on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
equivalences in/of $(\infty,1)$-categories
The model category structure on the category $SSet^+/S$ of marked simplicial sets over a given simplicial set $S$ is a presentation for the (∞,1)-category of Cartesian fibrations over $S$. Every object is cofibrant and the fibrant objects of $SSet^+/S$ are precisely the Cartesian fibrations over $S$.
Notably for $S = {*}$ this is a presentation of the (∞,1)-category of (∞,1)-categories: as a plain model category this is Quillen equivalent to the model structure for quasi-categories, but it is indeed an $sSet_{Quillen}$-enriched model category (i.e. enriched over the ordinary model structure on simplicial sets that models ∞-groupoids).
The $(\infty,1)$-categorical Grothendieck construction that exhibits the correspondence between Cartesian fibrations and (∞,1)-presheaves is in turn modeled by a Quillen equivalence between the model structure on marked simplicial over-sets and the projective global model structure on simplicial presheaves.
Marked simplicial sets are simplicial sets with a little bit of extra structure: a marking that remembers which edges are supposed to be Cartesian morphisms.
A marked simplicial set is
a pair $(S,E)$ consisting of
a simplicial set $S$
and a subset $E \subset S_1$ of edges of $S$, called the marked edges,
such that
A morphism $(S,E) \to (S',E')$ of marked simplicial sets is a morphism $f : S \to S'$ of simplicial sets that carries marked edges to marked edges in that $f(E) \subset E'$.
The category of marked simplicial sets is denoted $sSet^+$.
for $S$ a simplicial set let
$S^\flat$ or $S^{min}$ be the minimally marked simplicial set: only the degenerate edges are marked;
$S^\sharp$ or $S^{max}$ be the maximally marked simplicial set: every edge is marked.
for $p : X \to S$ a Cartesian fibration of simplicial sets let
Simple as the above definition is, for seeing some of its properties it is useful to think of $sSet^+$ in a more abstract way.
Let $\Delta^+$ be the category defined as the simplex category $\Delta$, but with one more object $[1^+]$ that factors the unique morphism $[1] \to [0]$ in $\Delta$
Equip this category with a coverage whose only non-trivial covering family is $\{p : [1] \to [1]^+\}$.
The category $sSet^+$ is the quasi-topos of separated presheaves on $\Delta^+$:
A presheaf $X : (\Delta^+)^{op} \to Set$ is separated precisely if the morphism
is a monomorphism, hence if $X_{1^+}$ is a subset of $X_1$. By functoriality this subset contains all the degenerate 1-cells
Therefore we may naturally identify $X$ as a simplicial set equipped with a subset of $X_1$ that contains all degenerate 1-cells.
Moreover, a morphism of separated preseheaves on $\Delta^+$ is by definition just a natural transformation between them, which means it is under this interpretation precisely a morphism of simplicial sets that respects the marked 1-simplices.
Notice that $sSet^+$ is a genuine quasi-topos:
$sSet^+$ is not a topos.
The canonical morphisms $X^\flat \to X^\sharp$ are monomorphisms and epimorphisms, but not isomorphisms. Therefore $sSet^+$ is not a balanced category, hence cannot be a topos.
The category $sSet^+$ is a cartesian closed category.
This is an immediate consequence of the above observation that $sSet^+$ is a quasitopos. But it is useful to spell out the Cartesian closure in detail.
By the general logic of the closed monoidal structure on presheaves we have that $PSh(\Delta^+)$ is cartesian closed. It remains to check that if $X,Y \in PSh(\Delta^+)$ are marked simplicial sets in that $X_{1^+} \to X_1$ is a monomorphism and similarly for $Y$, that then also $Y^X$ has this property.
We find that the marked edges of $Y^X$ are
and the morphism $(Y^X)_{1^+} \to (Y^X)_1$ sends $X \times [1^+] \stackrel{\eta}{\to} Y$ to
Now, by construction, every non-identity morphism $U \to [1^+]$ in $\Delta^+$ factors through $U \to [1]$, which implies that if the components of $p^* \eta_1$ and $p^* \eta_2$ coincide on $U \neq [1^+]$, then already the components of $\eta_1$ and $\eta_2$ on $U$ coincided. By assumption on $X$ the values of $\eta_1$ and $\eta_2$ on $U = [1^+]$ are already fixed, due to the inclusion $X_{1^+} \times [1^+]_{1^+} \hookrightarrow X_{1} \times [1^+]_{1}$. Hence $p^*$ is injective, and so $Y^X$ formed in $PSh(\Delta^+)$ is itself a marked simplicial set.
For $X$ and $Y$ marked simplicial sets let
$Map^\flat(X,Y)$ be the simplicial set underlying the cartesian internal hom $Y^X \in sSet^+$
$Map^\sharp(X,Y)$ the simplicial set consisting of all simplices $\sigma \in Map^\flat(X,Y)$ such that every edge of $\sigma$ is a marked edge of $Y^X$.
These mapping complexes are characterized by the fact that we have natural bijections
and
for $K \in sSet$ and $X,Y \in sSet^+$. In particular
and
In words we have
The $n$-simplices of the internal hom $Y^X$ are simplicial maps $X \times \Delta[n] \rightarrow Y$ such that when you restrict $X_1 \times \Delta[n]_1 \rightarrow Y_1$ to $E \times \Delta[n]_0$ (where $E$ is the set of marked edges of $X$), this morphism factors through the marked edges of $Y$.
The marked edges of $Y^X$ are those simplicial maps $X \times \Delta[1] \rightarrow Y$ such that the restriction of $X_1 \times \Delta[1]_1 \rightarrow Y_1$ to $E \times \Delta[1]_1$ factors though the marked edges of $Y$. In the presence of the previous condition, this says that when you apply the homotopy $X \times \Delta[1] \rightarrow Y$ to a marked edge of $X$ paired with the identity at $[1]$, the result should be marked.
We generalize all this notation from $sSet^+$ to the overcategory $sSet^+/S := sSet^+/(S^\sharp)$ for any given (plain) simplicial set $S$, by declaring
and
to be the subcomplexes spanned by the cells that respect that map to the base $S$.
Let $Y \to S$ be a Cartesian fibration of simplicial sets, and $X^\natural$ as above the marked simplicial set with precisely the Cartesian morphisms marked.
Then
$Map_S^\flat(X,Y^\natural)$ is an quasi-category;
$Map_S^\sharp(X, Y^\natural)$ is its core, the maximal Kan complex inside it.
This is HTT, remark 3.1.3.1.
The $n$-cells of $Map_S^\flat(X,Y^\natural)$ are morphisms $X \times \Delta[n]^\flat \to Y^\natural$ over $S$. This means that for fixed $x \in X_0$, $\Delta[n]$ maps into a fiber of $Y\to S$. But fibers of Cartesian fibrations are fibers of inner fibrations, hence are quasi-categories.
Similarly, the $n$-cells of $Map_S^\sharp(X,Y^\natural)$ are morphisms $X \times \Delta[n]^\sharp \to Y^\natural$ over $S$. Again for fixed $x \in X_0$, $\Delta[n]$ maps into a fiber of $Y\to S$, but now only hitting Cartesian edges there. But (as discussed at Cartesian morphism), an edge over a point is Cartesian precisely if it is an equivalence.
We have a sequence of adjoint functors
Observe that weak equivalences in the model structure for quasi-categories may be characterized as follows.
A morphism $f : C \to D$ between simplicial sets that are quasi-categories is a weak equivalence in the model structure for quasi-categories precisely if the following equivalent coditions hold:
For every simplicial set $K$, the morphism $sSet(K,f) : sSet(K,C) \to sSet(K,D)$ is a weak equivalence in the model structure for quasi-categories.
For every simplicial set $K$, the morphism $Core(sSet(K,f)) : Core(sSet(K,C)) \to Core(sSet(K,D))$ on the cores, the maximal Kan complexes inside, is a weak equivalence in the standard model structure on simplicial sets, hence a homotopy equivalence.
This is HTT, lemma 3.1.3.2.
This may be taken as motivation for the following definition.
For every Cartesian fibration $Z \to S$, we have that
is a quasi-category and
is the maximal Kan complex inside it.
A morphism $p : X \to Y$ in $sSet^+/S$ is a Cartesian equivalence if for every Cartesian fibration $Z$ we have
Or equivalently:
This is HTT, prop. 3.1.3.3 with HTT, remark 3.1.3.1.
Let
be a morphism in $sSet/S$ such that both vertical maps to $S$ are Cartesian fibrations. Then the following are equivalent:
$p$ is a homotopy equivalence.
The induced morphism $X^\natural \to Y^\natural$ in $sSet^+/S$ is a Cartesian equivalence.
The induced morphism on each fiber $X_s \to Y_{p(s)}$ is a weak equivalence in the model structure for quasi-categories.
This is HTT, lemma 3.1.3.5.
The model structure on marked simplicial over-sets $Set^+/S$ over $S \in SSet$ – also called the Cartesian model structure since it models Cartesian fibrations – is defined as follows.
(Cartesian model structure on $sSet^+/S$)
The category $SSet^+/S$ of marked simplicial sets over a marked simplicial set $S$ carries a structure of a proper combinatorial simplicial model category defined as follows.
The SSet-enrichment is given by
A morphism $f : X \to X'$ in $SSet^+/S$ of marked simplicial sets is
a cofibration precisely if underlying morphism of simplicial sets is a cofibration in the standard model structure on simplicial sets (i.e. a monomorphism).
(def. 3.1.2.2 of HTT)
a weak equivalences precisely if it is a Cartesian equivalence, as defined above.
The model structure is proposition 3.1.3.7 in HTT. The simplicial enrichment is corollary 3.1.4.4.
Using $Map_S^\flat(X,Y)$ for the mapping objects makes $sSet^+/S$ a $sSet_{Joyal}$-enriched model category (i.e. enriched in the model structure for quasi-categories). This is HTT, remark 3.1.4.5.
Notice that trivially every object in this model structure is cofibrant. The following proposition shows that the above model structure indeed presents the $(\infty,1)$-category $CartFib(S)$ of Cartesian fibrations.
An object $p : X \to S$ in $sSet^+/S$ is fibrant with respect to the above model structure precisely if it is isomorphic to an object of the form $Y^\natural$, for $Y \to S$ a Cartesian fibration in sSet.
This is HTT, prop. 3.1.4.1.
In particular, the fibrant objects of $sSet^+ \cong sSet^+/*$ are precisely the quasicategories in which the marked edges are precisely the equivalences. Note that the Cartesian model structure on $sSet^+/S$ is not the model structure on an over category induced on $sSet^+/S$ from the Cartesian model structure on $sSet^+$!
(coCartesian model structure on $sSet^+/S$)
There is another such model structure, with Cartesian fibrations replaced everywhere by coCartesian fibrations.
A class of morphisms with left lifting property again some class of fibrations is usually called anodyne . For instance a left/right/inner anodyne morphism of simplicial sets is one that has the left lifting property against all left/right fibrations or inner fibrations, respectively.
The class of marked anodyne morphisms in $sSet^+$ as defined in the following is something that comes close to having the left lifting property against all Cartesian fibrations. It does not quite, but is still useful for various purposes.
The collection of marked anodyne morphisms in $SSet^+/S$ is the class of morphisms $An^+ = LLP(RLP(An^+_0))$ where the generating set $An^+_0$ consists of
for $0 \lt i \lt n$ the minimally marked horn inclusions
for $i = n$ the horn inclusion with the last edge marked:
where $\mathcal{E}$ is the union of all degenerate edges in $\Delta[n]$ together with the edge $\Delta^{\{n-1,n\}} \to \Delta[n]$.
the inclusion
for every Kan complex $K$ the morphism $K^\flat \to K^\sharp$.
The crucial property of marked anodyne morphisms is the following characterization of morphisms that have the right lifting property with respect to them.
A morphism $p : X \to S$ in $SSet^+$ has the right lifting property with respect to the class $An^+$ of marked anodyne maps precisely if
$p$ is an inner fibration
an edge $e$ of $X$ is marked precisely if it is a $p$-Cartesian morphism and $p(e)$ is marked in $S$
for every object $y$ of $X$ and every marked edge $\bar e : \bar x \to p(y)$ in $S$ there exists a marked edge $e : x \to y$ of $X$ with $p(e) = \bar e$.
This is HTT, prop. 3.1.1.6
Thus, if $(X, E_X) \to (S,E_S)$ is a morphism in $sSet^+$ with RLP against marked anodyne morphisms, then its underlying morphism $X\to S$ in $sSet$ is almost a Cartesian fibration: it may fail to be such only due to missing markings in $E_S$.
However, if all morphisms in $S$ are marked, then $(X,E_X) \to S^\sharp$ has the RLP against marked anodyne morphisms precisely when the underlying morphism $X\to S$ is a Cartesian fibration and exactly the Cartesian morphisms are marked in $X$, $(X,E_X) = X^\natural$ — in other words, precisely if it is a fibrant object in the model structure on $sSet^+/S$.
See also HTT, remark. 3.1.1.11.
The following stability property of marked anodyne morphisms is important in applications. Recall that a cofibration in $sSet^+$ is a morphism whose underlying morphism in sSet is a monomorphism.
(stability under smash product with cofibrations)
Marked anodyne morphisms are stable under “smash product” with cofibrations:
for $f : X \to X'$ marked anodyne, and $g : Y \to Y'$ a cofibration, the induced morphism
out of the pushout in $sSet^+$ is marked anodyne.
This is HTT, prop. 3.1.2.3.
The Joyal model structure for quasi-categories $sSet_{Joyal}$ is an enriched category enriched over itself. So it is not a simplicial model category in the standard sense, which means $sSet_{Quillen}$-enriched.
Indeed, the full sSet-enriched subcategory $(sSet_{Joyal})^\circ$ on fibrant-cofibrant objects is a model for the (∞,2)-category (∞,1)Cat of (∞,1)-categories. For many applications it is more convenient to work just with the (∞,1)-category of (∞,1)-categories inside that, obtained by taking in each hom-object quasi-category the maximal Kan complex.
The resulting (∞,1)-category should have a presentation by a simplicial model category. And the model structure on marked simplicial sets does accomplish this.
Marked simplicial sets are introduced in section 3.1 of
The model structure on marked simplicial oversets is described in section 3.1.3