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\begin{document}

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\section*{red herring principle}

\hypertarget{the_red_herring_principle}{}\section*{{The red herring principle}}\label{the_red_herring_principle}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{notes}{Notes}\dotfill \pageref*{notes} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{nonexamples}{Non-examples}\dotfill \pageref*{nonexamples} \linebreak
\noindent\hyperlink{seminonexamples}{Semi-non-examples}\dotfill \pageref*{seminonexamples} \linebreak


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

The mathematical \textbf{red herring principle} is the principle that in [[mathematics]], a ``red herring'' need not, in general, be either red or a herring.

Frequently, in fact, it is conversely true that all herrings are red herrings. This often leads to mathematicians speaking of ``non-red herrings,'' and sometimes even to a redefinition of ``herring'' to include both the red and non-red versions.

\hypertarget{notes}{}\subsection*{{Notes}}\label{notes}

\begin{itemize}%
\item Clearly, the term ``red herring'' here is not to be confused with the usual meaning, which typically refers to a deliberate attempt to divert or throw one off track, as for example a rhetorical tactic for this purpose, or a novelistic device.


\item It should also not be thought that ``red herring'' as used here signals a pejorative, indicating for example ineptitude or lack of care in naming. For example, it may be that ``foo'' once meant a ``bar'', but over time the meaning of ``foo'' changed, so that while ``weak $*$-foo'' might seem strange if ``foo'' is taken in the modern sense, it made much better sense under the older (but mostly forgotten) meaning of ``foo''.



\end{itemize}
Thus, ``red herring'' as used here is to be interpreted neutrally: it refers to a name of a concept which might throw the reader off-track, by accident as it were.

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

\begin{itemize}%
\item A [[manifold with boundary]] is not a [[manifold]]. This leads to the use of ``manifold without boundary.''


\item In [[linear algebra]], an [[associative algebra|algebra]] is usually defined to be associative, so a [[nonassociative algebra]] is not an algebra in this sense. This leads to the technically redundant use of ``associative algebra''. Furthermore, a ``nonassociative algebra'' \emph{might} happen to be associative; it just doesn't \emph{have} to be!


\item Similarly, [[noncommutative geometry]] is really about \emph{not necessarily commutative} or \emph{possibly non-commutative} geometry. In fact, many tools developed in ``non-commutative geometry'' are also useful tools in ordinary commutative geometry.


\item A [[star-autonomous category|\emph{-autonomous category]] is not an [[autonomous category]], but the reverse is almost true: a}symmetric* autonomous (a.k.a. [[compact closed category|compact closed]]) category is a fairly special case of a $*$-autonomous one. This red herring is apparently an accident of history: at the time $*$-autonomous categories were invented, ``autonomous category'' was sometimes used to mean a [[closed monoidal category]], of which $*$-autonomous categories are indeed a special case, but nowadays that usage has mostly disappeared.


\item A [[linearly distributive category]] has essentially nothing to do with a [[distributive category]].


\item If a [[localizer]] refers to a class of maps in a [[presheaf category]], as it sometimes does, then a [[basic localizer]], being a class of maps in [[Cat]], is not a localizer.


\item A [[multivalued function]] is not a [[function]], but a function is a special case of a multivalued function.


\item A [[planar ternary ring]] is not a [[ring]]. A general ring is not a planar ternary ring either; their ``intersection'' is the class of [[division rings]].



\end{itemize}
Some adjectives are almost universally used as ``red herring adjectives,'' i.e. placing that adjective in front of something makes it \emph{more} general in some way. Some red herring adjectives almost always have the same meaning, such as ``pseudo'' and ``lax,'' but others, such as ``weak,'' have different meanings in different contexts.

\begin{itemize}%
\item A [[weak factorization system]] is not a factorization system, as originally defined, but rather the reverse. This has led some people to the use of [[orthogonal factorization system]] in place of the classical term ``factorization system.''


\item A [[weak limit]] is not a [[limit]] in a similar way: it satisfies only existence but not uniqueness.


\item Originally, [[n-category]] referred only to the strict version, so that a ``weak $n$-category'' was not an $n$-category. Nowadays some people (including many authors of the nLab) are trying to reverse this, so that $n$-category means the weak version, while the strict version needs an adjective added. Likewise for related concepts such as [[2-limit]] and [[strict 2-limit]].



\end{itemize}
Sometimes a red herring is only a red herring in degenerate case, perhaps only one case:

\begin{itemize}%
\item Fixing a [[natural number]] $n$, a [[homogeneous polynomial]] of degree $n$ is almost always a [[polynomial]] of degree $n$, but there is one exception: the [[zero]] polynomial has no degree (or is degree $-1$ or degree $-\infty$, depending on conventions), yet it is homogeneous of any degree.

\end{itemize}
\hypertarget{nonexamples}{}\subsection*{{Non-examples}}\label{nonexamples}

Some uses of terminology are similar in some ways, but don't quite fall under the same category. For instance, in a number of cases mathematicians working in a particular field tend to omit [[dichotomy between nice objects and nice categories|niceness]] adjectives, e.g.:

\begin{itemize}%
\item The generally accepted \emph{definition} of a [[ring]] does not include commutativity, but in [[algebraic geometry]] and [[commutative algebra]] it is an almost universal convention that all rings are taken to be commutative. However, almost every book concerning the subject announces its conventions early on.


\item Algebraic topologists usually work with a [[convenient category of topological spaces]] rather than the category of all [[topological spaces]], but usually they simply say ``space'' to mean an object of their chosen convenient category. Some algebraic topologists (particularly of the MIT school) take this even further and say ``space'' to mean [[simplicial set]]. In the Chicago school, this is occasionally stretched even further, with ``space'' meaning [[spectrum]] (although this is arguably more of a joke than a common usage).



\end{itemize}
These terminological uses can create situations that appear similar to actual red herrings, such as the use of ``noncommutative ring'' by people who are familiar with using ``ring'' to mean ``commutative ring.'' However, since the actual definitions of terms like ``ring'' and ``topological space'' is generally accepted to be unchanged (as opposed to the commonly used abbreviations), these are not true red herrings.

\hypertarget{seminonexamples}{}\subsubsection*{{Semi-non-examples}}\label{seminonexamples}

Sometimes, adding an adjective does not destroy the property that the noun has, but the noun now has its property in a different way, e.g.:

\begin{itemize}%
\item A [[contravariant functor]] from $C$ to $D$ \emph{is} still a functor, however not from $C$ to $D$, but from $C^{op}$ to $D$ (or equivalently from $C$ to $D^{op}$).

\end{itemize}
[[!redirects red herring]] [[!redirects red herring principle]] [[!redirects the red herring principle]]



\end{document}