equivalences in/of $(\infty,1)$-categories
homotopy theory, (∞,1)-category theory, homotopy type theory
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Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
An $n$-truncated ∞-groupoid is an n-groupoid.
An $n$-truncated topological space is a homotopy n-type: all homotopy groups above degree $n$ are trivial.
An $n$-truncated object in a general (∞,1)-category is an object such that all hom-∞-groupoids into it are $n$-truncated.
If an object in an (∞,1)-topos_ is $k$-truncated for any (possibly large) $k$, then it is $n$-truncated precisely if all its categorical homotopy groups above degree $n$ are trivial.
The complementary notion of $n$-truncated object is that of an n-connected object of an (∞,1)-category.
($n$-truncated $\infty$-groupoid)
An ∞-groupoid $A \in \infty Grpd$ is $n$-truncated for $n \in \mathbb{N}$ if it is an n-groupoid:
Precisely: in the model of ∞-groupoids given by Kan complexes $A$ is $n$-truncated if the simplicial homotopy groups $\pi_k(A,x)$ are trivial for all $x$ and all $k \gt n$.
It makes sense for the following to adopt the convention that $A$ is called
$(-1)$-truncated if it is empty or contractible – this is a (-1)-groupoid.
$(-2)$-truncated if it is non-empty and contractible – this is a (-2)-groupoid.
(following HTT, p. 6).
To generalize this, let now $C$ be an arbitrary (∞,1)-category. For $X,A$ objects in $C$ write $C(X,A) \in$ ∞ Grpd for the (∞,1)-categorical hom-space (if $C$ is given as a simplicially enriched category then this is just the SSet-hom-object which is guaranteed to be a Kan complex).
Using this, it shall be useful to slightly reformulate the above as follows:
An ∞-groupoid $A$ is $n$-truncated precisely for all other ∞-groupoids $X$ the hom-$\infty$-groupoid $\infty Grpd(X,A)$ is $n$-truncated.
In categorical terms this just says that (∞,k)-transformation between $X$ and $A$ whose components a k-morphisms in $A$ cannot be nontrivial for $k \gt n$ if there are no nontrivial k-morphisms with $k \gt n$ in $A$.
Using this fact we can transport the notion of $n$-truncation to any (∞,1)-category by testing it on hom-∞-groupoids:
($n$-truncated object in an $(\infty,1)$-category)
An object $A \in C$ of an (∞,1)-category $C$ is $n$-truncated, for $n \in \mathbb{N}$, if for all $X \in C$ the hom-∞-groupoid $C(X,A)$ is $n$-truncated.
This is HTT, def. 5.5.6.1.
Some terminology:
A 0-truncated object is also called discrete . Notice that this is categorically discrete as in discrete category, not discrete in the sense of topological spaces. An object in an (∞,1)-topos is discrete in this sense if, regarded as an ∞-groupoid with extra structure it has only trivial morphisms.
By the above convention on (-2)-truncated $\infty$-groupoids, it is only the terminal objects of $C$ that are (-2)-truncated.
Similarly, the (-1)-truncated objects are the subterminal objects.
($n$-truncated morphism in an $(\infty,1)$-category)
A morphism $f : X \to Y$ of ∞-groupoids is $n$-truncated if all of its homotopy fibers are $n$-truncated by def. 2.
A morphism $f : X \to Y$ in an (∞,1)-category $C$ is $n$-truncated if for all $W \in C$ the postcomposition morphism
is $n$-truncated in ∞Grpd.
By the characterization of homotopy fiber of functor categories this is equivalent to saying that $f$ is $k$-truncated when it is so regarded as an object of the over (∞,1)-category $C_{/Y}$.
At least if the ambient (∞,1)-category is even an ∞-stack (∞,1)-topos there is an alternative, more intrinsic, characterization of $n$-truncation in terms of categorical homotopy groups in an (∞,1)-topos:
Suppose that an object $X$ in an ∞-stack (∞,1)-topos is $k$-truncated for some $k \in \mathbb{N}$ (possibly very large).
Then for any $n \in \mathbb{N}$ this $X$ is $n$-truncated precisely if all the categorical homotopy groups above degree $n$ are trivial.
This is HTT, prop 6.5.1.7.
Notice that this expected statement does require the assumption that $X$ is $k$-truncated for some $k$. Without any a priori truncation assumption on $X$, there is no comparable statement about the relation to categorical homotopy groups. See HTT, remark 6.5.1.8.
In an $(\infty,1)$-category $C$ with finite limits, a morphism $f : X \to Y$ is $k$-truncated (for $k \geq -1$) precisely if the diagonal morphism $X \to X \times_Y X$ is $(k-1)$-truncated.
This is HTT, lemma 5.5.6.15.
By definition $f$ is $k$-truncated if for each object $d \in C$ we have that $C(d,f)$ is $k$-truncated in ∞Grpd. Since the hom-functors $C(d,-)$ preserve (∞,1)-limits, we have in particular that $X \to X \times_Y X$ in $C$ is $k$-truncated if $C(d,X) \to C(d,X) \times_{C(d,Y)} C(d,X)$ is $k$-truncated for all $d$ in ∞Grpd. Therefore it is sufficient to prove the statement for morphisms in $C =$ ∞Grpd.
So let now $f : X \to Y$ be a morphism of ∞-groupoids. We may find a fibration $\bar \phi : \bar X \to \bar Y$ between Kan complexes in sSet that models $f : X \to Y$ in the standard model structure on simplicial sets, and by the standard rules for homotopy pullbacks it follows that the object $X \times_Y X$ in $\infty$-Grpd is then modeled by the ordinary pullback $\bar X \times_{\bar Y} \bar X$ in sSet. And the homotopy fibers of $f$ over $y \in Y$ are then given by the ordinary fibers $\bar X_y$ of $\bar f$ in $sSet$.
This way the statement is reduced to the following fact: a Kan complex $\bar X_y$ is $k$-truncated precisely if the homotopy fibers of $\bar X_y \to \bar X_y \times \bar X_y$ are $(k-1)$-truncated.
We write now $X$ for $\bar X_y$, for simplicity. To see the last statement, let $(a,b) : * \to X \times X$ and compute the homotopy pullback
as usual by replacing the right vertical morphism by the fibration $(X \times X)^I \times_{X \times X} (a,b) \to X \times X$ and then forming the ordinary pullback. This shows that $Q$ is equivalent to the space of paths $P_{a,b}X$ in $X$ from $a$ to $b$. (Use that gluing of path space objects at endpoints of paths produces a new path space, see for instance section 4 of BrownAHT).
If $X$ is connected, then choosing any path $a \to b$ gives an isomorphism from the the homotopy groups of $P_{a,b} X$ to those of the loop space $\Omega_a X$. These latter are indeed those of $X$, shifted down in degree by one (as described for instance at fiber sequence).
If $X$ is not connected, we can easily reduce to the case that it is.
For $C$ an $(\infty,1)$-category and $k \geq -2$, the full sub-(∞,1)-category $\tau_{\leq k} C$ is stable under all limits in $C$.
This is HTT, prop. 5.5.6.5.
Let $\mathbf{H}$ be an (∞,1)-topos. For all $(-2) \leq n \leq \infty$ the class of $n$-truncated morphisms in $\mathbf{H}$ forms the right class in a orthogonal factorization system in an (∞,1)-category. The left class is that of n-connected morphisms in $\mathbf{H}$.
This appears as a remark in HTT, Example 5.2.8.16. A construction of the factorization in terms of a model category presentation is in (Rezk, prop. 8.5). See also n-connected/n-truncated factorization system.
There are model structures for homotopy n-types that presentable (∞,1)-category present the full sub-(∞,1)-categories of $n$-truncated objects in some ambient $(\infty,1)$-category. See there for more details.
Under mild conditions there is for each $n$ a universal way to send an arbitrary object $A$ to its $n$-truncation $\tau_{\leq n} A$. This is a general version of decategorification where n-morphisms are identified if they are connected by an invertible $(n+1)$-morphism.
For $C$ an (∞,1)-category and $n \geq -2$ in $\mathbb{Z}$ write $\tau_{\leq n} C$ for the full subcategory of $C$ on its $n$-truncated objects.
So for instance for $C =$ ∞Grpd we have $\tau_{\leq n} \infty Grpd = n Grpd$.
If $C$ is an (∞,1)-category that is presentable then the canonical inclusion (∞,1)-functor
has an accessible left adjoint
This is HTT 5.5.6.18.
Indeed, as the notation suggests, $C_{\leq n}$ is the essential image of $C$ under $\tau_{\leq n}$. The image $\tau_{\leq n} A$ of an object $A$ under this operation is the $n$-truncation of $A$.
So $n$-truncated objects form a reflective sub-(∞,1)-category
A left exact functor $F : C \to D$ between $(\infty,1)$-categories with finite limits sends $k$-truncated objects/morphisms to $k$-truncated objects/morphisms.
This is HTT, prop. 5.5.6.1.6.
Follows from the above recursive characterization of $k$-truncated morphisms by the $(k-1)$-truncation of their diagonal, which is preserved by the finite limit preserving $F$.
A presentable $(\infty,1)$-functor $F : C \to D$ between locally presentable (∞,1)-categories $C$ and $D$ commutes with truncation:
This is HTT, prop. 5.5.6.28.
By the above lemma, $F$ restricts to a functor on the truncations. So we need to show that the diagram
in (∞,1)Cat can be filled by a 2-cell. To see this, notice that the adjoint (∞,1)-functor of both composite morphisms exists (because that of $F$ exists by the adjoint (∞,1)-functor theorem and bcause adjoints of composites are composites of adjoints) and since the bottom morphism is just the restriction of the top morphism and the right adjoints of the vertical morphisms are full inclusions this adjoint diagram
evidently commutes, since it just expresses this restriction.
If $C$ is an (∞,1)-topos, then truncation $\tau_{\leq n} : C \to C$ preserves finite products.
This appears as HTT, lemma 6.5.1.2.
First notice that the statement is true for $C =$ ∞Grpd. For instance we can use the example In ∞Grpd and Top, model ∞-groupoids by Kan complexes and notice that then $\tau_{\leq n}$ is given by the truncation functor $tr_{n+1} : sSet \to [\Delta^{op}_{\leq n+1}, sSet]$. This is also a right adjoint and as such preserves in particular product in $sSet$, which are $(\infty,1)$-products in $\infty Grpd$.
From that we deduce that the statement is true for $C$ any (∞,1)-category of (∞,1)-presheaves $C = PSh_{(\infty,1)}(K) = Func_{(\infty,1)}(K^{op}, \infty Grpd)$ because all relevant operations there are objectwise those in $\infty Grpd$.
So far this shows even that on presheaf $(\infty,1)$-toposes all products (not necessarily finite) are preserved by truncation.
A general (∞,1)-topos $C$ is (by definition) a left exact reflective sub-(∞,1)-category of a presheaf $(\infty,1)$-topos,
Let $\prod_{j} i(X_j)$ be the product of the objects in question taken in $PSh(K)$. By the above there we have an equivalence
Now applying $L$ to this equivalence and using now that $L$ preserves the finite product, this gives an equivalence
in $C$. The claim follows now with the above result that $L \circ \tau_{\leq n} \simeq \tau_{\leq n} \circ L$.
By the fact that the truncation functor $\tau_{\leq n}$ is a left adjoint one obtains canonical morphisms
as the adjunct of the identity on $A$, and then by iteration also canonical morphisms
For any $A \in C$ the sequence
is the Postnikov tower in an (∞,1)-category of $A$. See there for more details.
Discussion of $n$-truncation of types in homotopy type theory via higher inductive types is in (Brunerie). This sends a type to an h-level $(n+2)$-type. The $(-1)$-truncation in the context is forming the bracket type hProp.
See at n-truncation modality.
In an $(\infty,1)$-topos $C$ there is a notion of categorical homotopy groups in an (∞,1)-topos. For the $(\infty,1)$-topos ∞Grpd given by the model of Kan complexes this coincides with the notion of simplicial homotopy groups:
An object $A$ in the (∞,1)-topos ∞Grpd is $n$-truncated precisely if its categorical homotopy groups $\pi_k(A)$ vanish for all $k \gt n$.
This simple relation between $n$-truncation and categorical homotopy groups is almost, but not exactly true in an arbitrary (∞,1)-topos.
Let $\mathbf{H}$ be an (∞,1)-topos and $A \in \mathbf{H}$ an $n$-truncated object.
Then
for $k \gt n$ we have for the categorical homotopy groups $\pi_k(A) = *$;
if (for $n \geq 0$) $\pi_n(A) = *$, then $X$ is in fact $(n-1)$-truncated.
This implies
If $A \in \mathbf{H}$ is truncated at all (for any value), then it is $n$-truncated precisely if all categorical homotopy groups vanish $\pi_k(A) = *$ for $k \gt n$.
Notice. If $A$ on the other hand is not truncated at all, then all its homotopy groups may be trivial and $A$ may still not be equivalent to the terminal object. This means that Whitehead's theorem may fail in a general (∞,1)-topos for untruncated objects. It holds, however, in hypercomplete (∞,1)-toposes.
A morphism $f : X \to 0$ is
(-2)-truncated precisely if it is an equivalence;
(-1)-truncated precisely if it is a monomorphism.
For morphisms between 1-groupoids, the notion of $n$-truncation for low $n$ reproduces standard concepts from ordinary category theory.
A functor $f : X \to Y$ between groupoids, is $n$-truncated precisely when regarded as a morphism in ∞Grpd it is
for $n = -2$ – an equivalence of categories;
for $n = -1$ – a full and faithful functor;
for $n = 0$ – a faithful functor.
Notice that $f$ being faithful means precisely that it induces a monomorphism on the first homotopy groups.
For $x : * \to X$ any point and $F_{f(x)}$ the corresponding homotopy fiber of $f$, the long exact sequence of homotopy groups gives that $\pi_1(F)$ is the kernel of an injective map
hence $\pi_1(F_{y}) = *$ for all points $y$ in the essential image of $f$. For $y$ not in the essential image we have $F_y \simeq \emptyset$. In either case it follows that $F$ is 0-truncated.
By def. 3 this is the defining condition for $f$ to be 0-truncated.
Let $C$ be a site and write $Sh_{(2,1)}(C) \hookrightarrow Sh_{(\infty,1)}(C)$ for the (2,1)-topos of stacks/(2,1)-sheaves inside the (∞,1)-sheaf (∞,1)-topos of all ∞-stacks/(∞,1)-sheaves.
Write $L_W [C^{op}, Grpd]$ for the simplicial localization of groupoid valued presheaves in $C$ and write $[C^{op}, sSet]_{proj,loc}$ for the local projective model structure on simplicial presheaves that presents $Sh_{(\infty,1)}(C)$.
Let $f : X \to Y$ be a morphism of stacks which has a presentation by a degreewise faithful functor that, under the nerve, goes between fibrant simplicial presheaves.
Then $f$ is 0-truncated as a morphism in $Sh_{(\infty,1)}(C)$.
We need to check that for any $\infty$-stack $A$ the morphism $Sh_\infty(A,f)$ is 0-truncated in ∞Grpd. We may choose a cofibrant model for $A$ in $[C^{op}, sSet]_{proj,loc}$ and by assumption that $X$ and $Y$ is fibrant we have that the ordinary hom of simplicial presheaves $[C^{op}, sSet](A, f)$ is the correct derived hom space morphism. This is itself (the nerve of) a faithful functor, hence the statement follows with prop. 9.
An object in ∞Grpd is $n$-truncated precisely if it is an n-groupoid. To some extent this is so by definition. Equivalently, an object in Top is $n$-truncated if it is (in the equivalence class of) a homotopy n-type.
So we have for $n \in \mathbb{N}$ a reflective sub-(∞,1)-category
If we model the (∞,1)-category ∞Grpd as the Kan complex-enriched category/fibrant simplicial category $KanCplx \subset$ sSet of Kan complexes, then the truncation adjunction
is modeled by the simplicial coskeleton sSet-enriched adjunction
where $KanCplx_{n+1}$ is the subcategory of $[\Delta^{op}_{\leq n+1}, Set]$ on those truncated simplicial sets that are truncations of Kan complexes, regarded as a Kan-complex-enriched category by the embedding via $cosk_{n+1}$.
Notice that every Kan complex $X$ which is $n$-truncated is homotopy equivalent to one in the image of $cosk_{n+1}$, namely to $cosk_{n+1} tr_{n+1} X$, because by one of the properties of $cosk_{n+1}$ we have that the unit
induces isomorphisms on homotopy groups $\pi_k$ for $k \leq n$.
This shows that $KanCplx_{n+1}$ is indeed a full sub-(∞,1)-category of $KanCplx$ on $n$-truncated objects
Moreover, by the fact discussed at Simplicial and derived adjunctions at adjoint (∞,1)-functor we have that the sSet-enriched adjunction $(tr_{n+1} \dashv cosk_{n+1})$ on $KanCplx$ indeed presents a pair of adjoint (∞,1)-functors on ∞Grpd. So $tr_{n+1} : KanCplx \to KanCplx$ indeed presents the left adjoint $\tau_{\leq} : \infty Grpd \to n Grpd$ to the inclusion $n Grpd \hookrightarrow \infty Grpd$.
In ordinary category theory we have that a morphism is a monomorphism (as discussed there), precisely if its diagonal is an isomorphism. Embedded into (∞,1)-category this becomes the special case of prop. 2 for $n = 0$: a morphism is (-1)-truncated (hence a monomorphism in an (∞,1)-category), precisely if its diagonal is (-2)-truncated (hence an equivalence in an (∞,1)-category).
Let $X$ be an object that is $n$-truncated. This means that $X \to *$ is an $n$-truncated morphism. So by prop. 2 the diagonal on that object
is an $(n-1)$-truncated morphism, and precisely if it is $(n-1)$-truncated is $X$ $n$-truncated.
In particular, the diaginal is a monomorphism in an (∞,1)-category, hence (-1)-truncated, precisely if $X$ is $0$-truncated (an h-set).
$n$-truncated morphism / n-connected morphism
homotopy level | n-truncation | homotopy theory | higher category theory | higher topos theory | homotopy type theory |
---|---|---|---|---|---|
h-level 0 | (-2)-truncated | contractible space | (-2)-groupoid | true/unit type/contractible type | |
h-level 1 | (-1)-truncated | contractible-if-inhabited | (-1)-groupoid/truth value | (0,1)-sheaf/ideal | mere proposition/h-proposition |
h-level 2 | 0-truncated | homotopy 0-type | 0-groupoid/set | sheaf | h-set |
h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid |
h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | (3,1)-sheaf/2-stack | h-2-groupoid |
h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | (4,1)-sheaf/3-stack | h-3-groupoid |
h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf/n-stack | h-$n$-groupoid |
h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |
The discussion of truncated objects in an $(\infty,1)$-category is in section 5.5.6 of
The discussion of categorical homotopy groups in an (∞,1)-topos is in section 6.5.1.
A discussion in terms of model category presentations is in section 7 of
A classical article that amplifies the expression of Postnikov towers in terms of coskeletons is
Discussion in the context of homotopy type theory is in