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\begin{document}

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\section*{n-types cover}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

\hypertarget{types_cover}{}\section*{{$n$-types cover}}\label{types_cover}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{in_a_higher_category}{In a higher category}\dotfill \pageref*{in_a_higher_category} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_homotopy_type_theory}{In homotopy type theory}\dotfill \pageref*{in_homotopy_type_theory} \linebreak
\noindent\hyperlink{InModels}{In models}\dotfill \pageref*{InModels} \linebreak
\noindent\hyperlink{relation_to_other_axioms}{Relation to other axioms}\dotfill \pageref*{relation_to_other_axioms} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The property that \textbf{$n$-types cover} is a property of a [[higher category theory|higher category]], or an [[axiom]] in its corresponding [[internal logic]], which says roughly that every [[object]] is [[cover|covered]] by one that is [[truncated object|truncated]] at some level.

\hypertarget{in_a_higher_category}{}\subsection*{{In a higher category}}\label{in_a_higher_category}

A [[n-category|higher category]] is said to satisfy the (external) property that \textbf{$n$-types cover}, or to have \textbf{enough $n$-types}, if for every [[object]] $X$ there exists an $n$-[[truncated object]] $Y$ and an [[effective epimorphism in an (∞,1)-category|effective epimorphism]] $Y\to X$. When $n=0$ one also says that \textbf{sets cover} or that there are \textbf{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 epimorphism|effective epimorphic]].

\hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples}

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 [[representable functor|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.

\hypertarget{in_homotopy_type_theory}{}\subsection*{{In homotopy type theory}}\label{in_homotopy_type_theory}

In [[homotopy type theory]], the (internal) axiom of \textbf{$n$-types cover} or \textbf{enough $n$-types} says that for any type $X$ there [[merely]] exists an $n$-type $Y$ and a surjection (i.e. a [[n-epimorphism|(-1)-connected map]]) $Y\to X$. In symbols:

\begin{displaymath}
\prod_{(X:Type)} {\Vert \sum_{(Y:n Type)} \sum_{(f:Y\to X)} surj(f) \Vert }
\end{displaymath}
where $\Vert-\Vert$ denotes the $(-1)$-truncation. As before, when $n=0$ we say that \textbf{[[h-set|sets]] cover} or that there are \textbf{enough sets}.

\hypertarget{InModels}{}\subsubsection*{{In models}}\label{InModels}

\begin{theorem}
\label{}\hypertarget{}{}
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.

\end{theorem}
\begin{proof}
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 $p^* X$.

\end{proof}
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.

\hypertarget{relation_to_other_axioms}{}\subsubsection*{{Relation to other axioms}}\label{relation_to_other_axioms}

The internal axiom that sets cover is related to some forms of the [[axiom of choice]]. Let us denote by

\begin{itemize}%
\item $AC_0 =$ every surjection between sets merely has a section.
\item $AC_\infty =$ every surjection with codomain a set merely has a section.

\end{itemize}
Then we have

\begin{theorem}
\label{}\hypertarget{}{}
$AC_\infty \Leftrightarrow (AC_0$ and sets cover).

\end{theorem}
\begin{proof}
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 $X\to Z$.

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[cover]]


\item [[atlas]]



\end{itemize}
category: foundational axiom

[[!redirects sets cover]] [[!redirects enough n-types]] [[!redirects enough sets]]



\end{document}