homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
The property that $n$-types cover is a property of a higher category, or an axiom in its corresponding internal logic, which says roughly that every object is covered by one that is truncated at some level.
A higher category is said to satisfy the (external) property that $n$-types cover, or to have enough $n$-types, if for every object $X$ there exists an $n$-truncated object $Y$ and an effective epimorphism $Y\to X$. When $n=0$ one also says that sets cover or that there are enough sets.
Usually the category in question is some sort of topos or higher topos, or at least a pretopos of an appropriate sort. In this case, the property that $n$-types cover means that the subcategory of $n$-truncated objects is “generating” in an appropriate sense.
When $n=-1$, however, having enough $n$-types in this sense is not really a useful notion, because $(-1)$-truncated objects (subterminal objects) are not closed under coproducts. In this case a better condition is that all maps out of subterminal objects are jointly effective epimorphic.
If $n\gt 0$, then any n-localic (∞,1)-topos has enough $(n-1)$-types, since every object is surjected onto by a coproduct of representables which are $n$-truncated. The converse seems plausible as well.
A specific example of a higher topos in which sets do not cover is the slice (∞,1)-topos $\infty Gpd / \mathbf{B}^2 \mathbb{Z}$ of ∞Grpd over the double delooping of the group of integers, which is equivalently the topos of ∞-actions of the ∞-group (2-group) $\mathbf{B} \mathbb{Z}$. From this second perspective, a 1-truncated object of this topos is a groupoid (1-groupoid) together with an automorphism of its identity functor, i.e. an element of its center, and a morphism of such is a functor that respects these central elements. Such an object is 0-truncated if it is essentially discrete, in which case its center is also trivial; thus the 0-truncated objects are just ordinary sets. However, since functors must respect the central elements, there can be no surjective map from a 0-truncated object to a 1-truncated one whose chosen central element is nontrivial.
In homotopy type theory, the (internal) axiom of $n$-types cover or enough $n$-types says that for any type $X$ there merely exists an $n$-type $Y$ and a surjection (i.e. a (-1)-connected map) $Y\to X$. In symbols:
where $\Vert-\Vert$ denotes the $(-1)$-truncation. As before, when $n=0$ we say that sets cover or that there are enough sets.
If an $(\infty,1)$-topos satisfies the external property that $n$-types cover, then its internal type theory satisfies the internal axiom that $n$-types cover.
By the Kripke-Joyal semantics of homotopy type theory, the conclusion requires that for any map $X\to \Gamma$ there is an effective epi $p:\Delta\to \Gamma$, an $n$-truncated map $Y\to \Delta$, and an effective epi $Y\to p^*X$ in the slice category over $\Delta$. Assuming the hypothesis, we can obtain this by letting $\Delta$ be an (external) $n$-type cover of $\Gamma$ and $Y$ an (external) $n$-type cover of $p^* X$.
The converse, however, is not true. For the internal axiom, like all internal statements, is preserved by passage to slices (i.e. introduction of a nonempty context), but we saw above that the slice topos $\infty Gpd / \mathbf{B}^2 \mathbb{Z}$ does not have enough sets, even though $\infty Gpd$ does.
For an example of a topos in whose internal logic sets do not cover, let $C$ be the $(2,1)$-category with two objects $a$ and $b$, one morphism $a\to b$, no morphisms $b\to a$, and $C(a,a) = \mathbf{B} \mathbb{Z}$ and $C(b,b)=1$, and consider the presheaf $(\infty,1)$-topos over $C$. A 1-truncated object therein consists of two groupoids $X_a$ and $X_b$, an element of the center of $X_a$, and a functor $X_b \to X_a$ which relates the chosen central element in $X_a$ with the identity in the center of $X_b$.
If we take $X_b$ to be empty, then $X_a$ is a groupoid with an arbitrary chosen central element, and we can choose it to be one which is nontrivial and hence admits no surjection from a 0-type. Let $\Gamma = 1$ for this $X$. Since the terminal object $1$ is a representable (it is $C(-,b)$), it is projective; thus given $\Delta$ and $Y$ as above, we could find a section of $p:\Delta\to\Gamma$ and pull back $Y$ along it to obtain a 0-type cover of $X$ itself, which does not exist.
The internal axiom that sets cover is related to some forms of the axiom of choice. Let us denote by
Then we have
$AC_\infty \Leftrightarrow (AC_0$ and sets cover).
This is a theorem inside homotopy type theory, and likewise for its proof. Clearly $AC_\infty \Rightarrow AC_0$. Given $X$, we have a map $X\to {\Vert X \Vert_0}$ which is 0-connected, hence also surjective, and its codomain is a set; thus by $AC_\infty$ it merely has a section. Any section of a 0-connected map is surjective; thus $X$ is merely covered by $\Vert X\Vert_0$. (Note that in this case we have a stronger version of “sets cover” in which the first $\sum$ is outside the truncation.)
Conversely, suppose $AC_0$ and sets cover, and let $X\to Z$ be a surjection with codomain a set. Then there merely exists a set $Y$ and a surjection $Y\to X$, and the composite surjection $Y\to X\to Z$ merely has a section by $AC_0$. The composite of this section with $Y\to X$ is a section of $X\to Z$.
Last revised on April 29, 2023 at 19:26:30. See the history of this page for a list of all contributions to it.