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\begin{document}

%-------------------------------------------------------------------


\section*{floor}

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

\hypertarget{arithmetic}{}\paragraph*{{Arithmetic}}\label{arithmetic}

[[!include arithmetic geometry - contents]]

\hypertarget{floors_and_ceilings}{}\section*{{Floors and ceilings}}\label{floors_and_ceilings}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{AsAdjointFunctors}{As adjoint functors}\dotfill \pageref*{AsAdjointFunctors} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The floor and ceiling of a [[real number]] are [[integers]], the result of rounding the real number down or up (respectively). When viewed as [[functions]] from $\mathbb{R}$ to itself (or to $\mathbb{Z}$), these are standard examples of functions exhibiting partial notions of [[continuity]]: the floor function is both [[right-continuous map|right-continuous]] and [[upper semicontinuous map|upper semicontinuous]], while the ceiling function is both [[left-continuous map|left-continuous]] and [[lower semicontinuous map|lower semicontinuous]]. They are [[step functions]] used to approximate definite [[integrals]] of [[continuous maps]] and otherwise to relate integrals and [[series]]. They provide convenient notation to express various notions of [[rounding]]. From the perspective of [[order theory]], the maps may be seen as the [[right adjoint]] and [[left adjoint]] (respectively) of the [[inclusion map]] from $\mathbb{Z}$ to $\mathbb{R}$.

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

Given a [[real number]] $x$, the \textbf{floor} of $x$, denoted $\lfloor{x}\rfloor$ or $[x]$, is the largest integer $n$ such that $n \leq x$, and the \textbf{ceiling} of $x$, denoted $\lceil{x}\rceil$, is the smallest integer $n$ such that $n \geq x$.

Typically the notation $\lfloor{x}\rfloor$ is used when both floor and ceiling appear, but $[x]$ is often easier when only the floor is considered. Since

\begin{displaymath}
\lceil{x}\rceil = -\lfloor{-x}\rfloor ,
\end{displaymath}
this is not actually a restriction.

The floor of $x$ is also called the \textbf{integer part} of $x$, and then one refers as well to the \textbf{fractional part} of $x$, denoted $\{x\}$, defined by

\begin{displaymath}
\{x\} = x - [x] .
\end{displaymath}
One must of course prove that the floor of $x$ exists; this fails in [[constructive mathematics]], although the floor of $x$ exists for [[full set|almost all]] $x$, and all of these functions can still be defined as continuous maps between appropriate [[locales]].

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

\hypertarget{AsAdjointFunctors}{}\subsubsection*{{As adjoint functors}}\label{AsAdjointFunctors}

We discuss how the floor and celing functions are left and right [[adjoint functors]] to the inclusion of the [[integers]] into the [[real numbers]], if both are regarded as [[posets]], and hence as [[categories]].

\begin{example}
\label{PartiallyOrderedSetsAsSmallCategories}\hypertarget{PartiallyOrderedSetsAsSmallCategories}{}
\textbf{([[preordered sets]] as [[thin categories]])}

Let $(S, \leq)$ be a [[preordered set]]. Then this induces a [[small category]] whose [[set]] of [[objects]] is $S$, and which has precisely one morphism $x \to y$ whenever $x \leq y$, and no such morphism otherwise:

\begin{equation}
x \overset{\exists !}{\to} y
  \phantom{AAA}
  \text{precisely if}
  \phantom{AAA}
  x \leq y
\label{RelationsAsMorphismInPartiallyOrderedSet}\end{equation}
Conversely, every [[small category]] with at most one morphism from any object to any other, called a \emph{[[thin category]]}, induces on its set of objects the [[structure]] of a [[partially ordered set]] via \eqref{RelationsAsMorphismInPartiallyOrderedSet}.

Here the [[axioms]] for [[preordered sets]] and for [[categories]] match as follows:

\newline | $\phantom{A}$[[partially ordered sets]]$\phantom{A}$ | $\phantom{A}$ $x \leq x$ $\phantom{A}$ | $\phantom{A}$ $(x \leq y \leq z) \Rightarrow (x \leq z)$ $\phantom{A}$ | | $\phantom{A}$[[thin categories]]$\phantom{A}$ | $\phantom{A}$[[identity morphisms]]$\phantom{A}$ | $\phantom{A}$[[composition]]$\phantom{A}$ |

\end{example}
\begin{prop}
\label{FloorAndCeilingAsAdjointFunctors}\hypertarget{FloorAndCeilingAsAdjointFunctors}{}
\textbf{([[floor]] and [[ceiling]] as [[adjoint functors]])}

Consider the canonical inclusion

\begin{displaymath}
\mathbb{Z}_{\leq} 
    \overset{\phantom{AA}\iota \phantom{AA}}{\hookrightarrow} 
  \mathbb{R}_{\leq}
\end{displaymath}
of the [[integers]] into the [[real numbers]], both regarded as [[preorders]] in the standard way (``lower or equal''). Regarded as [[full subcategory]]-inclusion of the corresponding [[thin categories]], via Example \ref{PartiallyOrderedSetsAsSmallCategories}, this inclusion functor has both a left and right [[adjoint functor]]:

\begin{itemize}%
\item the [[left adjoint]] to $\iota$ is the [[ceiling function]]


\item the [[right adjoint]] to $\iota$ is the [[floor function]]



\end{itemize}
\begin{equation}
\lceil(-)\rceil \;\;\dashv\;\; \iota \;\;\dashv\;\; \lfloor (-) \rfloor
  \,.
\label{AdjointTriple}\end{equation}
Hence this induces an [[adjoint modality]], as discussed there.

The [[adjunction unit]] and [[adjunction counit]] express that each real number is in between its floor and ceiling

\begin{displaymath}
\iota \lfloor x \rfloor \;\leq\; x \;\leq\; \iota \lceil x \rceil
\end{displaymath}
Hence the [[modal objects]] in both cases are the [[integers]] among the [[real numbers]], while every real number is ceiling-submodal and floor-supmodal.

\end{prop}
\begin{proof}
First of all, observe that we indeed have [[functors]]

\begin{displaymath}
\lfloor(-)\rfloor \;,\;
  \lceil(-)\rceil
  \;\;\colon\;
  \mathbb{R}
    \longrightarrow
  \mathbb{Z}
\end{displaymath}
since floor and ceiling preserve the ordering relation.

Now in view of the identification of [[preorders]] with [[thin categories]] in Example \ref{PartiallyOrderedSetsAsSmallCategories}, the hom-isomorphism defining [[adjoint functors]] of the form $\iota \dashv \lfloor(-)\rfloor$ says for all $n \in \mathbb{Z}$ and $x \in \mathbb{R}$, that we have

\begin{displaymath}
\underset{ \in \mathbb{Z}}{\underbrace{n \leq \lfloor x \rfloor}}
  \;\Leftrightarrow\;
  \underset{ \in \mathbb{R}}{\underbrace{n \leq x }}
  \,.
\end{displaymath}
This is clearly already the defining condition on the [[floor]] function $\lfloor x \rfloor$.

Similarly, the hom-isomorphism defining [[adjoint functors]] of the form $\lceil(-)\rceil \dashv \iota$ says that for all $n \in \mathbb{Z}$ and $x \in \mathbb{R}$, we have

\begin{displaymath}
\underset{ \in \mathbb{Z}}{\underbrace{\lceil x \rceil \leq n}}
  \;\Leftrightarrow\;
  \underset{ \in \mathbb{R}}{\underbrace{x \leq n }}
  \,.
\end{displaymath}
This is evidently already the defining condition on the [[floor]] function $\lfloor x \rfloor$.

Notice that in both cases the condition of a \emph{[[natural isomorphism]]} in both variables, as required for an [[adjunction]], is automatically satisfied: For let $x \leq x'$ and $n' \leq n$, then naturality means, again in view of the identifications in Example \ref{PartiallyOrderedSetsAsSmallCategories}, that

\begin{displaymath}
\itexarray{ 
    (n \leq \lfloor x \rfloor) &\Leftrightarrow& (n \leq x)
    \\
    \Downarrow && \Downarrow
    \\
    (n' \leq \lfloor x' \rfloor) &\Leftrightarrow& (n' \leq x')
    \\
    \\
    \in \mathbb{Z} && \in \mathbb{R}
  }
\end{displaymath}
where the logical implications are equivalently functions between [[sets]] that are either [[empty set|empty]] or [[singletons]]. But Functions between such sets are unique, when they exist.

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

Wikipedia summarizes the basic properties:

\begin{itemize}%
\item English Wikipedia. Floor and ceiling functions. \href{https://en.wikipedia.org/wiki/Floor_and_ceiling_functions}{Web}.

\end{itemize}
Chapter 6 of these notes uses the floor and ceiling functions throughout to relate [[integrals]] and [[series]] when teaching [[infinitesimal calculus]]:

\begin{itemize}%
\item Toby Bartels. One-variable Calculus. \href{http://tobybartels.name/MATH-1600/2017FA/calcbook/}{Web}.

\end{itemize}
[[!redirects floor]] [[!redirects floors]] [[!redirects floor operation]] [[!redirects floor operator]] [[!redirects floor function]] [[!redirects floor functor]] [[!redirects floor map]] [[!redirects floor mapping]]

[[!redirects ceiling]] [[!redirects ceilings]] [[!redirects ceiling operation]] [[!redirects ceiling operator]] [[!redirects ceiling function]] [[!redirects ceiling functor]] [[!redirects ceiling map]] [[!redirects ceiling mapping]]



\end{document}