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\begin{document}

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\section*{kernel}

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

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

[[!include category theory - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - 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{as_a_pullback}{As a pullback}\dotfill \pageref*{as_a_pullback} \linebreak
\noindent\hyperlink{as_an_equalizer}{As an equalizer}\dotfill \pageref*{as_an_equalizer} \linebreak
\noindent\hyperlink{as_a_weighted_limit}{As a weighted limit}\dotfill \pageref*{as_a_weighted_limit} \linebreak
\noindent\hyperlink{as_a_representing_object}{As a representing object}\dotfill \pageref*{as_a_representing_object} \linebreak
\noindent\hyperlink{in_an_category}{In an $(\infty,1)$-category}\dotfill \pageref*{in_an_category} \linebreak
\noindent\hyperlink{other_meanings}{Other meanings}\dotfill \pageref*{other_meanings} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

The \emph{kernel} of a [[morphism]] is that part of its [[domain]] which is sent to [[zero]].

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

There are various definitions of the notion of kernel, depending on the properties and structures available in the ambient category. We list a few definitions and discuss (in parts) when they are equivalent.

\hypertarget{as_a_pullback}{}\subsubsection*{{As a pullback}}\label{as_a_pullback}

\begin{defn}
\label{}\hypertarget{}{}
In a [[category]] with an [[initial object]] $0$ and [[pullbacks]], the \textbf{kernel} $ker(f)$ of a [[morphism]] $f: A \to B$ is the [[pullback]] $ker(f) \to A$ along $f$ of the unique morphism $0 \to B$

\begin{displaymath}
\itexarray{
    ker(f)
    &\to&
    0
    \\
    {}^{\mathllap{p}}\downarrow && \downarrow
    \\
    A &\stackrel{f}{\to}& B
  }
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
More explicitly, this characterizes the object $ker(f)$ as [[generalized the|the]] object (unique up to unique [[isomorphism]]) that satisfies the following [[universal property]]:

for every object $C$ and every morphism $h : C \to A$ such that $f\circ h = 0$ is the [[zero morphism]], there is a unique morphism $\phi : C \to ker(f)$ such that $h = p\circ \phi$.

\end{remark}
\hypertarget{as_an_equalizer}{}\subsubsection*{{As an equalizer}}\label{as_an_equalizer}

\begin{defn}
\label{}\hypertarget{}{}
In a [[category]] with [[zero morphism]]s (meaning: [[enriched category|enriched]] over the [[category of pointed sets]]), [[generalized the|the]] \textbf{kernel} $ker(f)$ of a [[morphism]] $f : c \to d$ is, if it exists, the [[equalizer]] of $f$ and the zero morphism $0_{c,d}$.

\end{defn}
\hypertarget{as_a_weighted_limit}{}\subsubsection*{{As a weighted limit}}\label{as_a_weighted_limit}

In any category enriched over pointed sets, the kernel of a morphism $f:c\to d$ is the universal morphism $k:a\to c$ such that $f \circ k$ is the basepoint. It is a [[weighted limit]] in the sense of [[enriched category theory]]. This applies in particular in any (pre)-[[additive category]].

This is a special case of the construction of [[generalized kernels]] in enriched categories.

\hypertarget{as_a_representing_object}{}\subsubsection*{{As a representing object}}\label{as_a_representing_object}

Let $Ab$ be the category of abelian groups. It is a category with kernels. In every $Ab$-enriched category $A$, for every morphism $f: X\to Y$ in $A$ there is a [[subfunctor]]

\begin{displaymath}
ker f : A^{op}\to Ab
\end{displaymath}
of the representable functor $hom(-,X)$, defined on objects by

\begin{displaymath}
(ker f)(Z) = ker(hom(Z,X)\to hom(Z,Y)),
\end{displaymath}
where $ker$ on the right-hand side is the kernel n the category of abelian groups.

If the category is in fact preabelian, $ker f$ is also representable with representing object $Ker f$. One has to be careful with $Coker f$ which does not represent the functor naive $coker f$ defined as $(coker f)(Z) = coker(hom(Z,X)\to hom(Z,Y))$ in $Ab$, which is often not representable at all, even in the simple example of the category of abelian groups. Instead, as a colimit construction, one should \emph{co}represent another functor, namely, the covariant functor $Z\mapsto ker(hom(Y,Z) \to hom(X,Z))$ (which is a quotient of the corepresentable functor $hom(X,-)$). In short, $Coker f$ is defined by the double dualization using the kernel in $Ab$: $Coker f = (Ker f^{op})^{op}$. This is a particular case of the dualization involved in defining any [[colimit]] from its corresponding [[limit]].

\hypertarget{in_an_category}{}\subsubsection*{{In an $(\infty,1)$-category}}\label{in_an_category}

The kernel of a morphism in an [[(∞,1)-category]] with $\infty$-categorical [[zero object]] is the [[homotopy pullback]] as in the pullback definition above: the [[homotopy fiber]].

See also [[stable (∞,1)-category]].

\hypertarget{other_meanings}{}\subsubsection*{{Other meanings}}\label{other_meanings}

In some fields, the term `kernel' refers to an [[equivalence relation]] that category theorists would see as a [[kernel pair]]. This is especially important in fields such as [[monoid]] theory where both notions exist but are not equivalent (while in [[group]] theory they are equivalent).

In [[ring]] theory, even when one assumes that rings have units preserved by ring homomorphisms, the traditional notion of kernel (an [[ideal]]) exists in the category of non-unital rings (and is not itself a unital ring in general). A purely category-theoretic theory of unital rings can be recovered either by using the kernel pair instead or (to fit better the usual language) moving to a category of [[modules]].

In [[universal algebra]], this may be handled in the framework of [[Malcev variety|Mal?cev varieties]].

Kashiwara-Schapira, following the terminology of EGA, uses kernel as a synonym of equalizer (and co-kernel of co-equalizer).

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\begin{prop}
\label{}\hypertarget{}{}
Let $C$ be a category with [[pullback]]s and [[zero object]].

In $C$, the kernel of a kernel is 0.

\end{prop}
\begin{proof}
By the  we have that the total square

\begin{displaymath}
\itexarray{
     ker ker f &\to& ker f  &\to& 0
     \\
     \downarrow && \downarrow && \downarrow
     \\
     0 &\to& c &\stackrel{f}{\to}& d
  }
\end{displaymath}
is a pullback. Since $0 \to c$ is a [[monomorphism]] and the pullback of a monomorphism along itself is the domain of the monomorphis, we have $ker ker f \simeq 0$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
This statement crucially fails to be true in [[higher category theory]]. There, the kernel of a kernel is the based [[loop space object]] of $d$. For this reason where one has [[short exact sequence]]s in 1-category theory, there are instead long [[fiber sequence]]s in higher category theory.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
In a category $C$ with [[pullback]]s and [[pushout]]s and [[zero object]], kernel and [[cokernel]] form a pair of [[adjoint functor]]s on the [[arrow category|arrow categories]]

\begin{displaymath}
(coker \dashv ker) : Arr(C) \stackrel{\overset{coker}{\leftarrow}}{\underset{ker}{\to}} Arr(C)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
We check the hom-isomorphism of a pair of [[adjoint functor]]s. An element in the [[hom-set]] $Arr_C(g,ker f)$ is a [[diagram]]

\begin{displaymath}
\itexarray{
     c &\to& ker(f) &\to& 0
     \\
     {}^{\mathllap{g}}\downarrow && \downarrow && \downarrow
     \\
     d &\to& a &\stackrel{f}{\to}& b
  }
  \,.
\end{displaymath}
By the universal property of the [[pullback]], this is the same as a diagram

\begin{displaymath}
\itexarray{
     c  &\to& &\to& 0
     \\
     {}^{\mathllap{g}}\downarrow && && \downarrow
     \\
     d &\to& a &\stackrel{f}{\to}& b
  }
  \,.
\end{displaymath}
By the dual reasoning, an element in $Arr_C(coker g, f)$ is a diagram

\begin{displaymath}
\itexarray{
    c &\stackrel{g}{\to}& d &\to& a
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{f}}
    \\
    0 &\to& coker g &\to& b
  }
  \,.
\end{displaymath}
By the universal property of the [[pushout]] this is equivalently a diagram

\begin{displaymath}
\itexarray{
    c &\stackrel{g}{\to}& d &\to& a
    \\
    \downarrow &&  && \downarrow^{\mathrlap{f}}
    \\
    0 &\to&  &\to& b
  }
  \,.
\end{displaymath}
\end{proof}
(This also follows from the general theory of [[generalized kernels]].)

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

\begin{example}
\label{}\hypertarget{}{}
In the [[category]] [[Ab]] of abelian groups, the kernel of a [[group homomorphism]] $f : A \to B$ is the [[subgroup]] of $A$ on the set $f^{-1}(0)$ of elements of $A$ that are sent to the zero-element of $B$.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
More generally, for $R$ any [[ring]], this is true in $R$[[Mod]]: the kernel of a morphism of modules is the [[preimage]] of the zero-element at the level of the underlying sets, equipped with the unique sub-module structure on that set.

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

\begin{itemize}%
\item \textbf{kernel}, [[fiber]], [[generalized kernel]]


\item [[homotopy fiber]]


\item [[cokernel]]



\end{itemize}
[[!redirects kernels]]



\end{document}