model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
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
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Philosophically, model structures allow one to localize a category at a particular collection of weak equivalences, which one would like to formally invert.
For topological spaces, there are two natural candidates for the collection $W$ of weak equivalences:
and the homotopy equivalences.
Both of these have accompanying model structures. Interestingly, these two model structures can also be combined to form what’s known as the mixed model structure.
All of these model structures exist not only on the category of all topological spaces, but also on most convenient categories of topological spaces. Using a nice category instead is sometimes important, such as if we want the model structure to be monoidal.
The first, and most prevalent, model category structure on Top, called the classical model structure on topological spaces (Quillen67, II.3) has
weak equivalences are the weak homotopy equivalences;
fibrations are the Serre fibrations, maps which have the right lifting property with respect to all inclusions of the form $i_0 : D^n \hookrightarrow D^n \times I$ that include the $n$-disk as $D^n \times \{0\}$.
cofibrations are the “retracts of relative cell complexes”.
This is a cofibrantly generated model category. The cofibrations $C$ are generated by the set of boundary inclusions $S^{n-1} \hookrightarrow D^n$ for all $n \in \mathbb{N}$ in the sense that they are the smallest saturated class containing these morphisms. As a consequence, of Quillen’s small object argument, all cofibrations have the form described above, where a relative cell complex is a transfinite composite of pushouts of coproducts of these generating maps.
This model structure is sometimes called the Quillen model structure on topological spaces or $q$-model structure on Top.
A second model structure – the Strøm model structure – has
weak equivalences are the homotopy equivalences;
fibrations are the Hurewicz fibrations, which are defined to be maps that have the right lifting property with respect to all inclusions $i_0 : A \hookrightarrow A \times I$ for any topological space $A$.
cofibrations are determined by these classes and are called the closed Hurewicz cofibrations.
This model structure is sometimes called the Hurewicz model structure, since it uses Hurewicz fibrations and cofibrations, or also the $h$-model structure, where $h$ can stand for either “Hurewicz” or “homotopy equivalence.” However, it is also sometimes called the Strøm model structure, since it was first proven to exist by Arne Strøm.
From the definitions, Hurewicz fibrations are necessarily Serre fibrations. It is well-known that homotopy equivalences are weak homotopy equivalences. If we write $(C_1, F_1, W_1)$ for the classes of the first model structure and $(C_2, F_2, W_2)$ for the classes of the second, we have $W_2 \subset W_1$ and $F_2 \subset F_1$.
In general given two model structures with these inclusions, we get a third mixed model structure $(C_m, F_2, W_1)$ where the cofibrations $C_m$ are determined by the other two classes.
On topological spaces, this model structure has
weak equivalences the weak homotopy equivalences
fibrations the Hurewicz fibrations;
cofibrant spaces (the m-cofibrant spaces) are precisely those spaces that are homotopy equivalent to CW complexes.
For the discussion of the homotopy theory given by the model structure on topological spaces, it is necessary or at least useful to pass to convenient categories of topological spaces.
Write
$k Top \hookrightarrow Top$ for the full subcategory of k-spaces;
$CGTop \hookrightarrow Top$ for the full subcategory of compactly generated spaces.
(model structure on compactly generated topological spaces)
There is a model category structure $kTop_{Qu}$ on $k Top$ in which a morphism is a cofibration, fibration or weak equivalence, respectively, precisely if it is so under the inclusion $k Top \hookrightarrow Top$ in the classical model structure on topological spaces. This inclusion is the left adjoint in a Quillen equivalence
This appears for instance as (Hovey, theorem 2.4.23)
There is a model category structure $CGTop_{Quillen}$ on $CGTop$ in which a morphism is a cofibration, fibration or weak equivalence, respectively, precisely if it is so under the inclusion $CGTop \hookrightarrow Top$. And this inclusion is the right adjoint in a Quillen equivalence
(e.g. Hovey 1999, Thm. 2.4.25)
To appreciate the following, notice that $Top_{Qu}$ is not a monoidal model category, because Top itself is not (cartesian) closed.
Both $k Top_{Qu}$ and $CGTop_{Qu}$ are symmetric monoidal model categories.
This appears as (Hovey, prop. 4.2.11).
In fact $CGTop_{Quillen}$ is a cartesian closed model category.
(This is briefly mentioned in Berger-Moerdijk 2003, Sec. 3.3.2.)
The Quillen model structure $Top_{Qullen}$ is Quillen equivalent to the standard (Quillen) model structure on simplicial sets via the total singular complex and geometric realization functors.
Since the standard model structure on simplicial sets is a presentation of the (∞,1)-category ∞Grpd of ∞-groupoids realized as Kan complexes, this identifies topological spaces with ∞-groupoids in an (∞,1)-categorical sense. Notably it says that every $\infty$-groupoid is, up to equivalence, the fundamental ∞-groupoid of some topological space.
This statement is called the homotopy hypothesis (which here is a theorem). See there for more details.
The identity functor constitutes a Quillen adjunction
between the Quillen model structure and the Strom model structure on $Top$. Here $Top_{Strom} \to Top_{Quillen}$ is the right Quillen functor.
The original “Quillen” or “q-” model structure is due to
An expository, concise and comprehensive writeup is in
Standard textbooks references include
Mark Hovey, Model Categories, Mathematical Surveys and Monographs, Volume 63, AMS (1999) (ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books)
Philip Hirschhorn, Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (ISBN:978-0-8218-4917-0, pdf toc, pdf)
Kate Ponto, Peter May, section 17 of More concise algebraic topology (pdf)
For the “Hurewicz,” “h-” or “Strøm model structure”:
For the “m-” or “mixed model structure”:
The generalization to the model structure on topological operads is due to
Last revised on November 30, 2021 at 13:22:28. See the history of this page for a list of all contributions to it.