homotopy theory, (∞,1)-category theory, homotopy type theory
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equivalences in/of $(\infty,1)$-categories
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
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hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
Where an ordinary category has a hom-set, an (∞,1)-category has an ∞-groupoid of morphisms between any two objects, a hom-space.
There are several ways to present an (∞,1)-category $\mathbf{C}$ by an ordinary category $C$ equipped with some extra structure: for instance $C$ may be a category with weak equivalences or a model category or even a simplicial model category. In all of these presentations, given two objects $X, Y \in C$, there is a way to construct a simplicial sets $\mathbb{R}\mathbf{C}(X,Y)$ that presents the hom-∞-groupoid $\mathbf{C}(X,Y)$. This simplicial set – or rather its homotopy type – is called the derived hom space or homotopy function complex and denoted $\mathbf{R}Hom(X,Y)$ or similarly.
There are many ways to present an (∞,1)-category by category theoretic data, and for each of these there are corresponding tools for explicitly computing the derived hom spaces.
The most basic data is that of a category with weak equivalences. Here the derived hom spaces can be constructed in terms of zig-zags of morphisms by a process called simplicial localization. This we discuss below in For a category with weak equivalences.
Particularly useful extra structure on a category with weak equivalences that helps with computing the derived hom spaces is the structure of a model category. Using this one can construct simplicial resolutions of objects – called framings – that generalize cylinder objects and path objects, and then construct the derived hom spaces in terms of direct morphisms between these resolutions. This we discuss below in For a model category.
Still a bit more helpful structure on top of a bare model category is that of a simplicial model category. Here, after a choice of cofibrant and fibrant resolutions of opjects, the derived hom spaces are given already by the sSet-hom objects. This we discuss below in For a simplicial model category.
Let $(C,W \subset Mor(C))$ be a category with weak equivalences.
Fix $n \in \mathbb{N}$. For $X,Y \in Obj(C)$, define a category $wMor_C^n(X,Y)$
whose objects are zig-zags of morphisms in $C$ of length $n$
such that each morphism going to the left, $X_{2k}\leftarrow X_{2k +1}$, is a weak equivalence, an element in $W$;
morphisms between such objects $(X,X_i,Y) \to (X',X'_i,Y')$ are collections of weak equivalences $(X_i \to X'_i)$ for all $0 \lt i \lt n$ such that all triangles and squares commute.
Write $N(wMor_C^n(X,Y))$ for the nerve of this category, a simplicial set.
The hammock localization $L_W^H C$ of $C$ is the simplicially enriched category with objects those of $C$ and hom-objects given by the colimit over the length of these hammock hom-categories
The Kan fibrant replacement of this simplicial set is the derived hom-space between $X$ and $Y$ of the $(\infty,1)$-category modeled by $(C,W)$.
The derived hom spaces of a model category $C$ may always be computed in terms of simplicial resolutions with respect to the Reedy model structure $[\Delta^{op}, C]_{Reedy}$. These resolutions are often called framings (Hovey). These constructions are originally due to (Dwyer-Hirschhorn-Kan).
Let $C$ be any model category.
There is an adjoint triple
where
$const X : [n] \mapsto X$;
$ev_0 X_\bullet = X_0$;
$X^{\times^\bullet} : [n] \mapsto X^{\times^{n+1}}$.
For $X \in C$ fibrant, $X^{\times^\bullet}$ is fibrant in the Reedy model structure $[\Delta^{op}, C]_{Reedy}$.
The matching morphisms are in fact isomorphisms.
Let $C$ be a model category.
For $X \in C$ any object, a simplicial frame on $X$ is a factorization of $const X \to X^{\times^\bullet}$ into a weak equivalence followed by a fibration in the Reedy model structure $[\Delta^{op}, C]_{Reedy}$.
A right framing in $C$ is a functor $(-)_\bullet : C \to [\Delta^{op}, C]$ with a natural isomorphism $(X)_0 \simeq X$ such that $X_\bullet$ is a simplicial frame on $X$.
Dually for cosimplicial frames.
This appears as (Hovey, def. 5.2.7).
By remark 1 a simplicial frame $X_\bullet$ in the above is in particular fibrant in $[\Delta^{op}, C]_{Reedy}$.
For $X \in C$ cofibrant and $A \in C$ fibrant, there are weak equivalences in $sSet_{Quillen}$
(where in the middle we have the diagonal of the bisimplicial set $Hom(X^\bullet, A_\bullet)$).
This appears as (Hovey, prop. 5.4.7).
Either of these simplicial sets is a model for the derived hom-space $\mathbb{R}Hom(X,A)$.
By developing these constructions further, one obtains a canonical simplicial model category-resolution of (left proper and combinatorial) model categories $C$, such that the simplicial resolutions given by framings are just the cofibrant$\to$fibrant $sSet$-hom objects as discussed below.
This is discussed at Simplicial Quillen equivalent models.
Let $C$ be a model category, let $\mathrm{c}_\mathrm{w} C$ be the full subcategory of $[\Delta, C]$ spanned by the cosimplicial objects whose coface and codegeneracy operators are weak equivalences, and let $\mathrm{s}_\mathrm{w} C$ be the full subcategory of $[\Delta^{op}, C]$ spanned by the simplicial objects whose face and degeneracy operators are weak equivalences.
We describe here in more detail properties of hom-objects in a simplicial model category for the case that the domain objects are cofibrant and the codomain objects are fibrant.
The crucial axiom used for this is the axiom of an enriched model category $C$ which says that
the tensor operation
is a Quillen bifunctor;
or equivalently that for $X \to Y$ a cofibration and $A \to B$ a fibration the induced morphism
is a fibration, which is acyclic if either $X \to Y$ or $A \to B$ is.
First of all the first statement directly implies that for $\emptyset \in C$ the initial object and $A \in C$ any object, the simplicial set $C(\emptyset,A) = {*}$ is the terminal simplicial set, because for any simplicial set $S$
where we use that the tensor Quillen bifunctor is required to respect colimits and that the empty colimit is the initial object. (All equality signs here denote isomorphisms, to distinguish them from weak equivalences.)
Similarly one has for all $X$ that $C(X,{*}) = {*}$.
Using this, the second equivalent form of the enrichment axiom has as a special case the following statement.
In a simplicial model category $C$, for $X \in C$ cofibrant and $A \in C$ fibrant, the simplicial set $C(X,A)$ is a Kan complex.
We apply the enriched model category axiom to the cofibration $\emptyset \to X$ and the fibration $A \to {*}$ to obtain a fibration
The right hand is the pullback of the terminal simplicial set ${*} = \Delta^0$ to itself, hence is itself the point. So we have a fibration $C(X,A) \to {*}$ and $C(X,A)$ is a fibrant object in the standard model structure on simplicial sets, hence a Kan complex. .
In a simplicial model category $C$, for $X \in C$ cofibrant and $f : A \to B$ a fibration, the morphism of simplicial sets $C(X,f) : C(X,A) \to C(X,B)$ is a Kan fibration that is a weak homotopy equivalence if $f$ is acyclic.
Dually, for $i : X \to Y$ a cofibration and $A$ fibrant, the morphism $C(i,A) : C(X,A) \to C(Y,A)$ is a cofibration of simplicial sets.
This is as before. Explicity, consider the first case, the second one is the formal dual of that:
We enter the enrichment axiom with the morphisms $\emptyset \to X$ and $A \to B$ and find for the required pullback that
and hence that $C(X,A) \to C(X,B)$ is, indeed, a fibration, which is acyclic if $A \to B$ is.
Let $C$ be a simplicial model category.
Then for $X$ a cofibant object and
a weak equivalence between fibrant objects, the enriched hom-functor
is a weak homotopy equivalence of Kan complexes.
Similarly, for $A$ a fibrant object and $j : X \stackrel{\simeq}{\to} Y$ a weak equivalence between cofibrant objects, the morphism
is a weak homotopy equivalence of Kan complexes.
The second case is formally dual to the first, so we restrict attention to the first one.
By the above, the axioms of an enriched model category ensure that the above statement is true when $f$ is in addition a fibration. So we reduce the situation to that case.
This is possible because both $A$ and $B$ are assumed to be fibrant. This allows to apply the factorization lemma that is described in great detail at category of fibrant objects. By this lemma, for every morphism $f : A \to B$ between fibrant objects there is a commutative diagram
Since $f$ is assumed a weak equivalence it follows by 2-out-of-3 that $\mathbf{E}_f B$ is also a weak equivalence.
Therefore by the above properties of simpliciall enriched categories we obtain a span of acyclic fibrations of Kan complexes
By the Whitehead theorem every weak equivalence of Kan complexes is a homotopy equivalence, hence there is a weak equivalence
that is homotopic to our $C(X,f)$. Therefore this is also a weak equivalence.
Let $C$ be a model category. We discuss how its simplicial function complexes from prop. 2 are related to the simplicial localization from def. 1 and def. 2.
Suppose now that $Q : C \to C$ is a cofibrant replacement functor and $R : C \to C$ a fibrant replacement functor, $\Gamma^\bullet : C \to (cC)_c$ a cosimplicial resolution functor and $\Lambda_\bullet : C \to (sC)_f$ a simplicial resolution functor in the model category $C$.
(Dwyer–Kan)
There are natural weak equivalences between the following equivalent realizations of this SSet hom-object:
The top row weak equivalences are those of prop. 2
There is also an explicit simplicial construction of the derived hom spaces for a homotopical category that is equipped with the structure of a category of fibrant objects. This is described in (Cisinksi 10) and (Nikolaus-Schreiber-Stevenson 12, section 3.6.2).
For $C$ a simplicial model category and $X$ an object, the delooping of the automorphism ∞-group
has the homotopy type of the component on $X$ of the nerve $N(C_W)$ of the subcategory of weak equivalences:
The equivalence is given by a finite sequence of zig-zags and is natural with respect to sSet-enriched functors of simplicial model categories that preserve weak equivalences and send a fibrant cofibrant model for $X$ again to a fibrant cofibrant object.
This is Dwyer-Kan 84, 2.3, 2.4.
For $C$ a model category, the simplicial set $N(C_W)$ is a model for the core of the (∞,1)-category determined by $C$.
That core, like every ∞-groupoid is equivalent to the disjoint union over its connected components of the deloopings of the automorphism $\infty$-groups of any representatives in each connected component.
hom-space, derived hom-space
homotopy | cohomology | homology | |
---|---|---|---|
$[S^n,-]$ | $[-,A]$ | $(-) \otimes A$ | |
category theory | covariant hom | contravariant hom | tensor product |
homological algebra | Ext | Ext | Tor |
enriched category theory | end | end | coend |
homotopy theory | derived hom space $\mathbb{R}Hom(S^n,-)$ | cocycles $\mathbb{R}Hom(-,A)$ | derived tensor product $(-) \otimes^{\mathbb{L}} A$ |
For some original references by William Dwyer and Dan Kan see simplicial localization. For instance
The theory of framings is due to
and in parallel section 5 of
and in sections 16, 17 of
A useful quick review of the interrelation of the various constructions of derived hom spaces is page 14, 15 of
Discussion of derived hom spaces for categories of fibrant objects is in
and section 3.6.2 of