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
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
A modelizer is a presentation of the (∞,1)-category of ∞-groupoids, or at least, the homotopy category thereof.
A modelizer is a category $M$ and a subcategory $W$ satisfying these conditions: * $(M, W)$ is a saturated homotopical category, meaning $W$ is precisely the class of morphisms in $M$ that become invertible in the localization $M [W^{-1}]$. * $M [W^{-1}]$ is equivalent to the category of weak homotopy types, i.e. Ho(Top) (with respect to weak homotopy equivalences).
More precisely, it is a category $M$ equipped with a functor $\pi : M \to Ho(Top)$ such that, for $W$ the class of morphisms inverted by $\pi$, the induced functor $M [W^{-1}] \to Ho(Top)$ is an equivalence of categories.
A morphism of modelizers $(M, W) \to (M', W')$ is a functor $F : M \to M'$ such that: * $F$ sends morphisms in $W$ to morphisms in $W'$. * The functor $M [W^{-1}] \to M' [W'^{-1}]$ so induced is an equivalence of categories. * The composite $M \overset{F}{\to} M' \overset{\pi}{\to} Ho(Top)$ is isomorphic to $M \overset{\pi}{\to} Ho(Top)$.
An elementary modelizer is a modelizer whose underlying category is the category of presheaves on a test category, with the weak equivalences the ones described at the linked page.
The main examples turn out to be model categories:
If $A$ is a test category, then there exists a model structure on $[A^{op}, Set]$ that is Quillen-equivalent to the standard model structure on $sSet$.
Last revised on June 19, 2022 at 14:31:08. See the history of this page for a list of all contributions to it.