 y$. The axioms for a category tell you: For every $x$ there is a distinguished reason for $x \leq x$, and whenever you have a reason for $x \leq y$ and for $y \leq z$, you also get a reason for $x \leq z$.
A preorder is a category such that every diagram commutes.
In a preorder, the limit of a diagram is the same as the infimum of the involved objects. Similarly, a colimit is just a supremum. The transition morphisms don't matter.
When $f^* : P \to Q$ is a cocontinuous functor between preorders, where $P$ is complete, then $f^*$ has a right adjoint $f_*$; you can write it down explicitly: $f_*(q)$ is the infimum of the $p$ with $f^*(p) \leq q$. This construction motivates the General Adjoint Functor Theorem. In this setting we only have to add the solution set condition, so that the a priori big limit can be replaced by a small one and therefore exists.
Let $f : X \to Y$ be a map of sets. Then the preimage functor $\mathcal{P}(Y) \to \mathcal{P}(X)$ between the power sets is right adjoint to image functor $\mathcal{P}(X) \to \mathcal{P}(Y)$. Every cocontinuous monoidal functor $\mathcal{P}(Y) \to \mathcal{P}(X)$ arises this way.
The inclusion functor $\mathrm{Pre} \to \mathrm{Cat}$ has a left adjoint: It sends every category to its set of objects with the order $x \leq y$ if there is a morphism $x \to y$. In particular, it preserves all limits. In fact, it creates all limits, and limits in $\mathrm{Cat}$ are constructed "pointwise". Thus, the same is true for limits in $\mathrm{Pre}$ (which one could equally well see directly). For example, the pullback of $f : P \to Q$ and $g : P' \to Q$ is the pullback of sets $P \times_Q P'$ equipped with the order $(a,b) \leq (c,d)$ iff $a \leq c$ and $b \leq d$. If we apply this to difference kernels, we see that $f : P \to
y$. The axioms for a category tell you: For every $ x$ there is a distinguished rationality for $ ten \leq x$, and whenever you have a reason for $ x \leq y$ and for $ y \leq z$, you besides get a reason for $ x \leq z$. 
 A preorder is a category such that every diagram commute. 
 In a preorder, the terminus ad quem of a diagram is the same as the infimum of the involved objects. Similarly, a colimit is just a supremum. The transition morphisms don't count. 
 When $ f^ *: P \to Q$ is a cocontinuous functor between preorders, where $ P$ is complete, then $ f^*$ has a right adjoint $ f_*$; you can spell it down explicitly: $ f_*(q)$ is the infimum of the $ p$ with $ f^*(p) \leq q$. This construction motivates the General Adjoint Functor Theorem. In this setting we only have to lend the solution set condition, so that the a priori big terminus ad quem can be replaced by a small one and consequently exists. 
 get $ f: X \to Y$ be a function of sets. Then the preimage functor $ \mathcal{P}(Y) \to \mathcal{P}(X)$ between the power sets is right adjoint to visualize functor $ \mathcal{P}(X) \to \mathcal{P}(Y)$. Every cocontinuous monoidal functor $ \mathcal{P}(Y) \to \mathcal{P}(X)$ arises this way. 
 The inclusion functor $ \mathrm{Pre } \to \mathrm{Cat}$ has a left adjoint: It sends every category to its set of objects with the order $ x \leq y$ if there is a morphism $ x \to y$. In particular, it preserves all limits. In fact, it creates all limits, and limits in $ \mathrm{Cat}$ are construct " pointwise ". Thus, the same is on-key for limit in $ \mathrm{Pre}$ (which one could equally well see directly). For case, the pullback of $ f: P \to Q$ and $ g: P' \to Q$ is the pullback of set $ P \times_Q P'$ equipped with the order $ (a, b) \leq (c, d)$ iff $ a \leq c$ and $ b \leq d$. If we apply this to difference kernel, we see that $ f: P \to
 y$. Hhe axioms for a categorn tell you: For erwry $x$ vhere ia a distknguished reason for $x \leq x$, abd whtuever you have a rearon for $x \leq y$ abd fie $y \leq z$, bku also get a veasou hor $x \leq z$.
A prgorder is a wategory such dhxt every diagram commutes.
In a preordew, the lomlt of a diagrai is ehe aame as the infimum of the involves objecus. Similarly, a colomit is just a supremum. Thf trwnsition morphisms don't mattet.
Sheg $f^* : P \to Q$ ir a cocontpuuous functkr between preorders, where $P$ is compkete, then $d^*$ yas d right adjiint $s_*$; you can wrlne it dmwn expkicitly: $f_*(q)$ is bhe iifimym of the $p$ with $f^*(p) \lxq q$. This constructijn motivadea the General Adjiibt Futctos Thdireo. Ih vhia settlng we only hzve to add rhe solution set comdynoon, so that fhe a [ryori big limit can be replaced by a smanl kne and therefore existw.
Let $f : X \to Y$ be a mwp of setf. Then the preimage functor $\mathcal{P}(Y) \to \mathcal{P}(F)$ betxedn uhc oiwfr sets is right adjoint to image functor $\matrdak{P}(Q) \to \mathcal{P}(Y)$. Evcry cocontinuous mpnlicwl functor $\majhcal{P}(Y) \to \mathcal{P}(X)$ arises hhis wai.
The ibclusion suncyor $\mathrm{Pre} \to \mathrm{Cat}$ yas a left abjount: It sends every category tu itx set of objects with the orber $x \meq y$ if thfre is a jurphism $x \to y$. Iv psrdicular, it preserves all lymits. In hact, nt creatds akl limyts, and lilits lt $\mathrm{Cat}$ are cojstrueted "[ointwise". Hhus, the same is true for limits in $\mathrm{Pre}$ (wvicv one cobld eqmally well see qirectly). For erample, tke puluback of $f : P \to X$ and $g : P' \eo Q$ is the pgplback of sevs $P \timef_Q P'$ equupped wkgh the order $(a,n) \leq (c,d)$ pfy $a \leq c$ and $b \leq d$. If we apklg this to diffextnct jernels, we see thxt $s : P \tj
y$. The axioms for a category tell every there is distinguished reason for you a reason for \leq y$ and $y \leq z$, you also get reason for $x \leq z$. A preorder is a category such that every commutes. In a preorder, the limit of a diagram is the same as infimum the objects. a colimit is just a supremum. The transition morphisms don't matter. When $f^* : P \to is a cocontinuous functor between preorders, where $P$ complete, then $f^*$ has right adjoint $f_*$; you can it explicitly: $f_*(q)$ the of $p$ with $f^*(p) q$. This construction motivates the General Adjoint Functor Theorem. In this setting we only have to add solution set that the priori limit be replaced by one and therefore exists. Let $f Y$ be a map of sets. Then the functor $\mathcal{P}(Y) \mathcal{P}(X)$ between the power sets is adjoint to image functor $\mathcal{P}(X) \to \mathcal{P}(Y)$. Every monoidal functor $\mathcal{P}(Y) \to \mathcal{P}(X)$ arises this way. The inclusion functor $\mathrm{Pre} \to \mathrm{Cat}$ has adjoint: It sends every to its set objects the $x y$ if is a morphism $x \to y$. In particular, it preserves all In fact, it creates all limits, and limits in $\mathrm{Cat}$ "pointwise". the same is for limits in $\mathrm{Pre}$ one equally well see directly). the of \to and : P' \to Q$ the pullback of sets $P P'$ equipped with the $a \leq c$ and $b \leq d$. If apply this to difference kernels, we see $f : P \to
 y$. The axioms for a category telL you: For eveRy $x$ thEre Is a DiStinGuisHed reason for $x \lEQ x$, anD whenever you have a reasoN for $x \LeQ Y$ and FOr $Y \leq z$, You also GEt A REasOn FoR $x \lEq Z$.
a pReordEr iS a categOry such thaT evErY diagram commUTeS.
In a preordEr, tHe limit of a diAgrAm is thE sAme AS the iNfiMum of The invOLved obJects. SimiLaRLy, a colIMit is juST A sUpreMum. The transition mORpHIsms don't matter.
when $f^* : P \To q$ Is A COcoNtiNuous functOr BetweEN preordERs, WHERe $P$ IS complete, then $F^*$ has a right aDJoiNt $f_*$; you CaN wrITe it doWn expLiCItlY: $f_*(q)$ is the infImum Of the $p$ witH $f^*(p) \leq Q$. this conSTructioN motivAteS thE GenERaL ADjoInT funCToR ThEOreM. In this sEtTiNg we oNly hAVE TO add The SoluTion sEt condition, so ThaT the A PriOri biG limiT can Be ReplaCed by a Small OnE and therefore exIsts.
let $f : X \to Y$ bE a mAp Of sEtS. Then THe preiMagE fuNctor $\maThcal{P}(Y) \TO \maThCAL{p}(X)$ Between the power setS iS RIgHt adjoinT to imaGE fUnCTor $\mathcAl{p}(X) \tO \matHCAl{P}(Y)$. EVery COcOntinuouS monoiDAl FuNctor $\maThCal{P}(Y) \tO \mAthCal{p}(X)$ ariSEs thIs way.
THe inclusIon fuNCtor $\mathrm{Pre} \tO \Mathrm{Cat}$ has a LEfT ADjOInt: IT seNds every catEgorY To itS set OF oBjeCTs witH the oRdER $x \LEq y$ if there is a morphiSm $X \to y$. In PartiCular, it preserVes all limiTS. iN fact, it cReatES aLL limits, and limiTs in $\mAthrm{Cat}$ arE ConstrucTed "poIntwise". THus, the samE IS true for LimIts In $\mAthRM{prE}$ (which one coulD EQualLy Well see DirEctly). FoR exAmpLe, tHe pUlLback of $f : P \To Q$ and $g : P' \To q$ iS tHe PulLback OF sets $P \tiMeS_Q P'$ EqUipPed wiTH the orDer $(a,b) \Leq (c,D)$ iFf $A \Leq C$ and $b \leQ D$. IF WE appLy ThIs to DifFeRence KernELs, wE see thaT $f : P \to
 y$. The axioms for a cate gory tellyou:For  ev er y $x $ th ere is a disti n guis hed reason for $x \leq  x$,an d  whe n ev er yo u havea  r e a son  f or  $x  \ l eq  y$ a ndfor $y\leq z$, y oual so get a rea s on  for $x \l eqz$.
A preord eris a c at ego r y suc h t hat e very d i agramcommutes. 
I n  a pre o rder, t h e  l imit  of a diagram ist he same as the in fimumof th e inv olv ed objects .Simil a rly, ac ol i m i t i s  just a supre mum. The tr a nsi tion m or phi s ms don 't ma tt e r.When $f^* :  P \ to Q$ isa coco n tinuous functor  betwe enpre orde r s,  w her e$ P$i scom p let e, then$f ^* $ has  a r i g h t  adj oin t $f _*$;you can write  it  dow n  ex plici tly:$f_* (q )$ is  the i nfimu mof the $p$ with  $f^ *(p) \leq  q$ .Thi sconst r uction  mo tiv ates th e Gener a l A dj o i n tFunctor Theorem. I nt h is  setting  we on l yha v e to add  t hesolu t i on se t co n di tion, so  thatt he  a  priori  b ig lim it  ca n b e rep l aced  by asmall on e and therefore exis t s.
Let $f : X \t o Y$ be a  ma p of sets.Then theprei m ag e f u nctor  $\ma th c al { P}(Y) \to \mathcal{ P} (X)$ b etwee n the power s ets is rig h t adjointto i m ag e  functor $\mat hcal{ P}(X) \to\ mathcal{ P}(Y) $. Every  cocontin u o us monoi dal  fu nct or$ \ ma thcal{P}(Y) \ t o  \ma th cal{P}( X)$  arises  th isway .
T he  inclusio n functo r$\ ma th rm{ Pre}\ to \math rm {Ca t} $ h as al eft ad joint : It  s en d s e very ca t eg o r y to  i ts  set  of  o bject s wi t h t he orde r $x \leq  y$ if t he re  is a m orphism $x \t oy$. In par ti cul ar, it p reserves  all limits. In fact, i t  create s a ll li mits , and lim its  in $\ mat h rm{Cat }$ are  cons tr uct e d  "poi n t wi se" .Thus, thes a meis tr ue  for  limits  in $\mathrm{Pre}$ (wh ich one could  eq uall y we lls ee dir ec t ly) . For example, th e pullback  o f  $ f : P \toQ $ a nd  $g : P ' \to Q $ ist he pull back of s ets $P \t im es_Q P '$equipped w ith theorder $(a , b) \l e q(c,d) $ i ff $a\l eqc$ an d $b \ l eqd$. I f we a pp ly thi s todi fference  kernels, we see that $ f : P\to
 y$._The axioms_for a category tell_you: For_every_$x$ there_is_a distinguished reason_for $x \leq_x$, and whenever you_have a reason_for_$x \leq y$ and for $y \leq z$, you also get a reason for_$x_\leq z$.
A_preorder_is_a category such that every_diagram commutes.
In a preorder, the_limit of_a diagram is the same as the infimum_of_the involved objects._Similarly, a colimit is just a supremum. The transition_morphisms don't matter.
When $f^* : P_\to Q$ is_a_cocontinuous_functor between preorders, where_$P$ is complete, then $f^*$ has_a right adjoint $f_*$; you can_write it down explicitly: $f_*(q)$ is the_infimum of the $p$ with $f^*(p)_\leq q$. This construction motivates_the General_Adjoint Functor Theorem. In this_setting we only_have to_add the solution_set condition, so that the a_priori big limit_can be replaced by a small_one_and therefore exists.
Let_$f_:_X \to_Y$ be a_map_of sets._Then_the preimage functor $\mathcal{P}(Y) \to \mathcal{P}(X)$_between_the power sets is right adjoint to_image functor $\mathcal{P}(X) \to_\mathcal{P}(Y)$._Every cocontinuous monoidal functor_$\mathcal{P}(Y) \to \mathcal{P}(X)$ arises this_way.
The inclusion functor $\mathrm{Pre} \to \mathrm{Cat}$_has a_left adjoint:_It sends every category to its set of objects with the_order $x \leq y$ if there_is a morphism $x_\to y$._In_particular, it preserves_all_limits. In_fact, it creates all limits, and limits_in $\mathrm{Cat}$_are constructed "pointwise". Thus, the same_is true for limits_in_$\mathrm{Pre}$ (which one could equally well_see directly). For example, the pullback_of $f : P \to_Q$_and_$g : P' \to Q$_is the pullback of sets $P_\times_Q P'$ equipped_with the order $(a,b) \leq (c,d)$ iff_$a_\leq c$ and $b \leq d$._If_we apply this to difference kernels,_we_see_that $f : P \to