Geometric realization is the operation that builds from a simplicial set a topological space obtained by interpreting each element in – each abstract -simplex in – as one copy of the standard topological -simplex 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 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.B.: in this article, denotes the ordinal with elements. The corresponding contravariant representable is denoted .)
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 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
This is a special case of a general notion of
There is an obvious functor
which sends the standard cellular shape (the standard cellular globe, simplex or cube, respectively) to the corresponding standard topological shape (for instance the standard -simplex ) with the obvious induced face and boundary maps.
Using this, in cases where can be regarded as enriched over and tensored over a base category , the geometric realization of a presheaf on – e.g., of a globular set, a simplicial set or a cubical set, respectively (when ) – 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 choice 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
In this section we consider topological geometric realization of simplicial sets, which is the best studied and perhaps most significant case.
Each is a CW complex (see lemma 1 below), and so geometric realization takes values in the full subcategory of CW complexes, and therefore in any convenient category of topological spaces, for example in the category of compactly generated Hausdorff spaces. Let be any convenient category of topological spaces, and let denote the inclusion.
For any simplicial set , there is a natural isomorphism , where the coend on the left is computed in .
This is obvious: more generally, if is a diagram and is a full replete subcategory, and if the colimit in of lands in , then this is also the colimit of in . (The dual statement also holds, with limits instead of colimits.)
Below, we let denote the geometric realization when considered as landing in .
We continue to assume is any convenient category of topological spaces. In this section we prove that geometric realization
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 . This gives the correct isomorphism in the case , where we have that ; the product on the right has been “kelleyfied” to the product appropriate for .)
We reiterate that denotes the geometric realization functor considered as valued in a convenient category of spaces, whereas is geometric realization viewed as taking values in .
Let be the underlying-set functor. Then the composite is left exact.
As described at the nLab article on triangulation here, the composite
can be described as the functor
where is the category of intervals (linearly ordered sets with distinct top and bottom). Because every interval, in particular , is a filtered colimit of finite intervals, and because finite intervals are finitely presentable intervals, it follows that 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 or .
If is a monomorphism of simplicial sets, then is a closed subspace inclusion, in fact a relative -complex. In particular, taking , is a -complex.
Any monomorphism in can be seen as the result of iteratively adjoining nondegenerate -simplices. In other words, there is a chain of inclusions , where is a functor from some ordinal (as preorder) that preserves directed colimits, and each inclusion fits into a pushout diagram
where is the inclusion. Now is identifiable as the inclusion , and since preserves pushouts (which are calculated as they are in ), we see by this lemma that is a closed subspace inclusion and evidently a relative CW-complex. By another lemma, it follows that is also a closed inclusion and indeed a relative CW-complex.
The equalizer of a pair of maps in is computed as the equalizer on the level of underlying sets, equipped with the subspace topology. So if
is an equalizer diagram in , then is the equalizer of the pair , , because the underlying function is the equalizer of , on the underlying set level by the preceding theorem, and because is a (closed) subspace inclusion by lemma 1. But this -equalizer lives in the full subcategory , and therefore is the equalizer of the pair , .
As the proof indicates, that realization preserves equalizers is not at all sensitive to whether we use or a convenient category of spaces .
That geometric realization preserves products is sensitive to whether we think of it as valued in or in a convenient category . In particular, the proof uses cartesian closure of in an essential way (in the form that finite products distribute over arbitrary colimits).
First, a small technical result about simplicial sets.
The product of two representables is the colimit of a finite diagram of representables, i.e., is the quotient of a finite coproduct of representables.
We describe a finite collection of monomorphisms (there are many to be exact) which collectively define an epimorphism
The product of ordinals can be pictured as a rectangular grid consisting of rectangles, and an order-preserving monomorphism can be pictured as a path of length along the grid, traveling north and east. A maximal such path is of length , traveling from to say, and is specified by a choice of which among many steps are steps north. These maximal paths give the monos .
To show the collective map is epic, we must show that any map lifts through . By factoring as an epi followed by a mono, it is sufficient to prove this in the case where is monic. In that case, is described by a path of length , which is embedded in a maximal path . Such an embedding is given by a monic , whereupon the composite
(where is the inclusion) provides the desired lift of , thus completing the proof.
The canonical map
is a homeomorphism.
The canonical map is continuous, and a bijection at the underlying set level by theorem 1. The codomain is the compact Hausdorff space , and the domain is also compact: by lemma 2, and using the fact that realization preserves finite colimits, the left side is the topological quotient of a coproduct of finitely many simplices, hence compact. But a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.
The key properties of needed for this subsection are (1) the fact it is compact Hausdorff, and (2) the order relation on the interval defines a closed subset of . These properties ensure that the affine -simplex is itself compact Hausdorff, so that the proof of lemma 3 goes through. The point is that in place of , we can really use any interval that satisfies these properties, thus defining an -based geometric realization instead of the standard (-based) geometric realization being developed here.
The functor preserves products.
The proof is purely formal. Let and 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 preserves colimits in its separate arguments (i.e., the fact that the nice category is cartesian closed), and the remaining lines used the fact that preserves colimits, and also products of representables by lemma 3.