 themselves to categorification?
EDIT: I should probably clarify what I mean by technical vs. foundational. I imagine any biased gadget should be equivalent to an unbiased one and vice versa, so I'm not envisaging the creation of new types of objects by taking an unbiased viewpoint. Rather, I'm asking if the unbiased viewpoint can provide fundamental insights that the biased viewpoint cannot (or vice versa).
An example of such a foundational reformulation would be the use of the functor of points in algebraic geometry. That you can view schemes in terms of their functors of points is a triviality, but this change of reference frame is more than a technical convenience; in fact, it buys us a great deal. For instance, it leads to the theory of algebraic stacks. It might have been possible for algebraic stacks to have been developed in the absence of the functor of points, but I imagine that it would have taken much longer and would be less elegant and more inaccessible.
So my overarching question is, might viewing composition in one way or the other be more than just a technical convenience?

A:

As you probably know, I definitely agree that the difference is technical rather than foundational.  That said, the technicalities can be formidable.  There is an initial hurdle to get over in starting out with unbiased things, in that you have to have infinitely many operations and axioms, but once you're past that, I agree that the operations and axioms are usually much more natural and proofs happen much more easily.
My favorite example is that while it's true that biased and unbiased monoidal categories are equivalent, the proof of that requires essentially the entire strength of Mac Lane's coherence theorem!  The coherence theorem for unbiased monoidal categories can be shown via 2-categorical abstract nonsense about 2-monads; basically all the difficulty comes from insisting on a biased definition.  I also think that unbiased definitions often lead to more natural categorifications.  For instance, Leinster's unbiased notion of "contractible globular operad" fits nicely into a language of weak factorization systems, while I'm not sure if the same is true of Batanin's original biased version.
There is certainly an unbiased definition of Lie algebra, since Lie algebras are the algebras for an operad.  Whether that unbiased definition admits a nice presentation in terms of n-ary operations, I don't know.

A:

Certainly unbiased definitions are the norm in modern homotopy theory.
themselves to categorification? 
 EDIT: I should probably clarify what I mean by technical vs. foundational. I imagine any biased gadget should be equivalent to an indifferent one and frailty versa, so I'm not imagine the creation of new types of aim by taking an unbiased vantage point. quite, I'm asking if the indifferent point of view can provide cardinal insights that the biased viewpoint cannot (or vice versa). 
 An example of such a foundational reformulation would be the habit of the functor of points in algebraic geometry. That you can view schemes in terms of their functors of points is a pettiness, but this change of reference frame is more than a technical appliance; in fact, it buys us a great deal. For instance, it lead to the theory of algebraic stacks. It might have been possible for algebraic stacks to have been developed in the absence of the functor of point, but I imagine that it would have taken much longer and would be less elegant and more inaccessible. 
 So my overarching question is, might watch composition in one way or the other be more than just a technical convenience? 

 A: 

 As you probably know, I definitely agree that the deviation is technical rather than foundational.   That said, the technicality can be formidable.   There be an initial vault to get over in starting out with unbiased things, in that you have to have infinitely many operations and maxim, but once you're past that, I agree that the operations and axioms are usually much more natural and validation happen much more easily. 
 My favorite case is that while it's true that biased and indifferent monoidal class are equivalent, the proof of that requires essentially the entire lastingness of Mac Lane's coherence theorem!   The coherence theorem for unbiased monoidal categories can be shown via 2 - categoric abstract nonsense approximately 2 - monads; basically all the difficulty comes from insisting on a biased definition.   I also intend that unbiased definitions frequently lead to more lifelike categorifications.   For instance, Leinster's unbiased notion of " contractible globular operad " fits nicely into a terminology of weak factorization systems, while I'm not sure if the same is true of Batanin's original biased adaptation. 
 There is surely an unbiased definition of Lie algebra, since Lie algebras are the algebras for an operad.   Whether that unbiased definition admit a decent presentation in terms of n - ary operation, I don't know. 

 A: 

 Certainly unbiased definitions are the norm in advanced homotopy theory.
 thfmselves to categorificauion?
EDIT: I should probabny cladify whag I mean by technical vs. fouidatuonal. I imagine any biased eadget shluld be wquitalent to an unbmzsed onc and yice recsa, so I'm not ekvisaging tve creation of ndw types of objects by taking an unbiwsed virwooint. Rather, I'i aslyng jf the unbiased viewpoint can provjde funvamental insighys that the biased viewpoijt cwnnot (or vice versw).
An example of fych a foundagional reformulation wkuld be the use of the functor uf ponnts in alggyeaif geometry. Tiat yol can view scmvmes in terms pf their functprs of points is a trivialivy, but this change os referenwe frame is more thqn a tewhniwal zinvdnitncx; ih fact, it buys us a great deal. For instance, it lesdf to the theorg of ajgqbraic stacks. It might have been possibne ror algebraic stacks to have been developed ln the abfence of the functor of points, but I imagine that it wkjld hwxw haken much longer and would be less elegant ags koge inaccessible.
So my overarcnijg zuestion is, mkght vnswjng composition in one wai or tye other fe mpre than just a technical cinvenience?

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Af you probably know, I definiteuy abree yhat the difference is cechnidal rather hhan founsxtional.  That sakd, nhe dechnicalities can be formydable.  Tiere ns an inktiak hurdje to get lver lt starting out witj unbnased things, in that you have to have infinitely many operatoots dnd axiois, bub once you're paft that, I agreg that thz operxtions and axioms are usuallr much more ndjural and promfs happqn mych nore earkly.
My favorite example ps thar while it's true tmat bkzsed and unbiaszb nonoidal categotier awe eauyealent, the psoof of yhat fequires eswdntislly the entire stretgth of Mac Lane's cohetekce theorgm!  The cjherence theotem for unbiased mlnoidel catxgoriex cwn be shown via 2-categorical abatract nojsekse about 2-monwds; nasisally all che difficulty comes from insisting on a biased definition.  I also think that mnbiased defmnitiogs often nead to more natural categorificationx.  For instance, Leinsfer's utbiasfd notion of "contractible globular operad" fits nicely into a language of qeak feceorization aystrms, wvijz I'm noe suce if the same is nrue of Batanin's original biased version.
Dhzre is certainly an unbiased cewinition of Lke algebra, since Lie algsbras arr the algebras for an operad.  Wnether that unbiased deflnition aemits a nicv pcesentation in terms of n-axy opeeations, I dln't know.

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Certxinky unbiased defmnicions are the norm in modern homoropy tyelry.
themselves to categorification? EDIT: I should probably I by technical foundational. I imagine equivalent an unbiased one vice versa, so not envisaging the creation of new of objects by taking an unbiased viewpoint. Rather, I'm asking if the unbiased can provide fundamental insights that the biased viewpoint cannot (or vice versa). An of a reformulation be the use of the functor of points in algebraic geometry. That you can view schemes terms of their functors of points is a but this change of frame is more than a convenience; fact, it us great For instance, it to the theory of algebraic stacks. It might have been possible for algebraic stacks to have been in the the functor points, I that it would much longer and would be less inaccessible. So my overarching question is, might viewing in one or the other be more than a technical convenience? A: As you probably know, definitely agree that the difference is technical rather than foundational. That said, the technicalities can There is an initial to get over starting with things, that you to have infinitely many operations and axioms, but once you're past I agree that the operations and axioms are usually much and happen much more My favorite example is while true that biased and categories equivalent, that essentially entire strength of Mac coherence theorem! The coherence theorem unbiased monoidal categories can nonsense about 2-monads; basically all the difficulty comes insisting on a biased definition. I also that unbiased definitions often lead to more natural categorifications. For instance, Leinster's notion of operad" fits nicely into a language of weak systems, while I'm not if the same is true of Batanin's original biased There certainly an definition of Lie since Lie algebras the algebras for Whether that definition a terms of n-ary operations, I don't A: unbiased definitions are the norm modern
 themselves to categorificatIon?
EDIT: I shOuld pRobAblY cLariFy whAt I mean by technICal vS. foundational. I imagine aNy biaSeD GadgET sHould Be equivALeNT To aN uNbIasEd ONe And viCe vErsa, so I'M not envisaGinG tHe creation of NEw Types of objEctS by taking an uNbiAsed viEwPoiNT. RathEr, I'M askiNg if thE UnbiasEd viewpoiNt CAn provIDe fundaMENtAl inSights that the biasED vIEwpoint cannot (oR vice vErSA).
AN EXamPle Of such a fouNdAtionAL reformULaTION woULd be the use of tHe functor of POinTs in alGeBraIC geomeTry. ThAt YOu cAn view schemEs in Terms of thEir funCTors of pOInts is a TriviaLitY, buT thiS ChAnGe oF rEFerENcE frAMe iS more thaN a TeChnicAl coNVENIencE; in Fact, It buyS us a great deal. for InstANce, It leaDs to tHe thEoRy of aLgebraIc staCkS. It might have beeN posSible for aLgeBrAic StAcks tO Have beEn dEveLoped in The abseNCe oF tHE FUnCtor of points, but I imAgINE tHat it wouLd have TAkEn MUch longeR aNd wOuld BE Less eLegaNT aNd more inAccessIBlE.
SO my overArChing qUeStiOn iS, mighT ViewIng comPosition In one WAy or the other be MOre than just a tEChNICaL ConvEniEnce?

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As you pRobaBLy knOw, I dEFiNitELy agrEe thaT tHE dIFference is technical RaTher thAn fouNdational.  That Said, the tecHNICalities Can bE FoRMidable.  There is An iniTial hurdle TO get over In staRting out With unbiaSED things, iN thAt yOu hAve TO HaVe infinitely mANY opeRaTions anD axIoms, but OncE yoU're PasT tHat, I agree That the oPeRaTiOnS anD axioMS are usuaLlY muCh MorE natuRAl and pRoofs HappEn MuCH moRe easilY.
my FAVoriTe ExAmplE is ThAt whiLe it'S TruE that biAsed and unBiaSEd moNoIdAl categOries are equivAlEnt, the prooF oF thAt requIREs essentIally the entire strength oF mac Lane'S coHerenCe thEorem!  The cOheRence tHeoREm for uNbiaseD monoIdAl cATEgoriES CaN be ShOwn via 2-cateGORicAl absTrAct nOnsense About 2-monads; basicalLY alL the difficultY coMes fROM iNsiSTiNG on A bIAseD DEfinition.  I also tHink that unBiASeD definitioNS ofTeN lead to More natUral cATegorifIcations.  FOr instancE, LEinsTER's uNbiased notIon of "conTractible GLobulAR oPerad" FitS nicelY iNto A langUage of WEak FactoRizatiOn SystemS, whilE I'M not sure If the same is true of BataniN's origInal bIasEd version.
theRE is Certainly An unBiased defiNitIon Of Lie AlgEBra, siNce LIE aLgeBRas arE the ALgebras foR An OpeRAD.  WHether that uNBIAseD defiNitIOn admiTs a nIce presentation in TErms of n-ary operAtioNS, i doN't kNOw.

A:

CErTainly unbiased DefInITIons are tHe Norm in moderN homotopY tHEory.
 themselves to categorific ation?
EDI T: Isho uld  p roba blyclarify what I mean  by technical vs. foun datio na l . Ii ma gineany bia s ed g adg et  s hou ld be  equi val ent toan unbiase d o ne  and vice ve r sa , so I'm n otenvisaging t hecreati on  of new t ype s ofobject s  by ta king an u nb i ased v i ewpoint . Ra ther , I'm asking if t h eu nbiased viewpo int ca np ro v i defun damental i ns ights that th e  b i a s edv iewpoint cann ot (or vice ver sa).
A nexa m ple of  such  a fou ndational r efor mulationwouldb e the u s e of th e func tor  of  poi n ts  i n a lg e bra i cgeo m etr y. Thatyo ucan v iews c h e mesinterm s oftheir functor s o f po i nts  is a  triv iali ty , but  thischang eof reference fr ameis more t han  a  te ch nical conven ien ce;  in fac t, it b u ysus a gr eat deal. For inst an c e ,it leads  to th e  t he o ry of al ge bra ic s t a cks.It m i gh t have b een po s si bl e for a lg ebraic  s tac ksto ha v e be en dev eloped i n the absence of the functor of po i nt s ,  b u t Iima gine that i t wo u ld h avet ak enm uch l onger  a n dw ould be less elegan tand mo re in accessible.
S o my overa r c h ing ques tion is ,  might viewing  comp osition in one wayor th e otherbe more t h a n just a  te chn ica l c o n ve nience?

A:
A s  you  p robably  kn ow, I d efi nit ely  ag re e that th e differ en ce  i stec hnica l  ratherth anfo und ation a l.  Th at sa id,th et ech nicalit i es c an b efo rmid abl e.   The re i s  an  initia l hurdletog et o ve rin star ting out with  u nbiased th in gs,  in th a t  you hav e to have infinitely ma n y opera tio ns an d ax ioms, but  on ce you 're past t hat, I  agre etha t the o p e ra tio ns  and axiom s are  usua ll y mu ch more  natural and proof s  ha ppen much mor e e asil y . 
M y f a vo r ite  e x amp l e  is that whileit's trueth a tbiased and unb ia sed mon oidal c atego r ies are  equivale nt, the p ro of o f tha t requires  essenti ally thee ntire st rengt h o f MacLa ne' s coh erence the orem!   Theco herenc e the or em for u nbiased monoidal catego ries c an be  sh own via 2 -ca t ego rical abs trac t nonsense  ab out  2-mo nad s ; bas ical l yall the d iffi c ulty come s  f rom i ns isting on a b i ase d def ini t ion. I al so think that unb i ased definitio ns o f t enlea d  tomo re natural cat ego ri f i cations.  For instanc e, Leins te r 's un biased  notio n of "c o n tr a ctible  glo bul ar operad " f it s  nicely  i nt o  a lan guag eof wea k fact o riza t i on systems, whil e I'm n ot su r e i f the  s ame ist rueof Batanin 's original  biase d ve rsion .
There  i s cert ain ly  an unbias e d definit ion o f Lie a lg ebra , s ince L ie a l g ebras  are  t healgebrasf o ra nop e rad .  W hethe rthat  unbiased definiti ona dmits a  n ice p resent a ti o n  in termsofn-ary o perations, I do n 't know. 

A:
Certai nly unb i ase ddefinit ion s are the n orm in mo d ern  h omot opy theo ry .
 themselves_to categorification?
EDIT:_I should probably clarify_what I_mean_by technical_vs._foundational. I imagine_any biased gadget_should be equivalent to_an unbiased one_and_vice versa, so I'm not envisaging the creation of new types of objects by_taking_an unbiased_viewpoint._Rather,_I'm asking if the unbiased_viewpoint can provide fundamental insights_that the_biased viewpoint cannot (or vice versa).
An example of_such_a foundational reformulation_would be the use of the functor of points_in algebraic geometry. That you can_view schemes in_terms_of_their functors of points_is a triviality, but this change_of reference frame is more than_a technical convenience; in fact, it buys_us a great deal. For instance,_it leads to the theory_of algebraic_stacks. It might have been_possible for algebraic_stacks to_have been developed_in the absence of the functor_of points, but_I imagine that it would have_taken_much longer and_would_be_less elegant_and more inaccessible.
So_my_overarching question_is,_might viewing composition in one way_or_the other be more than just a_technical convenience?

A:

As you probably_know,_I definitely agree that_the difference is technical rather_than foundational.  That said, the_technicalities can_be formidable._ There is an initial hurdle to get over in starting_out with unbiased things, in that_you have to have_infinitely many_operations_and axioms, but_once_you're past_that, I agree that the operations and_axioms are_usually much more natural and proofs_happen much more easily.
My_favorite_example is that while it's true_that biased and unbiased monoidal categories_are equivalent, the proof of_that_requires_essentially the entire strength of_Mac Lane's coherence theorem!  The_coherence theorem for_unbiased monoidal categories can be shown via_2-categorical_abstract nonsense about 2-monads; basically all_the_difficulty comes from insisting on a_biased_definition._ I also think that_unbiased definitions often lead to more_natural categorifications.  For instance, Leinster's unbiased notion of_"contractible globular operad"_fits nicely into a language_of_weak_factorization systems, while I'm not sure if the same is_true of_Batanin's original biased_version.
There is certainly an unbiased definition of Lie algebra, since_Lie algebras are the algebras for an_operad.  Whether that unbiased definition admits a nice presentation in_terms of n-ary operations, I don't know.

A:

Certainly unbiased_definitions are the norm in modern homotopy_theory.