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
Let $A$ be a small category, and let $Psh(A)=Set^{A^{op}}$ be the category of presheaves on $A$. Since $Psh(A)$ is a Grothendieck topos, it has a unique subobject classifier, $L$.
Let $\mathbf{0}$ and $\mathbf{1}$ denote the initial object and terminal object, respectively, of $Psh(A)$. The terminal presheaf $\mathbf{1}$ has two distinguished subobjects $\mathbf{0}\hookrightarrow \mathbf{1}$ and $\mathbf{1}\hookrightarrow \mathbf{1}$, which correspond to global points $\lambda^0,\lambda^1\in L(\mathbf{1})=Hom(\mathbf{1},L)$ of the subobject classifier.
The triple $\mathfrak{L}=(L,\lambda^0,\lambda^1)$ is called the Lawvere interval for the topos $Psh(A)$. This object determines a cylinder functor given by taking the cartesian product with $L$, called the Lawvere cylinder.
By Prop. below, the Lawvere interval $\mathfrak{L}$ may be regarded as the universal cylinder object for Cisinski model structures on presheaf toposes.
The subobject classifier in any topos is an injective object, whence the Lawvere interval $L$ (Def. ) is a fibrant resolution of the terminal object in any Cisinski model structure on $Psh(A)$.
Recall that for $L$ to be an injective object means that every solid span as below, where the vertical map is a monomorphism, admits a dashed lifting as shown:
Since in a Cisinski model structure, by definition, the monomorphisms are precisely the cofibrations, injective objects here are equivalently those for which the terminal map $L \to \ast$ is an acyclic cofibration.
Hence assuming the solid diagram above, we show the existence of $\underline{f}$:
Here $f \colon B \to L$ classifies a subobject of $B$, which we denote $C \hookrightarrow B$. The point now is that, with $B \hookrightarrow A$ being a monomorphism, the composite
exhibits $C$ also as a subobject of $A$, which as such is classified by some map $\underline{f} \colon A \to L$, and we claim that this serves as the desired extension. To see that indeed this $\underline{f}$ makes the above triangle commute, consider the following commuting diagram:
Here
the right rectangle is the pullback square that witnesses $\underline{f}$ as the classifying map of $C \hookrightarrow A$, by definition,
the left rectangle is the fiber product $B \cap_A C$ computed via the pasting law for pullbacks to be isomorphic to $C$, using that $C$ is already a subobject of $A$ (which gives the factorization in the middle) and then using (for the two squares on the left) that:
the fiber product of a monomorphism with itself is its domain (this Prop.),
isomorphisms are preserved by pullback (this Prop.).
This implies, again by the pasting law, that the total square is a pullback, hence that the total bottom map classifies $C$. But $C$ was defined to be classified by $f$, and so the uniqueness of subobject classifying maps implies that the total bottom map equals $f$, which was to be shown.
Given any small set of monomorphisms in $Psh(A)$, there exists the smallest Cisinski model structure for which those monomorphisms are acyclic cofibrations.
By applying a theorem of Denis-Charles Cisinski. (…)
Suppose $A=\Delta$ is the simplex category, and let $S$ consist only of the inclusion $\{1\}:\Delta^0\to\Delta^1$. Applying Cisinski’s anodyne completion of $S$ by Lawvere’s cylinder $\mathbf{\Lambda}_\mathfrak{L}(S,M)$, we get exactly the contravariant model structure on the category of simplicial sets.
Last revised on June 11, 2023 at 11:20:15. See the history of this page for a list of all contributions to it.