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\begin{document}

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\section*{core in a 2-category}

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

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{core}{Core}\dotfill \pageref*{core} \linebreak
\noindent\hyperlink{enough_groupoids}{Enough groupoids}\dotfill \pageref*{enough_groupoids} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{subobjects}{Subobjects}\dotfill \pageref*{subobjects} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \textbf{[[core]]} of a [[category]] $A$ is the maximal [[subcategory]] of $A$ that is a [[groupoid]], consisting of all its [[objects]] and all [[isomorphisms]] between them. The core is not a [[functor]] on the [[2-category]] [[Cat]], but it is a functor on the [[(2,1)-category]] $Cat_g$ of categories, functors, and [[natural isomorphisms]]. In fact, it is a [[coreflective subcategory|coreflection]] of $Cat_g$ into [[Gpd]].

In general, for any [[2-category]] $K$ we write $K_g$ for its ``homwise core:'' the sub-2-category determined by all the objects and morphisms but only the iso [[2-morphisms]]. Of course, $K_g$ is a [[(2,1)-category]]. Then $gpd(K)$ is a full [[sub-2-category]] of $K_g$, and we can ask whether it is coreflective. In a [[regular 2-category]], however, it turns out that there is a stronger and more useful notion which implies this coreflectivity.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{core}{}\subsubsection*{{Core}}\label{core}

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{core} of an object $A$ in a [[regular 2-category]] is a [[morphism]] $J\to A$ which is [[eso morphism|eso]], [[pseudomonic functor|pseudomonic]], and where $J$ is groupoidal.

\end{defn}
\hypertarget{enough_groupoids}{}\subsubsection*{{Enough groupoids}}\label{enough_groupoids}

We say that a regular 2-category has \textbf{enough groupoids} if every object admits an eso from a groupoidal one. Thus, a (2,1)-exact 2-category has cores if and only if it has enough groupoids.

Likewise, we say that a regular 2-category has \textbf{enough discretes} if every object admits an eso from a discrete one. Clearly this is a stronger condition. Note, though, that if $K$ has enough groupoids, then $pos(K)$ has enough discretes, since the core of any posetal object is discrete.

Having enough discretes, or at least enough groupoids, is a very familiar aspect of 2-categories such as $Cat$. It also turns out to make the [[2-internal logic|internal logic]] noticeably easier to work with. However, in a sense none of the really ``new'' 2-categories have enough groupoids or discretes.

\begin{itemize}%
\item The 2-exact 2-categories having enough discretes are precisely the categories of internal categories and anafunctors in 1-exact 1-categories; see [[exact completion of a 2-category]]. Likewise, any 2-exact 2-categories having enough groupoids consists of certain internal categories in a (2,1)-category.


\item Basically the only [[Grothendieck 2-toposes]] having enough discretes are the 2-categories of stacks on [[2-site|1-sites]], and the only ones having enough groupoids are the 2-categories of stacks on (2,1)-sites. The ``honestly 2-dimensional'' case of stacks on 2-sites (almost?) never have enough of either.



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{lemma}
\label{}\hypertarget{}{}
In a regular 2-category $K$, any core $J\to A$ is a coreflection of $A$ from $K_g$ into $gpd(K)$.

\end{lemma}
\begin{proof}
We must show that for any groupoidal $G$, the functor

\begin{displaymath}
gpd(K)(G,J)=K_g(G,J)\to K_g(G,A)
\end{displaymath}
is an equivalence. Since $J\to A$ is pseudomonic in $K$, it is ff in $K_g$, so this functor is full and faithful; thus it remains to show it is eso. Given any map $G\to A$, take the pullback

\begin{displaymath}
\itexarray{P & \to & J\\ \downarrow && \downarrow \\ G & \to & A}
\end{displaymath}
and let $P_1\;\rightrightarrows\; P$ be the kernel of $P\to G$. Since the composite $P\to A$ descends to $G$, it comes equipped with an action by this kernel. However, since $G$ is groupoidal, $P_1\;\rightrightarrows\; P$ is a [[n-congruence|(2,1)-congruence]], so the 2-cell defining the action must be an isomorphism. Therefore, it must factor uniquely through the pseudomonic $J\to A$, so $P\to J$ has an action as well; thus it descends to a map $G\to J$, as desired.

\end{proof}
In particular, cores in a regular 2-category are unique up to unique equivalence. We write $J(A)$ for the core of $A$, when it exists.

\begin{lemma}
\label{}\hypertarget{}{}
An object $A$ of a (2,1)-exact 2-category has a core if and only if there is an eso $C\to A$ where $C$ is groupoidal.

\end{lemma}
\begin{proof}
``Only if'' is clear, so suppose that $p:C\to A$ is eso and $C$ is groupoidal. Let $C_1 = C\times_A C$ be the pullback. Then $C_1$ is also groupoidal and is a (2,1)-congruence on $C$, so by exactness it is the kernel of some eso $q:C\to J$. There is an evident induced map $m:J\to A$; we claim that this is a core of $A$.

Evidently $m:J\to A$ is eso, since the eso $C\to A$ factors through it. And since $C_1$ is a (2,1)-congruence, the classification of [[n-congruence|congruences]] implies that $J$ is groupoidal; thus it remains to show that $m$ is pseudomonic.

First suppose given $f,g: X\;\rightrightarrows\; J$. Pulling back $q$ along $f$ and $g$ gives esos $P_1\to X$ and $P_2\to X$, whose pullback $P = P_1\times_X P_2$ comes with an eso $r:P \to T$ and two morphisms $h,k:P \to C$ with $q h \cong f r$ and $q k \cong g r$. Then any pair of 2-cells $\alpha,\beta: f\to g$ induce maps $P\;\rightrightarrows\; C_1$, since $C_1$ is the kernel $(q/q)$. But if $m\alpha = m\beta$, then these two maps must be equal, since $C_1$ is also the kernel $(p/p)$. Therefore, $\alpha r=\beta r$, and since $r$ is eso, $\alpha=\beta$; thus $m$ is faithful.

On the other hand, again given $f,g: X\;\rightrightarrows\; J$, any isomorphism $\alpha: m f\cong m g$ induces a map $P\to C_1$ and therefore a 2-cell $\beta: f r\to g r$ with $m\beta = \alpha r$. To show that $\beta$ descends to a 2-cell $f\to g$, we must verify that it is an action 2-cell for the actions of $P\;\rightrightarrows\; J$ on $f r$ and $g r$. But $m\beta$ is an action 2-cell, since $m\beta = \alpha r$, and we now know that $m$ is faithful, so it reflects the axiom for an action 2-cell. Therefore, $m$ is full-on-isomorphisms, and hence pseudomonic.

\end{proof}
\hypertarget{subobjects}{}\subsubsection*{{Subobjects}}\label{subobjects}

We also remark that cores, when they exist, ``capture all the information about [[subobjects]].''

\begin{prop}
\label{}\hypertarget{}{}
If $K$ is a regular 2-category and $A$ is an object having a core $j:J\to A$, then $j^*:Sub(A)\to Sub(J)$ is an equivalence.

\end{prop}
\begin{proof}
It is a general fact in a regular 2-category that for any eso $f:X\to Y$, $f^*:Sub(Y)\to Sub(X)$ is full (and faithful, which of course is automatic for a functor between posets). For if $f^*U \le f^*V$, then we have a square

\begin{displaymath}
\itexarray{f^*U & \to & f^*V & \to & V\\
  \downarrow &&&& \downarrow\\
  U & & \to & & Y}
\end{displaymath}
in which $f^*U \to U$ is eso and $V\to Y$ is ff, hence we have $U\to V$ over $Y$.

Thus, in our case, $j^*$ is full and faithful since $j$ is eso, so it suffices to show that for any ff $U\to J$ we have $j^* \exists_j U \le U$ in $Sub(J)$. But we have a commutative square

\begin{displaymath}
\itexarray{U & \to & \exists_j U\\
  \downarrow && \downarrow\\
  J & \to & X}
\end{displaymath}
where the vertical arrows are ff and the bottom arrow $J\to X$ is faithful and pseudomonic, from which it follows that $U\to \exists_j U$ is also faithful and pseudomonic. Since $U\to \exists_j U$ is eso by definition, $U$ is a core of $\exists_j U$, and since $j^*\exists_j U$ is a groupoid mapping to $\exists_j U$, it factors through $U$, as desired.

\end{proof}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The material on this page is taken from

\begin{itemize}%
\item [[Mike Shulman]], \emph{[[michaelshulman:core]]}

\end{itemize}
[[!redirects cores in a 2-category]]



\end{document}