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\begin{document}

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\section*{homomorphism}

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

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

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

[[!include category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{traditional_magmas_semigroups_groups_rngs}{Traditional (magmas, semigroups, groups, rngs)}\dotfill \pageref*{traditional_magmas_semigroups_groups_rngs} \linebreak
\noindent\hyperlink{identitypreserving_monoids_rings}{Identity-preserving (monoids, rings)}\dotfill \pageref*{identitypreserving_monoids_rings} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \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}

In a restrictive sense, a homomorphism is a [[function]] between (the [[underlying sets]]) of two [[algebras]] that preserves the [[algebraic structure]].

More generally, a homomorphism is a function between [[structured sets]] that preserves whatever [[structure]] there is around. Even more generally, `homomorphism' is just a synonym for `[[morphism]]' in any [[category]], the structured sets being generalised to arbitrary [[object]]s.

\emph{Note: The word ``homomorphism'' has also traditionally been used for what we call a (weak) [[2-functor]] between [[bicategories]]}.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{traditional_magmas_semigroups_groups_rngs}{}\subsubsection*{{Traditional (magmas, semigroups, groups, rngs)}}\label{traditional_magmas_semigroups_groups_rngs}

Traditionally, a \textbf{homomorphism} between two [[magmas]] $A$ and $B$ is a [[function]]

\begin{displaymath}
\phi\colon A \to B
\end{displaymath}
of the underlying [[set]]s that respects the binary operation in that for all $a_1, a_2$ in $A$ we have

\begin{displaymath}
\phi(a_1 \cdot a_2) = \phi(a_1) \cdot \phi(a_2)
  \,.
\end{displaymath}
This definition gives us the correct notion of \textbf{magma homomorphism}, \textbf{[[semigroup]] homomorphsim}, and \textbf{[[group]] homomorphism}, but it is actually a bit of a coincidence that it works for groups. It does \emph{not} give the correct definition of [[monoid]] homomorphism, since it doesn't properly treat the [[identity elements]]. (However, the correct notion of monoid [[isomorphism]] can still be constructed from this inadequate definition of homomorphism.)

A \textbf{rng homomorphism} is a function between [[rngs]] that is a homomorphism for both the additive group and the multiplicative semigroup. (For [[rings with identity]], this is again inadequate.)

\hypertarget{identitypreserving_monoids_rings}{}\subsubsection*{{Identity-preserving (monoids, rings)}}\label{identitypreserving_monoids_rings}

A \textbf{homomorphism} between two [[monoids]] $A$ and $B$ is a semigroup homomorphism

\begin{displaymath}
\phi\colon A \to B
\end{displaymath}
of the underlying [[semigroup]]s that preserves [[identity elements]] in that we have

\begin{displaymath}
\phi(1_A) = 1_B
  \,.
\end{displaymath}
This definition of \textbf{monoid homomorphism} is a special case (via [[delooping]]) of the definition of [[functor]].

It is a theorem that a semigroup homomorphism between [[groups]] must be a monoid homomorphism (and additionally must preserve [[inverse elements]], which is also necessary to be the \emph{correct} definition of \textbf{group homomorphism}.)

A \textbf{[[ring]] homomorphism} is a function between rings that is a homomorphism for both the additive group and the multiplicative monoid. Traditional [[ring theory]] sometimes actually uses rng homomorphisms even when the rngs in question are assumed to have identity elements, so be careful when reading old books.

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

More generally, a \textbf{homomorphism} between sets equipped with any algebraic structure is a map preserving this structure. This can be made precise using [[Lawvere theories]], [[monads]], etc.

Generalizing further, we may simply treat `homomorphism' as a synonym for `[[morphism]]' in any [[category]], although there is a strong tendency to use `homomorphism' in the case of `algebraic' categories: for example, nobody seems to speak of a homomorphism between [[topological space]]s ([[continuous map]]s), or between [[manifold]]s. Here we put `algebraic' in scare quotes to indicate the \emph{field} of [[algebra]]; even morphisms in [[algebraic categories]] from other fields (such as the category of [[compacta]]) are not usually called homomorphisms.

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

\begin{itemize}%
\item an [[action]] homomorphism is an [[equivariant function]]


\item \ldots{}



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

\begin{itemize}%
\item homomorphisms are to [[functions]] as [[logical relations]] are to [[relations]]

\end{itemize}
[[!redirects homomorphism]] [[!redirects homomorphisms]]

[[!redirects magma homomorphism]] [[!redirects magma homomorphisms]] [[!redirects semigroup homomorphism]] [[!redirects semigroup homomorphisms]] [[!redirects monoid homomorphism]] [[!redirects monoid homomorphisms]] [[!redirects group homomorphism]] [[!redirects group homomorphisms]] [[!redirects rng homomorphism]] [[!redirects rng homomorphisms]] [[!redirects ring homomorphism]] [[!redirects ring homomorphisms]] [[!redirects algebra homomorphism]] [[!redirects algebra homomorphisms]]



\end{document}