topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Geometric realization is the operation that builds from a simplicial set $X$ a topological space $|X|$ obtained by interpreting each element in $X_n$ – each abstract $n$-simplex in $X$ – as one copy of the standard topological $n$-simplex $\Delta^n_{Top}$ and then gluing together all these along their boundaries to a big topological space, using the information encoded in the face and degeneracy maps of $X$ on how these simplices are supposed to be stuck together. It generalises the geometric realization of simplicial complexes as described at that entry.
This is the special case of the general notion of nerve and realization that is induced from the standard cosimplicial topological space $[n] \mapsto \Delta^n_{Top}$. (N.B.: in this article, $[n]$ denotes the ordinal with $n+1$ elements. The corresponding contravariant representable is denoted $\Delta(-, n)$.)
In the context of homotopy theory geometric realization plays a notable role in the homotopy hypothesis, where it is part of the Quillen equivalence between the model structure on topological spaces and the standard model structure on simplicial sets.
The construction generalizes naturally to a map from simplicial topological spaces to plain topological spaces. For more on that see geometric realization of simplicial spaces.
The dual concept is totalization .
There are various levels of generality in which the notion of (topological) geometric realization makes sense. The basic definition is
A generalization of this of central importance is the
Up to homotopy, this is a special case of a general notion of
At the point-set level, it is also a special case of a general notion of
Let $S$ be one of the categories of geometric shapes for higher structures, such as the globe category or the simplex category or the cube category.
There is an obvious functor
$st : S \to$ Top
which sends the standard cellular shape $[n]$ (the standard cellular globe, simplex or cube, respectively) to the corresponding standard topological shape (for instance the standard $n$-simplex $st([n]) := \{ (x_1, \cdots, x_n) | \sum_{i=1}^n x_i = 1, x_i \geq 0 \} \subset \mathbb{R}^{n}$ ) with the obvious induced face and boundary maps.
Using this, in cases where $Top$ can be regarded as enriched over and tensored over a base category $V$, the geometric realization of a presheaf $K^\bullet : S^{op} \to V$ on $S$ – e.g., of a globular set, a simplicial set or a cubical set, respectively (when $V= Set$) – is the topological space given by the coend, weighted colimit, or tensor product of functors
In the case of simplicial sets, see for more discussion also
Via simplicial nerve functors geometric realization of simplicial sets induces geometric realizations of many other structures, for instance
For the case of cubical sets, see cubical geometric realisation.
See
Every cohesive (∞,1)-topos $\mathbf{H}$ (in fact every locally ∞-connected (∞,1)-topos) comes with its intrinsic notion of geometric realization.
The general abstract definition is at cohesive (∞,1)-topos in the section Geometric homotopy.
For the choice $\mathbf{H} =$ ∞Grpd this reproduces the geometric realization of simplicial sets, see at discrete ∞-groupoid the section
For the choice $\mathbf{H} =$ ETop∞Grpd and Smooth∞Grpd this reproduces geometric realization of simplicial topological spaces. See the sections ETop∞Grpd – Geometric homotopy and Smooth ∞-groupoid – Geometric homotopy
Let $M$ be a cocomplete simplicially enriched category with copowers. A simplicial object in $M$ is a functor $X:\Delta^{op}\to M$, where $\Delta$ is the simplex category. Its geometric realization is defined similarly to the classical case as a coend:
where $\odot$ denotes the copower in $M$. This operation is a left adjoint which is even a simplicially enriched functor; see simplicial object for more details.
In this section we consider topological geometric realization of simplicial sets, which is the best studied and perhaps most significant case.
Each ${|X|}$ is a CW complex (see lemma below), and so geometric realization ${|(-)|}: Set^{\Delta^{op}} \to Top$ takes values in the full subcategory of CW complexes, and therefore in any convenient category of topological spaces, for example in the category $CGHaus$ of compactly generated Hausdorff spaces. Let $Space$ be any convenient category of topological spaces, and let $i \colon Space \to Top$ denote the inclusion.
For any simplicial set $X$, there is a natural isomorphism $i(\int^{n: \Delta} X(n) \cdot \sigma(n)) \cong {|X|}$, where the coend on the left is computed in $Space$.
This is obvious: more generally, if $F: J \to A$ is a diagram and $i: A \hookrightarrow B$ is a full replete subcategory, and if the colimit in $B$ of $i \circ F$ lands in $A$, then this is also the colimit of $F$ in $A$. (The dual statement also holds, with limits instead of colimits.)
Below, we let $R: Set^{\Delta^{op}} \to Space$ denote the geometric realization when considered as landing in $Space$.
We continue to assume $Space$ is any convenient category of topological spaces. In this section we prove that geometric realization
is a left exact functor in that it preserves finite limits.
It is important that we use some such “convenience” assumption, because for example
valued in general topological spaces, does not preserve products. (To get a correct statement, one usual procedure is to “kelley-fy” products by applying the coreflection $k \colon Haus \to CGHaus$ to Hausdorff and compactly generated topological spaces. This gives the correct isomorphism in the case $Space = CGHaus$, where we have that ${|X \times Y|} \cong {|X|} \times_k {|Y|} \coloneqq k({|X|} \times {|Y|})$; the product on the right has been “kelleyfied” to the product appropriate for $CGHaus$.)
We reiterate that $R$ denotes the geometric realization functor considered as valued in a convenient category of spaces, whereas ${|(-)|}$ is geometric realization viewed as taking values in $Top$.
Let $U = \hom(1, -): Space \to Set$ be the underlying-set functor. Then the composite $U R: Set^{\Delta^{op}} \to Set$ is left exact.
As described at the nLab article on triangulation here, the composite
can be described as the functor
where $Int$ is the category of intervals (linearly ordered sets with distinct top and bottom). Because every interval, in particular $I$, is a filtered colimit of finite intervals, and because finite intervals are finitely presentable intervals, it follows that $U \sigma \colon \Delta \to Set$ is a flat functor (a filtered colimit of representables). But on general grounds, tensoring with a flat functor is left exact, which in this case means
is left exact.
Obviously the preceding proof is not sensitive to whether we use $Space$ or $Top$.
If $i: X \to Y$ is a monomorphism of simplicial sets, then $R(i): R(X) \to R(Y)$ is a closed subspace inclusion, in fact a relative CW-complex. In particular, taking $X = \emptyset$, $R(Y)$ is a $CW$-complex.
Any monomorphism $i \colon X \to Y$ in $Set^{\Delta^{op}}$ can be seen as the result of iteratively adjoining nondegenerate $n$-simplices. In other words, there is a chain of inclusions $X = F(0) \hookrightarrow F(1) \hookrightarrow \ldots Y = colim_i F(i)$, where $F: \kappa \to Set^{\Delta^{op}}$ is a functor from some ordinal $\kappa = \{0 \leq 1\leq \ldots\}$ (as preorder) that preserves directed colimits, and each inclusion $F(\alpha \leq \alpha + 1): F(\alpha) \to F(\alpha + 1)$ fits into a pushout diagram
where $j$ is the inclusion. Now $R(j)$ is identifiable as the inclusion $S^{n-1} \to D^n$, and since $R$ preserves pushouts (which are calculated as they are in $Top$), we see by this lemma that $R F(\alpha) \to R F(\alpha+1)$ is a closed subspace inclusion and evidently a relative CW-complex. By another lemma, it follows that $X \to Y$ is also a closed inclusion and indeed a relative CW-complex.
$R: Set^{\Delta^{op}} \to Space$ preserves equalizers.
The equalizer of a pair of maps in $Top$ is computed as the equalizer on the level of underlying sets, equipped with the subspace topology. So if
is an equalizer diagram in $Set^{\Delta^{op}}$, then ${|i|}$ is the equalizer of the pair ${|f|}$, ${|g|}$, because the underlying function $U({|i|})$ is the equalizer of $U({|f|})$, $U({|g|})$ on the underlying set level by the preceding theorem, and because ${|i|}$ is a (closed) subspace inclusion by lemma . But this $Top$-equalizer ${{|i|}}: {{|E|}} \to {{|X|}}$ lives in the full subcategory $Space$, and therefore $R(i) = {|i|}$ is the equalizer of the pair $R(f) = {|f|}$, $R(g) = {|g|}$.
As the proof indicates, that realization preserves equalizers is not at all sensitive to whether we use $Top$ or a convenient category of spaces $Space$.
That geometric realization preserves products is sensitive to whether we think of it as valued in $Top$ or in a convenient category $Space$. In particular, the proof uses cartesian closure of $Space$ in an essential way (in the form that finite products distribute over arbitrary colimits).
First, an easy result on products of simplices.
The realization of a product of two representables $\Delta(-, m) \times \Delta(-, n)$ is compact.
It suffices to observe that $\Delta[m] \times \Delta[n]$ has finitely many non-degenerate simplices. That is clear since non-degenerate $k$-simplices in the nerve of a poset $P$ are exactly injective order preserving maps $[k] \to P$.
The canonical map
is a homeomorphism.
The canonical map is continuous, and a bijection at the underlying set level by theorem . The codomain is the compact Hausdorff space $\sigma(m) \times \sigma(n)$, and the domain is compact by Lemma . But a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.
The key properties of $I$ needed for this subsection are (1) the fact it is compact Hausdorff, and (2) the order relation $\leq$ on the interval $I$ defines a closed subset of $I \times I$. These properties ensure that the affine $n$-simplex $\{(x_1, \ldots, x_n) \in I^n: x_1 \leq \ldots \leq x_n\}$ is itself compact Hausdorff, so that the proof of lemma goes through. The point is that in place of $I$, we can really use any interval $L$ that satisfies these properties, thus defining an $L$-based geometric realization instead of the standard ($I$-based) geometric realization being developed here.
The functor $R: Set^{\Delta^{op}} \to Space$ preserves products.
The proof is purely formal. Let $X$ and $Y$ be simplicial sets. By the co-Yoneda lemma, we have isomorphisms
and so we calculate
where in each of the second and penultimate lines, we twice used the fact that $- \times -$ preserves colimits in its separate arguments (i.e., the fact that the nice category $Space$ is cartesian closed), and the remaining lines used the fact that $R$ preserves colimits, and also products of representables by lemma .
Drinfeld, On the notion of geometric realization provides a conceptual explanation of preserving finite limits, and “reformulates the definitions so that the following facts become obvious:
geometric realization commutes with finite projective limits (e.g., with Cartesian products);
the geometric realization of a simplicial set … (resp. cyclic set) is equipped with an action of the group of orientation preserving homeomorphisms of the segment $I := [0, 1]$ … (resp. the circle $S^ 1$).“ (quote from the paper)
A draft M.Gavrilovich, K.Pimenov. Geometric realisation as a Skorokhod semi-continuous path space endofunctor attempts to further reformulate this by showing that, in a certain precise sense, geometric realisation is an endofunctor of a certain category $sF$ of simplicial sets equipped with extra structure of topological nature (a notion of smallness). The underlying endofunctor of $sSets$ is
The category $sF$ contains simplicial sets, topological and uniform spaces as full subcategories, and has forgetful functors $sF\to sSets$, $sF\to Top$, and $sF\to UniformSpaces$ such that the following compositions are identity: $sSets\to sF\to sSets$, $Top\to sF\to Top$, and $UniformSpaces\to sF\to UniformSpaces$. Moreover, this endofunctor seems to have the right adjoint, defined by the usual construction.
Here are some details. The category $sF$ may be thought as the category of simplicial sets with extra structure of topological nature, a notion of smallness. Formally it is just the category of simplicial objects in the category of filters. The endofunctor $HHom ( Hom_{preorders}(-, [0,1]_\leq), Y_.):sF\to sF$ above is the inner hom of sSets equipped with an extra structure motivated by Skorokhod/Levi-Prokhorov convergence. The precise claim is that the geometric realisation of sSets factors as
To gain some intuition, consider $Y_.=\Delta_n=Hom(-,[n])$ the standard simplex. Then by Remark 2.4.1-2 of Grayson the standard geometric simplex is the set of monotone functions $[0.1]\to [n]$
equipped with a metric
reminiscent of Skorokhod metric in probability theory. Now instead of $\Delta_n$ take an arbitrary simplicial set, and rewrite the definition of Skorokhod convergence in terms of the notion of smallness in $sF$.
The geometric realization of a Kan fibration is a Serre fibration.
This is shown in Quillen 68.
This result implies that the geometric realization functor preserves all five classes of maps in a model category: weak equivalences, cofibrations, acyclic cofibrations, fibrations, and acyclic fibrations.
In fact the geometric realization of a Kan fibration is even a Hurewicz fibration (at least relative to a convenient category of spaces in which it lives). This follows from the fact that a Serre fibration between CW-complexes is a Hurewicz fibration; a direct proof along the lines of Quillen’s can be found in Fritch and Piccinini, Theorem 4.5.25.
The previous two sections show that the geometric realization preserves finite limits and fibrations. Since its right adjoint, the singular complex functor $Top \to sSet$, also preserves both (much more trivially), and since all objects of $Top$ are fibrant and the adjunction is simplicially enriched, it follows that the composite $sSet \to Top \to sSet$ is a simplicially enriched fibrant replacement functor on $sSet$ that additionally preserves both finite limits and fibrations.
is the topological space that is the classifying space for $G$-principal bundles (covering spaces), as long as we give $G$ the discrete topology.
geometric realization
singular complex functor?
Daniel Quillen, The geometric realization of a Kan fibration is a Serre fibration. Proc. Amer. Math. Soc. 19 1968 1499–1500. pdf
Fritsch and Piccinini, Cellular Structures in Topology, Cambridge University Press, 1990, ISBN 0521327849
Discussion of sufficient conditions for homotopy geometric realization to be compatible with homotopy pullback (see also at geometric realization of simplicial topological spaces):
D. Anderson, Fibrations and geometric realization , Bull. Amer. Math. Soc. Volume 84, Number 5 (1978), 765-788. (euclid:1183541139)
Charles Rezk, When are homotopy colimits compatible with homotopy base change?, 2014 (pdf, pdf)
Edoardo Lanari, Compatibility of homotopy colimits and homotopy pullbacks of simplicial presheaves (pdf, pdf)
(expanded version of Rezk 14)