
0-cells in Topoi are Grothendieck topoi.
1-cells are geometric morphisms and have the direction of the right adjoint.
2-cells are invertible natural transformation (between the left adjoints).

Johnstone (and Lurie) proves that this 2-category has pseudo colimits that can be computed in Cat. To be precise, given a diagram in Topoi its colimit is the limit of the corresponding diagram in Cat associated to the inverse images.

Q. Does the representable functor $\mathsf{pt}:= \text{Topoi(Set,  )} $ preserve directed colimits?

After some naive optimism, I have the feeling that the answer is no, but I suspect that it might be known.

A candidate counterexample would be a topos $\mathcal{E}$ whose subtopoi lattice is not infinitary distributive, in fact a point is an atom in this coHeyting algebra.
Another idea is to build a topos with with some points as a directed colimit of topoi without points, but I am not sure that this is possible at all.
If a counterexample exist, I expect that it's possible to exhibit a localic counterexample.

A:

If $\mathcal{C}$ is a small category with finite limits then geometric morphisms from ${\rm Set}$ to the presheaf topos ${\rm PSh}(\mathcal{C})$ are in bijection with left exact functors $\mathcal{C} \to {\rm Set}$, or, equivalently, with objects in the pro-category ${\rm Pro}({\cal C})$. More explicitly, if ${\bf X} = \{X_i\}_{i \in {\cal I}}$ is a pro-object in ${\cal C}$ then the corresponding point ${\bf X}^*:{\rm PSh}(\mathcal{C}) \to {\rm Set}$ sends a presheaf $F :\mathcal{C}^{\rm op} \to {\rm Set}$ to ${\rm colim}_{i \in {\cal I}} F(X_i)$. If ${\cal C}_1,{\cal C}_2$ are two categories with finite limits and $f: {\cal C}_1 \to {\cal C}_2$ is a functor (which does not necessarily preserve finite limits) then we have a geometric morphism $f_*: {\rm PSh}(\mathcal{C}_1) \leftrightarrows {\
0 - cells in Topoi are Grothendieck topoi. 
 1 - cells are geometric morphisms and have the direction of the proper adjoint. 
 2 - cell are invertible natural transformation (between the left adjoints). 

 Johnstone (and Lurie) proves that this 2 - class has pseudo colimits that can be computed in Cat. To be precise, give a diagram in Topoi its colimit is the limit of the corresponding diagram in Cat associate to the inverse images. 

 Q. Does the representable functor $ \mathsf{pt}:= \text{Topoi(Set,  ) } $ preserve address colimits? 

 After some naive optimism, I have the spirit that the solution is no, but I suspect that it might be known. 

 A campaigner counterexample would be a topos $ \mathcal{E}$ whose subtopoi lattice is not infinitary distributive, in fact a distributor point is an atom in this coHeyting algebra. 
 Another mind is to build a topos with with some points as a directed colimit of topos without points, but I am not sure that this is potential at all. 
 If a counterexample exist, I have a bun in the oven that it's possible to show a localic counterexample. 

 A: 

 If $ \mathcal{C}$ is a small category with finite terminus ad quem then geometric morphisms from $ { \rm Set}$ to the presheaf topos $ { \rm PSh}(\mathcal{C})$ are in bijection with left exact functors $ \mathcal{C } \to { \rm Set}$, or, equivalently, with objects in the pro - category $ { \rm Pro}({\cal C})$. More explicitly, if $ { \bf X } = \{X_i\}_{i \in { \cal I}}$ is a pro - object in $ { \cal C}$ then the comparable detail $ { \bf X}^*:{\rm PSh}(\mathcal{C }) \to { \rm Set}$ sends a presheaf $ F: \mathcal{C}^{\rm op } \to { \rm Set}$ to $ { \rm colim}_{i \in { \cal I } } F(X_i)$. If $ { \cal C}_1,{\cal C}_2 $ are two categories with finite limits and $ f: { \cal C}_1 \to { \cal C}_2 $ is a functor (which does not necessarily preserve finite limits) then we have a geometric morphism $ f _ *: { \rm PSh}(\mathcal{C}_1) \leftrightarrows { \

0-celps in Topoi are Grothendleck topoi.
1-cells cee geoketric morphisos and have the direction of tye ritht adjoint.
2-cells are ivvertible natural traisformation (betwxsn the left asmointv).

Oohnstone (and Lorie) proves dhat this 2-catecofy has pseudo colimits that can be coiputed on Cat. To be presise, diveh a diagram in Topoi its colimit ia the lpmit of the correxponding diagram in Cat asdociwted to the inversf images.

Q. Dies eye representxble functor $\mathsf{pt}:= \jext{Topoi(Set,  )} $ preserve directea colnmits?

After wone jdive optimiwm, I rave the feeling thad the amswer is no, bub I snspext that it might be kiown.

A candidate counjerexample wkuld be a topos $\mqtycal{E}$ whove sjvtoooi lettjce is nov infinitarg distributuve, in fact a point if an atom in tgis coRerting algebra.
Another idea is to build a tolos with with some poinrs as a directed colilit of to[oi without points, but I am not sure that this is possmbue cb xol.
Lf a counterexample exist, I expect that it's pjasobke to exhibit c localic countetedakkle.

A:

If $\mathcal{Z}$ is a smzll category with vinite jimitw then gejmettic morphisms from ${\rm Set}$ ti the preshecf ropos ${\rm PSh}(\mathcap{C})$ are in bnjectipn wiyh left exact functors $\oathdal{C} \to {\rm Det}$, or, eqhkvalently, with ocjebts hn the pro-category ${\rm Pro}({\cwl C})$. More expkicitly, if ${\nf X} = \{V_i\}_{i \in {\cal I}}$ is a pro-object in ${\cal C}$ thgn the corresponfing point ${\bf X}^*:{\rm PSh}(\mathcal{C}) \to {\rm Set}$ sends a prvsheaf $F :\iathcsl{C}^{\rm op} \to {\ri Set}$ to ${\rm cokim}_{i \in {\cal K}} F(X_i)$. If ${\czl C}_1,{\cal C}_2$ are two sategories widj finite limmts and $f: {\cal C}_1 \ti {\cal C}_2$ ks a functor (wnich does not neceswarily preserve fikite ujmits) then we hcre a geometric motphksm $f_*: {\rk [Vh}(\mathcal{C}_1) \laftrkghgsrrowr {\
0-cells in Topoi are Grothendieck topoi. 1-cells morphisms have the of the right transformation the left adjoints). (and Lurie) proves this 2-category has pseudo colimits that be computed in Cat. To be precise, given a diagram in Topoi its is the limit of the corresponding diagram in Cat associated to the inverse Q. the functor \text{Topoi(Set, )} $ preserve directed colimits? After some naive optimism, I have the feeling that the is no, but I suspect that it might known. A candidate counterexample be a topos $\mathcal{E}$ whose lattice not infinitary in a is an atom this coHeyting algebra. Another idea is to build a topos with with some points as a directed of topoi but I not that is possible at a counterexample exist, I expect that exhibit a localic counterexample. A: If $\mathcal{C}$ is small category finite limits then geometric morphisms from Set}$ to the presheaf topos ${\rm PSh}(\mathcal{C})$ are bijection with left exact functors $\mathcal{C} \to {\rm Set}$, or, equivalently, with objects in the Pro}({\cal C})$. More explicitly, ${\bf X} = \in I}}$ a in ${\cal then the corresponding point ${\bf X}^*:{\rm PSh}(\mathcal{C}) \to {\rm Set}$ sends presheaf $F :\mathcal{C}^{\rm op} \to {\rm Set}$ to ${\rm colim}_{i I}} If ${\cal C}_1,{\cal are two categories with limits $f: {\cal C}_1 \to is functor necessarily finite then we have a morphism $f_*: {\rm PSh}(\mathcal{C}_1) \leftrightarrows

0-cells in Topoi are GrothendieCk topoi.
1-celLs are GeoMetRiC morPhisMs and have the diREctiOn of the right adjoint.
2-celLs are InVErtiBLe NaturAl transFOrMATioN (bEtWeeN tHE lEft adJoiNts).

JohnStone (and LuRie) PrOves that this 2-CAtEgory has psEudO colimits thaT caN be comPuTed IN Cat. TO be PreciSe, giveN A diagrAm in Topoi ItS ColimiT Is the liMIT oF the Corresponding diagRAm IN Cat associated To the iNvERsE IMagEs.

Q. does the repReSentaBLe functOR $\mATHSf{pT}:= \Text{Topoi(Set,  )} $ pReserve direCTed ColimiTs?

aftER some nAive oPtIMisM, I have the feElinG that the aNswer iS No, but I sUSpect thAt it miGht Be kNown.

a CaNdIdaTe COunTErExaMPle Would be a ToPoS $\mathCal{E}$ WHOSE subTopOi laTtice Is not infinitaRy dIstrIButIve, in Fact a PoinT iS an atOm in thIs coHEyTing algebra.
AnotHer iDea is to buIld A tOpoS wIth wiTH some pOinTs aS a direcTed coliMIt oF tOPOI wIthout points, but I am NoT SUrE that thiS is posSIbLe AT all.
If a cOuNteRexaMPLe exiSt, I eXPeCt that it'S possiBLe To Exhibit A lOcalic CoUntEreXamplE.

a:

If $\mAthcal{c}$ is a smalL cateGOry with finite lIMits then geomeTRiC MOrPHismS frOm ${\rm Set}$ to thE preSHeaf TopoS ${\Rm pSh}(\MAthcaL{C})$ are In BIjECtion with left exact fUnCtors $\mAthcaL{C} \to {\rm Set}$, or, eqUivalently, WITH objects In thE PrO-Category ${\rm Pro}({\cAl C})$. MoRe explicitLY, if ${\bf X} = \{X_i\}_{I \in {\caL I}}$ is a pro-Object in ${\cAL c}$ then the CorResPonDinG POiNt ${\bf X}^*:{\rm PSh}(\matHCAl{C}) \tO {\rM Set}$ senDs a PresheaF $F :\mAthCal{c}^{\rm Op} \To {\rm Set}$ to ${\Rm colim}_{i \In {\CaL I}} f(X_I)$. If ${\Cal C}_1,{\cAL C}_2$ are two CaTegOrIes With fINite liMits aNd $f: {\cAl c}_1 \tO {\Cal c}_2$ is a funCToR (WHich DoEs Not nEceSsArily PresERve Finite lImits) then We hAVe a gEoMeTric morPhism $f_*: {\rm PSh}(\maThCal{C}_1) \leftriGhTarRows {\

0-cells in Topoi are Grot hendieck t opoi. 
1- cel ls  are  geo metric morphis m s an d have the direction o f the  r i ghta dj oint. 
2-cell s  a r e  in ve rt ibl en at uraltra nsforma tion (betw een  t he left adjo i nt s).

Johns ton e (and Lurie ) p rovesth att his 2 -ca tegor y hasp seudocolimitsth a t canb e compu t e din C at. To be precise ,  g i ven a diagramin Top oi it s col imi t is the l im it of the cor r es p o n din g  diagram in C at associat e d t o thein ver s e imag es.
Q. Doe s the repre sent able func tor $\ m athsf{p t }:= \te xt{Top oi( Set ,  ) }  $  p res er v e d i re cte d  co limits?
A ft er so me n a i v e  opt imi sm,I hav e the feeling  th at t h e a nswer  is n o, b ut  I su spectthatit  might be known .

A  candidat e c ou nte re xampl e  would  be  atopos $ \mathca l {E} $w h o se  subtopoi latticeis n ot  infinit ary di s tr ib u tive, in  f act  a p o i nt is  ana to m in thi s coHe y ti ng  algebr a. 
Anoth er  id eais to buil d a to pos with  with some points as a directed co l im i t  o f  top oiwithout poi nts, butI am no t s u re th at th is is possible at all.
If  a  count erexa mple exist, I  expect th a t it's pos sibl e  t o  exhibit a loc aliccounterexa m ple.

A: 

If$\mathca l{C}$ isa small ca teg ory  wi thf i ni te limits the n geom et ric mor phi sms fro m $ {\r m S et} $to the pr esheaf t op os  $ {\ rmPSh}( \ mathcal{ C} )$ar e i n bij e ctionwithleft  e xa c t f unctors $\ m a thca l{ C}  \to  {\ rm  Set} $, o r , e quivale ntly, wit h o b ject sin  the pr o-category ${ \r m Pro}({\c al  C} )$. Mo r e  explici tly, if ${\bf X} = \{X_ i \}_{i \ in{\cal  I}} $ is a pr o-o bjectin$ {\calC}$ th en th ecor r e spond i n gpoi nt  ${\bf X}^ * : {\r m PSh }( \mat hcal{C} ) \to {\rm Set}$ s e nds  a presheaf $ F : \mat h c al {C} ^ {\ r m o p} \to { \rm Set}$ to ${ \rm colim} _{ i  \ in {\cal I } } F (X _i)$. I f ${\ca l C}_ 1 ,{\calC}_2$ are  two cate go ries w ith  finite li mits and  $f: {\ca l  C}_1 \t o {\c alC}_2$is  afunct or (wh i chdoesnot ne ce ssaril y pre se rve fini te limits) then we have  a geo metri c m orphism $ f_* :  {\ rm PSh}(\ math cal{C}_1)\le ftr ighta rro w s {\

0-cells in_Topoi are_Grothendieck topoi.
1-cells are geometric_morphisms and_have_the direction_of_the right adjoint.
2-cells_are invertible natural_transformation (between the left_adjoints).

Johnstone (and Lurie)_proves_that this 2-category has pseudo colimits that can be computed in Cat. To be_precise,_given a_diagram_in_Topoi its colimit is the_limit of the corresponding diagram_in Cat_associated to the inverse images.

Q. Does the representable_functor_$\mathsf{pt}:= \text{Topoi(Set, _)} $ preserve directed colimits?

After some naive optimism, I_have the feeling that the answer_is no, but_I_suspect_that it might be_known.

A candidate counterexample would be a_topos $\mathcal{E}$ whose subtopoi lattice is_not infinitary distributive, in fact a point_is an atom in this coHeyting_algebra.
Another idea is to build_a topos_with with some points as_a directed colimit_of topoi_without points, but_I am not sure that this_is possible at_all.
If a counterexample exist, I expect_that_it's possible to_exhibit_a_localic counterexample.

A:

If_$\mathcal{C}$ is a_small_category with_finite_limits then geometric morphisms from ${\rm_Set}$_to the presheaf topos ${\rm PSh}(\mathcal{C})$ are_in bijection with left_exact_functors $\mathcal{C} \to {\rm_Set}$, or, equivalently, with objects_in the pro-category ${\rm Pro}({\cal C})$._More explicitly,_if ${\bf_X} = \{X_i\}_{i \in {\cal I}}$ is a pro-object in ${\cal_C}$ then the corresponding point ${\bf_X}^*:{\rm PSh}(\mathcal{C}) \to {\rm_Set}$ sends_a_presheaf $F :\mathcal{C}^{\rm_op}_\to {\rm_Set}$ to ${\rm colim}_{i \in {\cal I}}_F(X_i)$. If_${\cal C}_1,{\cal C}_2$ are two categories_with finite limits and_$f:_{\cal C}_1 \to {\cal C}_2$ is_a functor (which does not necessarily_preserve finite limits) then we_have_a_geometric morphism $f_*: {\rm PSh}(\mathcal{C}_1)_\leftrightarrows {\