) = id$ follows from functoriality of each relative homotopy group in the map of pairs. $□$

Stated without proof — see Brown-Higgins-Sivera §14 ^{[Brown-Higgins-Sivera §B.7]}. The higher homotopy van Kampen theorem: the functor $Π$ from connected filtered spaces to crossed complexes preserves the colimits arising from gluing a connected filtered space along a connected filtered subspace, and the resulting equivalence of homotopy categories between crossed complexes and the cubical $ω$ -groupoids of filtered spaces.

Connections Master

CW complex 03.12.10. The skeletal filtration is the prototype filtered space and the only example needed to motivate the whole framework: $X_{n} = X^{(n)}$ . The functoriality of $X \mapsto X_{*}$ on cellular maps proved here is what lets later constructions read a CW complex one skeleton at a time, and the connectivity condition $ϕ$ is verified for it using cellular approximation imported from that unit.
Fundamental crossed complex (forward unit) 03.12.55. The crossed complex $Π X_{*}$ is the direct algebraic successor of this unit: it takes a connected filtered space as input and outputs the graded relative-homotopy data on which the higher van Kampen theorem operates. The colimit computation in $FTop$ established here is precisely the geometric side that $Π$ must preserve.
Cubical $ω$ -groupoid of a filtered space (forward unit) 03.12.57. The cubical model $ρ X_{*}$ is the second invariant built from a filtered space, assembled from filtered maps of cubes $I_{*}^{n} \to X_{*}$ . Its multiple compositions make the same staged information explicit and calculable, and it is equivalent to $Π X_{*}$ ; the filtered space is their common source.
Spectral sequence 03.13.01. A filtered chain or cochain complex — a different but cognate filtration, decreasing rather than increasing — drives a spectral sequence whose pages are the successive associated gradeds. The conceptual parallel is exact: both turn a filtration into a sequence of computable pieces, and both recover the whole from how each stage sits inside the next.

Historical & philosophical context Master

Filtration by skeleta is implicit in J. H. C. Whitehead's 1949 introduction of CW complexes, where the dimension grading was the organising device for combinatorial homotopy. The free-standing notion of a filtered space as the input to a functorial algebraic invariant emerged from the work of Ronald Brown and Philip J. Higgins through the 1970s and early 1980s on higher-dimensional analogues of the van Kampen theorem ^{[Brown-Higgins 1981]}. Their 1981 paper Colimit theorems for relative homotopy groups fixed the connectivity condition under which relative homotopy groups of a filtered space assemble into a crossed complex with good colimit behaviour, and identified the filtered space — not the bare space — as the carrier on which a nonabelian, higher-dimensional van Kampen theorem could be both stated and proved.

The synthesis appears in the 2011 monograph of Brown, Higgins, and Sivera, Nonabelian Algebraic Topology ^{[Brown-Higgins-Sivera §B.7]}, which takes the filtered space as the basic geometric object of an entire approach to algebraic topology, develops the equivalence between crossed complexes and cubical $ω$ -groupoids, and derives classical results — the relative Hurewicz theorem, the Blakers-Massey description of the first non-vanishing relative homotopy group — as consequences of the colimit theorem for filtered spaces.

Bibliography Master

@book{BrownHigginsSivera2011NAT,
  author    = {Brown, Ronald and Higgins, Philip J. and Sivera, Rafael},
  title     = {Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids},
  series    = {EMS Tracts in Mathematics},
  volume    = {15},
  publisher = {European Mathematical Society},
  year      = {2011}
}

@article{BrownHiggins1981Colimit,
  author  = {Brown, Ronald and Higgins, Philip J.},
  title   = {Colimit theorems for relative homotopy groups},
  journal = {J. Pure Appl. Algebra},
  volume  = {22},
  year    = {1981},
  pages   = {11--41}
}

@book{HatcherAlgebraicTopology,
  author    = {Hatcher, Allen},
  title     = {Algebraic Topology},
  publisher = {Cambridge University Press},
  year      = {2002}
}

@article{Whitehead1949CombinatorialHomotopyI,
  author  = {Whitehead, J. H. C.},
  title   = {Combinatorial homotopy. {I}},
  journal = {Bull. Amer. Math. Soc.},
  volume  = {55},
  year    = {1949},
  pages   = {213--245}
}

Prerequisites

03.12.10
02.01.01
02.01.06

Used in

03.12.55
03.12.57

Tier anchors

beginner: Hatcher *Algebraic Topology* §0 (the skeletal filtration of a CW complex); the dots-sticks-patches build-up picture
intermediate: Brown-Higgins-Sivera *Nonabelian Algebraic Topology* §B.7 (filtered spaces, filtered maps, the category FTop); Hatcher §0 and §1
master: Brown-Higgins-Sivera *Nonabelian Algebraic Topology* §B.7 and Part III (connected filtered spaces, the connectivity condition $\phi$); Brown-Higgins on crossed complexes

References

Brown, Higgins, Sivera — Nonabelian Algebraic Topology · §B.7 filtered spaces; Part II-III connected filtered spaces and the connectivity condition
Hatcher — Algebraic Topology · §0 (CW complexes and the skeletal filtration), §1 (skeleta and $\pi_1$)
Brown, Higgins — Colimit theorems for relative homotopy groups · J. Pure Appl. Algebra 22 (1981) 11-41, connected filtered spaces

Estimated time

beginner: 12m
intermediate: 30m
master: 55m