# Adjoint functors

I'm attempting to cover my mind around adjoint functors. Among the instances I've seen is the groups $\bf IntLE \bf = (\mathbb{Z}, ≤)$ and also $\bf RealLE \bf = (\mathbb{R}, ≤)$, where the ceiling functor $ceil : \bf RealLE \rightarrow IntLE$ is left adjoint to the incorporation functor $incl : \bf IntLE \rightarrow RealLE$. I intend to examine that the adhering to hold true, as they appear to be:

- $floor : \bf RealLE \rightarrow IntLE$ would certainly be appropriate adjoint to $incl$
- Between the twin groups of $\bf IntGE \bf = (\mathbb{Z}, ≥)$ and also $\bf RealGE \bf = (\mathbb{R}, ≥)$, $ceil$ would certainly be appropriate adjoint to $incl$
- Between $\bf RealGE$ and also $\bf IntGE$, $floor$ would certainly be left adjoint to $incl$

Is my understanding deal with on these factors?

Arturo has actually currently uploaded a wonderful solution. I 'd just such as to stress that such global interpretations usually enable glossy evidence, as an example see listed below. For a far more striking instance see the theory in my post here, which offers a glossy one - line evidence of the LCM * GCD regulation using their global interpretations.

**LEMMA ** $\rm\: \ \lfloor x/(mn)\rfloor\ =\ \lfloor{\lfloor x/m\rfloor}/n\rfloor\ \ $
for $\rm\ \ n > 0$

**Proof ** $\rm\quad\quad\quad\quad\quad\quad\quad k\ \le \lfloor{\lfloor x/m\rfloor}/n\rfloor$

$\rm\quad\quad\quad\quad\quad\iff\quad\ \ k\ \le\ \:{\lfloor x/m\rfloor}/n$

$\rm\quad\quad\quad\quad\quad\iff\ \ nk\ \le\ \ \lfloor x/m\rfloor$

$\rm\quad\quad\quad\quad\quad\iff\ \ nk\ \le\:\ \ \ x/m$

$\rm\quad\quad\quad\quad\quad\iff\ \ \ \ k\ \le\:\ \ \ x/(mn)$

$\rm\quad\quad\quad\quad\quad\iff\ \ \ \ k\ \le\ \ \lfloor x/(mn)\rfloor $

Compare the above unimportant evidence to even more typical evidence, as an example the special case $\rm\ m = 1\ $ here.

Given groups $\mathcal{C}$ and also $\mathcal{D}$, and also functors $\mathbf{F}\colon\mathcal{C}\to\mathcal{D}$ and also $\mathbf{U}\colon\mathcal{D}\to\mathcal{C}$, $\mathbf{F}$ is the left adjoint of $\mathbf{U}$ if and also just if for every single things $C\in\mathcal{C}$ and also $D\in\mathcal{D}$ there is an all-natural bijection in between $\mathcal{C}(C,\mathbf{U}(D))$ and also $\mathcal{D}(\mathbf{F}(C),D)$.

Allow me make use of $\leq$ and also $\geq$ for the relationship amongst reals, and also $\preceq, \succeq$ for the relationship amongst integers.

For $ceil$ to be the left adjoint of the incorporation functor, you would certainly require that for all actual numbers $r$ and also all integers $z$, $\lceil r\rceil \preceq z$ if and also just if $r\leq z$. This holds, so you do have an ajunction (this functions due to the fact that in these groups, the morphism set **IntLE **$(a,b)$ is vacant if $a\not\preceq b$, and also has an one-of-a-kind arrowhead if $a\preceq b$ ; and also in a similar way with **RealLE ** ; so you get an all-natural bijection in between the collections of arrowheads if and also just if they are either both vacant or both are singletons at the very same time).

For $floor$ to be an appropriate adjoint to $incl$, you would certainly require that for all actual numbers $r$ and also all integers $z$, $z\preceq \lfloor r\rfloor$ if and also just if $z\leq r$, which once more holds true ; so $floor$ is an appropriate adjoint to the incorporation functor.

For $ceil$ to be the appropriate adjoint to the incorporation functor in the twin groups, you would certainly require $z\succeq \lceil r \rceil$ if and also just if $z\geq r$ ; and also for $floor$ to be the left adjoint, you would certainly require $\lfloor r\rfloor \succeq z$ if and also just if $r\geq z$. Both hold, so your assertions 1 via 3 are proper.

P.S. Let me 2nd Mariano is pointer in the remarks to remember the instance of the underlying set functor and also the free team functor for thinking of right and also left adjoints. I locate myself returning to those 2 every single time I require to advise myself of just how points collaborate with adjoints, what adjoints regard or do not regard, and also specifically when thinking of several of the various other equal interpretations, specifically the one in regards to the device and also carbon monoxide - device of the adjunction (which are all-natural makeovers in between the identification functors and also the functors $\mathbf{FU}$ and also $\mathbf{UF}$).

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