return

Celebratio Mathematica

Saunders Mac Lane

Saunders Mac Lane, 1909–2005: Meeting a grand leader

by Walter Tholen

Preamble

It was in Au­gust of 2000 at the cat­egory the­ory meet­ing in Como, Italy, when I saw Saun­ders Mac Lane for the last time. The am­bu­lance turned in front of the con­fer­ence site, the stately Villa Olmo at the lake shore, and Saun­ders, who had just col­lapsed and looked very pale in­deed, waved gra­ciously from in­side the vehicle to the wor­ried bystand­ers. Im­me­di­ately it al­most felt to me like the roy­al hand wave by a lead­er par ex­cel­lence, who was say­ing “Good bye” to his people; to the fol­low­ers of the field that he had helped to cre­ate al­most sixty years earli­er, that is. Al­though it turned out that Saun­ders had only suffered a tem­por­ary weak­ness, prob­ably caused by the stress of over­seas travel, he in fact did not at­tend any of the ma­jor cat­egory the­ory con­fer­ences af­ter­wards. When I spoke to him a few times on the tele­phone in 2002 to in­vite him to be the guest of hon­our at a meet­ing at the Fields In­sti­tute, he was ob­vi­ously temp­ted to ac­cept, but even­tu­ally con­cluded that travel at the age of 93 would just be too hard on him. Saun­ders died al­most three years later, on April 14, 2005. An out­spoken math­em­atician of ex­traordin­ary vis­ion, de­term­in­a­tion, and un­com­prom­ising prin­ciples had left the stage.

It was in the sum­mer of 1972 when I first met Saun­ders Mac Lane; more pre­cisely, I just saw him. As a gradu­ate stu­dent two years pri­or to com­ple­tion of my Ph.D. thes­is, I stayed down in the vil­lage of Ober­wolfach and just sat in on the lec­tures giv­en up the hill at the cat­egory the­ory con­fer­ence in the Math­em­at­ic­al In­sti­tute. A year later I fi­nally had the cour­age to in­tro­duce my­self to him, and was very im­pressed that he gave me all his at­ten­tion. The lead­er, who could be equally strong in his en­cour­age­ment and cri­ti­cism of any math­em­at­ic­al en­deav­ours, was in­deed al­ways will­ing to listen and learn, no mat­ter from whom, and give clear dir­ec­tion and ad­vice. Like many of his stu­dents, I felt these qual­it­ies even in my first brief dis­cus­sion with him, and I quickly dis­covered how kind and down to earth Saun­ders was, des­pite his cel­eb­rated status as one of the two great old men of cat­egory the­ory. Of course, his cowork­er since the 1940s, Samuel Ei­len­berg, also at­ten­ded these Ober­wolfach meet­ings of the 1970s, or­gan­ized by Horst Schubert and John Gray, but nor­mally kept a much lower pro­file than Saun­ders. Sammy’s com­ments, however, made from the back rather than the front row, and of­ten wrapped in subtle hu­mour, could be equally sharp as Saun­ders’ more dir­ect re­ac­tions. Still, even if, in the heat of the of­ten very emo­tion­al dis­cus­sions on al­geb­ra­ic the­or­ies, mon­ads, and topos and sheaf the­ory, with people like Bill Lawvere, Peter Freyd, Mi­chael Barr, Max Kelly, John Is­bell, An­dré Joy­al and Jean Bén­abou im­press­ing the audi­ence with their of­ten ex­treme views, one may have for­got­ten who the true lead­er was, then one would cer­tainly be re­minded at the hike on Wed­nes­day af­ter­noon. I will nev­er for­get the scene when, very shortly after leav­ing the In­sti­tute, at the fork of the trail go­ing fur­ther up the hill, John Gray was head­ing in one dir­ec­tion and Saun­ders was point­ing his cane in the oth­er while vehe­mently dis­put­ing John’s claims about the right dir­ec­tion. Of course, neither of the two men re­treated, leav­ing every­body else with the dif­fi­cult de­cision of wheth­er to fol­low what would most likely be the bet­ter dir­ec­tion, or to simply fol­low the boss. With most people choos­ing the lat­ter op­tion, only few ar­rived back in time for din­ner at the In­sti­tute.

Throughout the years I had my best en­coun­ters with Saun­ders on those con­fer­ence hikes, wherever in the world they took place, since we both seemed to like the chal­lenge of a moun­tain hike. In his won­der­ful es­say about “Con­cepts and cat­egor­ies in per­spect­ive,” he in­deed uses the im­age of hik­ing to ex­plain the very nature of re­search in math­em­at­ics:

The pro­gress of math­em­at­ics is like the dif­fi­cult ex­plor­a­tion of pos­sible trails up a massive in­fin­itely high moun­tain, shrouded in a heavy mist which will oc­ca­sion­ally lift a little to af­ford new and charm­ing per­spect­ives. This or that route is ex­plored a bit more, and we hope that some will lead on high­er up, while in­deed many routes may join and re­in­force each oth­er. ([17], p. 359)

This es­say, writ­ten for “A cen­tury of math­em­at­ics in Amer­ica” and pub­lished by the Amer­ic­an Math­em­at­ic­al So­ci­ety, gives any­thing but just an Amer­ic­an view. Rather, it reads like the mani­festo of a world cit­izen who re­ports glob­ally on the de­vel­op­ment and state of cat­egory the­ory, an area of math­em­at­ics that he nev­er saw in isol­a­tion, but al­ways as a tool or me­di­at­or for gain­ing new math­em­at­ic­al in­sights, wherever they needed to be made. There are nu­mer­ous oth­er pub­lic­a­tions by Saun­ders Mac Lane, in which he de­scribes the many math­em­at­ic­al de­vel­op­ments he was in­volved in, es­pe­cially the de­vel­op­ment of cat­egory the­ory; see par­tic­u­larly his 1965 art­icle on “Cat­egor­ic­al al­gebra” [9], his 1976 Re­tir­ing Pres­id­en­tial Ad­dress “To­po­logy and lo­gic as a source of al­gebra” [12], his 1995 sur­vey art­icle “Cat­egor­ies in geo­metry, al­gebra and lo­gic” [21], and his own “Math­em­at­ic­al auto­bi­o­graphy” [23], pub­lished only after his death. Taken in con­junc­tion with the many ex­cel­lent art­icles writ­ten about Saun­ders Mac Lane, such as Max Kelly’s ac­count [e1] of his en­coun­ters with Saun­ders that ap­peared in the 1979 Spring­er book “Saun­ders Mac Lane: Se­lec­ted pa­pers” [13] (ed­ited by Irving Ka­plansky, Saun­ders’ first of a total of 39 suc­cess­ful Ph.D. stu­dents), there seems to be little that one could add. Hence, in this art­icle, I can only at­tempt to give a very small glimpse of Saun­ders’ per­son­al­ity and work, from the per­spect­ive of someone who met and ob­served him just at con­fer­ences dur­ing the last third of his life. And just like Max did at the time, I must apo­lo­gize that this ap­proach al­most “ne­ces­sar­ily in­cludes more about my­self than is de­cent.”

Göttingen

The ques­tion that in­trigued me first when gath­er­ing my thoughts for this art­icle was how the young man, born on Au­gust 4, 1909, in small-town Con­necti­c­ut, grew in­to the role of a math­em­at­ic­al world lead­er, with such a strong sense of caring for the well-be­ing of his do­mains of in­terest any­where around the globe. Yes, as a stu­dent at Yale and Chica­go he was deeply in­flu­enced by fam­ous and broadly edu­cated teach­ers such as E. H. Moore, but his stay in Göt­tin­gen 1931–1933 seems to have been pivotal in shap­ing the math­em­atician as we know and re­mem­ber him today. At the dawn of the dark ages of Nazi rule, the Math­em­at­ic­al In­sti­tute in this pretty pro­vin­cial town was for a very short time still the place to be in world math­em­at­ics. The gi­ant of the time, Dav­id Hil­bert, was still giv­ing lec­tures and prob­ably helped to shape the highly in­ter­na­tion­al per­spect­ive of the young Mac Lane. Of his teach­ers, Ed­mund Land­au and Her­mann Weyl seem to have had the greatest math­em­at­ic­al in­flu­ence on the gradu­ate stu­dent from Amer­ica. In his short art­icle “A late re­turn to a thes­is in lo­gic” [14], Mac Lane re­calls how im­pressed he was with Land­au’s lec­tures on Di­rich­let series, the style of which may have giv­en him some un­in­ten­ded in­spir­a­tion for the choice of his thes­is top­ic:

As al­ways, Land­au’s proofs were simply care­ful lists of one de­tail after the oth­er, but he gave this de­tail with such ex­em­plary care that I could both copy down in my note­book all the needed de­tail and enter in the mar­gin some over­arch­ing de­scrip­tion of the plan of his proof (a plan which he nev­er dir­ectly re­vealed).

But, as he had re­marked earli­er,

a good proof con­sist[s] of more than just rig­or­ous de­tail, be­cause there [is] also an im­port­ant ele­ment of plan for the proof — the cru­cial ideas, which, over and above the care­ful de­tail make the proof func­tion.

In the brief cur­riculum vitae at the end of Saun­ders’ thes­is “Abgekürzte Be­weise im Lo­gikkalkul” [Ab­bre­vi­ated proofs in the cal­cu­lus of lo­gic], he thanks his ad­visor Paul Bernays “für seine Kritik” [for his cri­ti­cism], “und vor al­lem Pro­fess­or Weyl für seine Ratschläge und für die An­re­gung sein­er Vor­le­sun­gen” [and most of all Pro­fess­or Her­mann Weyl for his ad­vice and for the in­spir­a­tion of his lec­tures], a very strong in­dic­a­tion of the ad­mir­a­tion Saun­ders must have felt for Hil­bert’s bril­liant suc­cessor. Weyl is in fact lis­ted of­fi­cially as the “Ref­er­ent” [ref­er­ee] of the thes­is, after Paul Bernays had been chased out by the Nazis, who could bank on far too many act­ive and pass­ive help­ers with­in the uni­versity sys­tem, some of whom will­ing to im­ple­ment their no­tori­ous ideo­lo­gies at any cost. But already be­fore these in­cred­ible events that leveled a world class in­sti­tu­tion to av­er­age qual­ity with­in a year and made for­eign­ers like Mac Lane rush to fin­ish their de­grees (see his lively art­icle “Math­em­at­ics at Göt­tin­gen un­der the Nazis” [20]), Weyl must have as­sumed the role of a co-su­per­visor, as is in­dic­ated when he writes in [14]:

I had already star­ted work on an earli­er thes­is idea, also in lo­gic, early in the aca­dem­ic year 1932–33. I no longer know what was in­ten­ded as the con­tent of the thes­is, but I do clearly re­call that it did not find fa­vor with either Pro­fess­or Bernays or Pro­fess­or Weyl when I ex­plained it to them in Feb­ru­ary of 1933.

Saun­ders re­ports that, after this dis­ap­point­ment, he briefly tossed around the idea of leav­ing Göt­tin­gen for Vi­enna to work with Rudolph Carnap, but that he then ex­per­i­enced “a de­cis­ive spurt” in April 1933 and pro­duced “an ex­uber­ant first draft (in Eng­lish) of the thes­is: long-win­ded, full of rash philo­soph­ic­al as­ser­tions” shortly af­ter­wards. “The thes­is it­self (re­writ­ten later, first in Eng­lish and then trans­lated in­to Ger­man) is more math­em­at­ic­al and busi­ness­like,” Saun­ders re­ports. In­deed, a care­ful read­er of the thes­is (which was de­fen­ded on Ju­ly 19, 1933, prin­ted (as re­quired in Ger­many) in 1934 and re­prin­ted in [13]), will re­cog­nize early traces of Mac Lane’s trade­mark math­em­at­ic­al style. Al­though the goal of the thes­is, namely to give an “ana­lys­is of the struc­ture of lo­gic­al proofs and the con­firm­a­tion that this ana­lys­is will yield ab­bre­vi­ated proofs” (my trans­la­tion of the Ger­man thes­is text, [13], pp. 56-57), could have eas­ily led him to a more form­al­ist­ic present­a­tion, as seen in Rus­sell and White­head’s “Prin­cipia math­em­at­ica” and in many oth­er works of the time, Mac Lane makes re­peated ef­forts to al­ways re­con­nect with the av­er­age math­em­at­ic­al read­er with easy but il­lus­trat­ive ex­amples, a prin­ciple that is ap­plied in all of his math­em­at­ic­al writ­ings and that has made books like “Cat­egor­ies for the work­ing math­em­atician” [11] so suc­cess­ful.

I can­not res­ist men­tion­ing here one of the Göt­tin­gen an­ec­dotes, which I re­mem­ber Saun­ders telling me about with a big smile on his face, and which should be es­pe­cially ap­pre­ci­ated by all of us who struggle with the daily pit­falls when hav­ing to op­er­ate in a second lan­guage. In those days it was (and still is) quite cus­tom­ary in Ger­many for pro­fess­ors to in­vite some col­leagues and their gradu­ate stu­dents at least once a term for more or less in­form­al after-din­ner re­cep­tions, the pur­pose of which is to have free-ran­ging dis­cus­sions and, as a kind of co­in­cid­ent­al by-product, in­tro­duce the young men to the pro­fess­ors’ daugh­ters. To­wards the end of one of those oc­ca­sions, Saun­ders in­ten­ded to of­fer to take one of the young wo­men home, but ended up say­ing “Darf ich Sie zu Hause neh­men,” mean­ing “May I take you at home.”

I gath­er that this event must have pred­ated the ar­rival of his fiancée Dorothy Jones in Göt­tin­gen, who typed his thes­is and whom he mar­ried there. As Saun­ders tells us in [23], she was very fond of trav­el­ing and may there­fore have greatly helped to shape Saun­ders’ de­cidedly in­ter­na­tion­al views. Even when con­fined to a wheel­chair, she ac­com­pan­ied her hus­band not only on con­fer­ence travel but also on more cum­ber­some trips, such as a vis­it to China in 1981. Saun­ders would nor­mally refer to her re­spect­fully as Mrs. Mac Lane, just as he would refer to any sig­ni­fic­ant res­ult of his as that of Mac Lane. Al­though be­ing very un­as­sum­ing him­self and usu­ally stay­ing with every­body else in low-cost con­fer­ence ac­com­mod­a­tions (tol­er­at­ing the com­mon bath­room of stu­dent houses as late as 1993 at a MSRI con­fer­ence at Berke­ley — in his mid-eighties, that is!), he cer­tainly made sure that Mrs. Mac Lane was ap­pro­pri­ately cared for and en­ter­tained. When I co-or­gan­ized a con­fer­ence in 1981 at Gum­mers­bach (Ger­many), one of my col­leagues had to skip the lec­tures and spend the af­ter­noon driv­ing her around in search of loc­al col­lect­ors’ tea­spoons.

There is an­oth­er life-long fal­lout of Mac Lane’s early stud­ies and his Göt­tin­gen times, namely his pro­found in­terest in the philo­sophy and the found­a­tions of math­em­at­ics, doc­u­mented by bril­liant writ­ings such as “Proof, truth and con­fu­sion” [15], as well as his many con­tri­bu­tions about the status of cat­egory the­ory vis-à-vis set the­ory. His 1986 book “Math­em­at­ics, form and func­tion” [16] de­scribes much of his gen­er­al per­cep­tion and philo­sophy of math­em­at­ics.

The clash

At the Prague con­fer­ence on “cat­egor­ic­al to­po­logy” in Au­gust of 1988, Saun­ders Mac Lane gave a talk on the “De­vel­op­ment and clash of ideas,” ex­amin­ing vari­ous math­em­at­ic­al mile­stones of his life­time in terms of the ques­tion wheth­er they came about as an im­port­ant de­vel­op­ment with­in a giv­en dis­cip­line, or wheth­er they arose as the res­ult of a (pos­it­ive) clash of ideas from dif­fer­ent dis­cip­lines. At many oc­ca­sions he in fact re­ferred to the birth of the no­tions of cat­egory, func­tor and nat­ur­al trans­form­a­tion as a clash between al­gebra and to­po­logy. Hav­ing re­turned from Göt­tin­gen to the U.S. in 1933, while hold­ing fel­low- and in­struct­or­ships con­sec­ut­ively at Yale, Har­vard, Cor­nell and Chica­go, and fi­nally as­sum­ing an as­sist­ant pro­fess­or­ship at Har­vard in 1938, Saun­ders quite quickly shif­ted his in­terests away from lo­gic to pre­dom­in­antly (but by no means ex­clus­ively) al­gebra, as wit­nessed most vis­ibly by his 1941 book “A sur­vey of mod­ern al­gebra” [1] with Gar­rett Birk­hoff which, a dec­ade after van der Waer­den’s “Mo­d­erne Al­gebra,” changed un­der­gradu­ate courses in al­gebra fun­da­ment­ally and made ab­stract al­gebra “both ac­cess­ible and at­tract­ive” (as Mac Lane’s stu­dent Al­fred Put­nam writes in the pre­face to [13]; vari­ous edi­tions and trans­la­tions ap­peared as late as 1997, in ad­di­tion to the 1967 book on “Al­gebra” [18]). The ini­tial clash of al­gebra and to­po­logy, per­son­i­fied by Mac Lane and Ei­len­berg, oc­curred at a Michigan con­fer­ence on to­po­logy in 1940, when Saun­ders lec­tured on group ex­ten­sions. In “Samuel Ei­len­berg and cat­egor­ies” [22], he re­calls what happened:

I set out the de­scrip­tion of a group ex­ten­sion by means of factor sets and com­puted the group of such ex­ten­sions for the case of an in­ter­est­ing abeli­an factor group defined for any prime \( p \) and giv­en by gen­er­at­ors \( a_n \) with \( pa_{n+1} = a_n \) for all \( n \). When I presen­ted this res­ult in my lec­ture, Sammy im­me­di­ately poin­ted out that I had found Steen­rod’s cal­cu­la­tion of the ho­mo­logy group of the \( p \)-ad­ic solen­oid. This solen­oid, already stud­ied by Sammy in Po­land, can be de­scribed thus: In­side a tor­us \( T_1 \), wind an­oth­er tor­us \( T_2 \) \( p \)-times, then an­oth­er tor­us \( T_3 \) \( p \)-times in­side \( T_2 \), and so on. What is the co­homo­logy of the fi­nal in­ter­sec­tion? Sammy ob­served that the \( \operatorname{Ext} \) group I had cal­cu­lated gave ex­actly Steen­rod’s cal­cu­la­tion of the ho­mo­logy of the solen­oid! The co­in­cid­ence was highly mys­ter­i­ous. Why in the world did a group of abeli­an group ex­ten­sions come up in ho­mo­logy? We stayed up all night try­ing to find out “why.” Sammy wanted to get to the bot­tom of this co­in­cid­ence.

Mac Lane and Ei­len­berg were about to find the “Uni­ver­sal Coef­fi­cient The­or­em,” which forms the core of their first joint pa­per “Group ex­ten­sions and ho­mo­logy” [2], pub­lished “with the steady en­cour­age­ment of Lef­schetz” (Saun­ders’ words) in the An­nals of Math­em­at­ics in 1942. Then, in a let­ter to Sammy, dated May 10, 1942, Saun­ders points to the sim­il­ar be­ha­viour of vari­ous math­em­at­ic­al ob­jects which they had con­sidered in their pa­per (see [22]):

This in­dic­ates that it is pos­sible to give a pre­cise defin­i­tion of a nat­ur­al iso­morph­ism between func­tions of groups. Then it will be pos­sible to have all the iso­morph­isms in any such in­vest­ig­a­tion proved at once to be nat­ur­al.

With the term “nat­ur­al” trans­form­a­tion already around in vec­tor-space the­ory (as­so­ci­ated with maps like the one from a space to its double dual), Ei­len­berg and Mac Lane “pur­loined” (Saun­ders’ word) the terms “cat­egory” from Kant and “func­tor” from Carnap (who had used it in a dif­fer­ent sense in his book on “Lo­gic­al syn­tax of lan­guage”), and pub­lished their first pa­per on the be­gin­nings of cat­egory the­ory, first as a short note on “Nat­ur­al iso­morph­isms in group the­ory” [3] in 1942, and then as a longer pa­per (that takes a de­cis­ive step away from groups) in their “Gen­er­al the­ory of nat­ur­al equi­val­ences” [4] of 1945. As Saun­ders re­calls in [22],

That pa­per was cer­tainly off beat, but hap­pily it was ac­cep­ted for pub­lic­a­tion. At the time, Sammy stated firmly that this would be the only pa­per needed for cat­egory the­ory. Prob­ably what he had in mind was that the trio of no­tions — cat­egory, func­tor, and nat­ur­al trans­form­a­tion was enough to make good ap­plic­a­tions pos­sible.

Why should we mar­vel at the cre­ation of three re­l­at­ively easy no­tions that seem to be just in­stances of a lan­guage, rather than a the­ory? After all, giv­en that “nat­ur­al­ity” was already around, a crit­ic may claim that al­most any­body may have been able to find the right no­tions for the struc­ture giv­en by a gen­er­al­ized mon­oid with a par­tially defined bin­ary op­er­a­tion and the ap­pro­pri­ate struc­ture-pre­serving maps between them! But Ei­len­berg and Mac Lane’s re­volu­tion­ary achieve­ment does not lie in the mere cre­ation of yet an­oth­er al­geb­ra­ic struc­ture, but in dar­ing to think of prop­er classes as ob­jects with struc­ture them­selves and to study the ap­pro­pri­ate maps between such mon­sters, just as if these were ho­mo­morph­isms of groups, and in fi­nally em­bed­ding nat­ur­al­ity in­to this frame­work. The bold­ness of this in­tel­lec­tu­al step is com­par­able only to Georg Can­tor’s dar­ing to think dif­fer­ent de­grees of in­fin­ity. Of course, be­ing them­selves em­bed­ded in the per­ceived safe haven of set the­ory, they had to over­come their own scruples to ven­ture bey­ond it. In fact, they try to by­pass them when they state in typ­ic­al Mac Lanean prag­mat­ism (in Sec­tion 6 “Found­a­tions” of [4]):

Hence we have chosen to ad­opt the in­tu­it­ive stand­point, leav­ing the read­er free to in­sert whatever type of lo­gic­al found­a­tion (or ab­sence there­of) he may prefer.

(Spot­ting this sen­tence only re­cently for the first time re­minded me that I made an al­most identic­al dis­claim­er in my own Ph.D. thes­is.) In fact, they go on to ex­plain that “the concept of a cat­egory is es­sen­tially an aux­il­i­ary one,” and that one could avoid it since, in prac­tice, one is deal­ing only with a few ob­jects at a time. In my own ex­per­i­ence, if you hide quan­ti­fic­a­tion over a prop­er class by say­ing “let \( G \) be a group,” nobody even thinks about that, but “let \( G \) be an ob­ject of the cat­egory of groups” still sends shivers through a big part of your gen­er­al math­em­at­ic­al audi­ence.

The die-hard be­lief that set the­ory provides the sole found­a­tion of math­em­at­ics still pre­vents the no­tion of cat­egory to en­joy un­re­served gen­er­al ac­cept­ance. Saun­ders, be­ing al­ways very con­cerned about stay­ing in touch with main­stream math­em­at­ics, wrote re­peatedly about the “Found­a­tions for cat­egor­ies and sets,” for in­stance, un­der this title, in [10] in 1969. Be­fore em­bra­cing topos the­ory more con­sequently he fa­voured the (very lim­ited) use of Grothen­dieck uni­verses, in or­der to re­con­cile set-the­or­et­ic prob­lems nat­ur­ally arising when form­ing func­tor cat­egor­ies and the like, and this is what the read­er finds in [11]. But the second edi­tion of [11], in ad­di­tion to the old sec­tion on “Found­a­tions” of the main text, con­tains also a new three-page ap­pendix, again en­titled “Found­a­tions,” in which he sum­mar­izes topos-the­or­et­ic ax­ioms for sets and their stand­ing vis-à-vis Zer­melo–Fraen­kel. (An ele­ment­ary ac­count of such an ap­proach is giv­en in the book by F. W. Lawvere and R. Rosebrugh, en­titled “Sets for math­em­at­ics,” Cam­bridge 2003).

As a math­em­at­ic­al achieve­ment, the cre­ation of bold no­tions that open the gates for a new the­ory be­comes, after some time, eas­ily un­der­es­tim­ated. Like Can­tor’s first steps in­to new ter­rit­ory, Ei­len­berg and Mac Lane’s cat­egor­ies also had to face not only dis­missal but of­ten out­right op­pos­i­tion. In fact, they did not make it in­to Bourbaki, who in­stead stuck with an ill-fated gen­er­al no­tion of math­em­at­ic­al struc­ture. Saun­ders writes in [23]:

At the time, we some­times called our sub­ject “gen­er­al ab­stract non­sense.” We didn’t really mean the non­sense part, and we were proud of its gen­er­al­ity.

Un­for­tu­nately, even today many math­em­aticians use that la­bel to not only por­tray cat­egory the­ory as a mere “lan­guage,” but to claim to “know cat­egory the­ory,” es­sen­tially based on fa­mili­ar­ity with the ori­gin­al three no­tions — a mani­fest­a­tion of ig­nor­ance com­par­able really only to someone claim­ing know­ledge of set the­ory after hav­ing learned na­ively the lan­guage of sets, and how to form uni­ons and power sets.

The big leaps

While equal cred­it must be giv­en to Ei­len­berg and Mac Lane for the cre­ation of the trio of ba­sic no­tions, it was Saun­ders alone who took the de­cis­ive step of trans­form­ing a cat­egory from (in Max Kelly’s words) “a rather struc­ture­less do­main or codo­main of a func­tor [to] be­come something more tan­gible and in­di­vidu­al” [e1]. Hav­ing as­sumed his fi­nal reg­u­lar aca­dem­ic ap­point­ment at the Uni­versity of Chica­go in 1947, in his “Du­al­ity of groups” [6] of 1950 (pre­ceded by the short 1948 note “Groups, cat­egor­ies and du­al­it­ies” [5]) he makes the big leap of char­ac­ter­iz­ing cartesian and free products of groups by their cat­egor­ic­al prop­er­ties. The ob­jects \( G \times H \) and \( G\mathbin{*}H \), that look so dif­fer­ent when (with­in tra­di­tion­al found­a­tions) one tries to say what they “phys­ic­ally” are, be­came all of a sud­den so sim­il­ar (in fact, dual to each oth­er) when one con­cen­trates on what they do with­in the league of all groups. This was the start­ing point of a fun­da­ment­ally nov­el breed of think­ing that lies at the core of cat­egory the­ory and that was bound to blos­som in many vari­ations: Step in­side a cat­egory and char­ac­ter­ize vari­ous play­ers solely by means of their in­ter­ac­tion with their team­mates! The ap­par­ent de­fect of be­ing able to char­ac­ter­ize math­em­at­ic­al ob­jects or con­struc­tions in this way uniquely “only up to (a unique) iso­morph­ism” ac­tu­ally turns in­to an un­deni­able as­set, since it frees us from hav­ing to stare at ir­rel­ev­ant dif­fer­ences. In fact, since nobody wants to ser­i­ously dis­tin­guish between, for ex­ample, the ob­jects \( \mathbb{R} \) and \( \mathbb{R}\times\{0\} \) (as em­bed­ded in­to \( \mathbb{R}^2 \)), so that most people would de­clare that the two ob­jects should be “iden­ti­fied,” only few seem to real­ize that they are ac­tu­ally ad­opt­ing the cat­egor­ic­al view­point.

The second big leap ac­com­plished in [6] is the real­iz­a­tion that a cat­egory may need ad­di­tion­al prop­er­ties or struc­ture to af­ford res­ults of sub­stance. Mac Lane comes close to giv­ing the no­tion of abeli­an cat­egory (the def­in­ite form of which was giv­en by Buchs­baum in 1955, al­beit un­der a dif­fer­ent name), and he proves a rep­res­ent­a­tion the­or­em, with the help of a pre­de­cessor of Grothen­dieck’s fam­ous AB5 ax­iom. In de­fin­ing his “abeli­an bic­at­egor­ies,” Saun­ders also presents a defin­i­tion equi­val­ent to the no­tion of fac­tor­iz­a­tion sys­tem in gen­er­al cat­egor­ies, as pro­moted primar­ily by Freyd and Kelly more than twenty years later, a fact that I had to learn the very hard way more than forty years after Mac Lane’s pa­per. When in an art­icle with M. Koros­tenski I re­marked that the roots of fac­tor­iz­a­tion sys­tems were “already present in Is­bell’s work” (since John Is­bell was the first to use the “unique di­ag­on­al­iz­a­tion prop­erty,” the weak form of which be­came known as the right or left “lift­ing prop­erty” in Quil­len’s mod­el cat­egor­ies), im­me­di­ately after its ap­pear­ance I re­ceived a hast­ily scribbled and hardly legible note from Saun­ders in the mail, dated April 9, 1993, that left me feel­ing dev­ast­ated:

Dear Wal­ter

Just saw your joint pa­per with Koros­tenski JPAA 85 (1993) 57
What do you mean!
Fac­tor­iz­a­tion sys­tems in Is­bell 1957
They were in Mac Lane Du­al­ity for Groups BULL AMS 1950
Look it up!!!

Saun­ders

I nev­er for­gave my­self for hav­ing re­lied just on sec­ond­ary sources in this mat­ter. For­tu­nately, Saun­ders did. Like to many oth­er writers of art­icles or presenters at con­fer­ences, the eagle eye of the field had taught me a les­son, but he also quickly turned to his kind and cheer­ful self after the job was done.

Throughout the 1950s and early 1960s, abeli­an cat­egory the­ory found its def­in­ite form through the works of Al­ex­an­dre Grothen­dieck, Peter Gab­ri­el and Peter Freyd. Mean­while Mac Lane pur­sued primar­ily his mo­nu­ment­al work on ho­mo­logy and co­homo­logy the­ory that, inter alia, boos­ted a total of fif­teen joint pa­pers with Ei­len­berg and cul­min­ated in the 1963 pub­lic­a­tion of his book on “Ho­mo­logy” [7]. The ax­io­mat­iz­a­tion of these sub­jects through cat­egory the­ory is a ma­jor mile­stone of twen­ti­eth cen­tury math­em­at­ics. But that year turned also out to be a vin­tage year for great pro­gress in cat­egory the­ory it­self, fea­tur­ing the ap­pear­ance of the mi­meo­graphed notes of SGA IV by Grothen­dieck and his school (that re­defined al­geb­ra­ic geo­metry al­to­geth­er), of Bill Lawvere’s Ph.D. thes­is (which in­tro­duces al­geb­ra­ic the­or­ies as the ul­ti­mate set­ting for the syn­tax of Birk­hoff’s gen­er­al al­geb­ras, and which makes the first, but de­cis­ive steps to­wards a cat­egor­ic­al found­a­tion of math­em­at­ics), of Peter Freyd’s ad­joint func­tor the­or­em (that showed the ubi­quity of this bril­liant cat­egor­ic­al concept, in­ven­ted by Daniel Kan in 1958), of Charles Ehresmann’s “Catégor­ies struc­turées” (that, among many oth­er things, in­tro­duces cat­egory ob­jects in cat­egor­ies), and last, but not least, of Mac Lane’s first co­her­ence the­or­em in “Nat­ur­al as­so­ci­ativ­ity and com­mut­ativ­ity” [8]. His fam­ous pentagon dia­gram for the as­so­ci­ativ­ity iso­morph­isms of tensor products took centre stage in many of his talks dur­ing the fol­low­ing years, and he must be largely cred­ited for hav­ing ini­ti­ated the de­vel­op­ment of mon­oid­al cat­egor­ies, and there­fore of en­riched cat­egory the­ory and of high­er cat­egory the­ory, ir­re­spect­ive of the fact that the sub­ject seemed to be some­what in the air: Jean Bén­abou’s “Catégor­ies avec mul­ti­plic­a­tion” ap­peared that very same year.

But 1963, also the year of Co­hen’s in­de­pend­ence proof for the con­tinuum hy­po­thes­is, was just the be­gin­ning of a dec­ade of ex­traordin­ary activ­ity in cat­egory the­ory. The first widely cir­cu­lated book on the “The­ory of cat­egor­ies” by Barry Mitchell ap­peared in 1965, and the pro­ceed­ings of the “Con­fer­ence on cat­egor­ic­al al­gebra” held the same year in La Jolla give first testi­mony to the many fa­cets of cat­egory the­ory, a new dis­cip­line that was be­gin­ning to grow far bey­ond the do­mains of its ori­gins and that was at­tract­ing the in­terest of re­search­ers around the globe. The year 1967 turned out to be an­oth­er vin­tage year for cat­egory the­ory, es­pe­cially as it ap­plies to ho­mo­topy the­ory, fea­tur­ing Jon Beck’s Ph. D. thes­is (which made mon­ad the­ory the hot top­ic of the time, yield­ing many un­ex­pec­ted dis­cov­er­ies with­in a short peri­od of time, such as Ernie Manes’ proof of the mon­adicity of com­pact Haus­dorff spaces over sets), the in­flu­en­tial book by Gab­ri­el and Zis­man on cat­egor­ies of frac­tions, and Dan Quil­len’s Spring­er Lec­ture Notes (which re­shaped ab­stract ho­mo­topy the­ory). Then, in 1969, Bill Lawvere and Myles Tier­ney defined the no­tion of ele­ment­ary topos and opened the doors for the dis­cov­ery of fas­cin­at­ing con­nec­tions between geo­metry and lo­gic. In his Prague lec­ture, Saun­ders labeled the event as a “triple clash” of the ele­ment­ary the­ory of sets, of Co­hen’s for­cing meth­ods, and of ax­io­mat­ic sheaf the­ory. The gen­er­al ex­cite­ment that fol­lowed may have pre­ven­ted people from fully ap­pre­ci­at­ing oth­er im­port­ant de­vel­op­ments of the time, such as the ap­pear­ance of Peter Gab­ri­el and Friedrich Ulmer’s Spring­er Lec­ture Notes on “Lokal präsen­ti­erbare Kat­egori­en” in 1971, per­haps not just be­cause of the lan­guage bar­ri­er, but also be­cause the mes­sage that “size mat­ters” may have run against the cur­rent of get­ting away from sets. Also in that year, two art­icles by Os­wald Wyler as well as Guil­laume Brüm­mer’s Ph.D. thes­is ap­peared that made the people around Horst Her­r­lich try to cre­ate a cat­egor­ic­al frame­work for point-set to­po­logy. Mean­while, in Prague, the school around Vera Trnková and Aleš Pul­tr, also firmly based on set-the­or­et­ic­al found­a­tions, dug deep­er in­to their study of set-val­ued func­tors and were about to em­bark on the study of it­er­at­ive meth­ods that quickly drew cat­egory the­ory in­to the do­main of the­or­et­ic­al com­puter sci­ence. In 1972 John Is­bell pub­lished his “Atom­less parts of spaces,” the start­ing point of the the­ory of “loc­ales” (or com­plete Heyt­ing al­geb­ras) which, in a sense, re­con­ciles topos the­ory and gen­er­al to­po­logy. Ross Street’s “Form­al the­ory of mon­ads” ap­peared the same year, mak­ing him, to­geth­er with Max Kelly, the core of what be­came known as the Aus­trali­an School in cat­egory the­ory.

The worker

Saun­ders found him­self in the midst of these de­vel­op­ments and count­less oth­ers, par­ti­cip­at­ing vig­or­ously in most of them, and try­ing to make his care­ful se­lec­tion of what seemed worthy of be­ing treated in his 1971 book [11], and what did not. “Cat­egor­ies for the work­er,” as it was quickly dubbed by the 1968 gen­er­a­tion (I first heard the ex­pres­sion from John Is­bell but don’t know wheth­er he is re­spons­ible for it; Saun­ders’ short title was “Cat­egor­ies work”) makes the most ele­ment­ary no­tion of ter­min­al (or, du­ally, ini­tial) ob­ject in a cat­egory cent­ral to the the­ory. By draw­ing from a wide ar­ray of ex­amples and il­lus­tra­tions, he is nev­er in danger of los­ing touch with the av­er­age read­er. So very much un­like the long jour­neys in­to the land of ab­strac­tion by the prot­ag­on­ists of the great two (but dis­joint) French schools, lead by Grothen­dieck and Ehresmann, he nev­er stretches the pa­tience of the av­er­age read­er. While hav­ing great re­spect for their achieve­ments, one senses the ab­sence of “math­em­at­ic­al chem­istry” between Saun­ders and any of the two. In fact, Grothen­dieck’s fibred cat­egor­ies, in spite of hav­ing been pop­ular­ized to Eng­lish read­ers by John Gray in the La Jolla pro­ceed­ings, are worth only one sen­tence in [11], and Ehresmann’s double cat­egor­ies fare only slightly bet­ter. However, the second edi­tion of 1997 treats them in more de­tail, as part of a new chapter on “Struc­tures in cat­egor­ies,” which in fact leads the read­er in­to many do­mains of cur­rent re­search in cat­egory the­ory. En­riched cat­egor­ies get men­tioned in that chapter only briefly and, sur­pris­ingly, Gab­ri­el and Ulmer’s more hands-on no­tion of loc­ally present­able cat­egory re­mains ab­sent al­to­geth­er. Again and again, we see a man ready to make choices and to stand by them.

Peter John­stone’s de­mand­ing book on “Topos the­ory” of 1977 was the first uni­ver­sally re­spec­ted text on the sub­ject, pre­ceded only by in­form­al re­ports (by An­ders Kock and Gav­in Wraith in 1971, for ex­ample), journ­al art­icles (such as Peter Freyd’s 1972 “As­pects of topoi”) and by Gav­in Wraith’s Spring­er Lec­ture Notes of 1975. The more ele­ment­ary book by Robert Gold­blatt of 1979 gained wide cir­cu­la­tion but was met with con­sid­er­able re­ser­va­tions by the ex­perts. In the 1985 book on “Toposes, triples and the­or­ies” by Mi­chael Barr and Charles Wells, topos the­ory ap­pears only as part of a three-theme story, side by side with mon­ads and Ehresmann sketches. Mean­while the sub­ject had grown in­to oth­er ex­cit­ing branches, in the form of syn­thet­ic dif­fer­en­tial geo­metry (Lawvere, Kock, Dubuc, Lav­end­homme, Mo­er­dijk, Reyes) and of the lambda cal­cu­lus (Lam­bek, Seely, Scott). With nobody in the large com­munity of highly tal­en­ted ex­perts step­ping for­ward to write a book that would treat the geo­met­ric and lo­gic­al as­pects of the sub­ject with equal weight, Saun­ders Mac Lane, now in his eighties, set out to do just that. In Ieke Mo­er­dijk, born al­most half a cen­tury after he was, he found a coau­thor, highly gif­ted in both as­pects, so that in 1992 “Sheaves in geo­metry and lo­gic” [19] was ready to go. In [23] Saun­ders writes about the col­lab­or­a­tion: “The nov­elty of the sub­ject very much re­quired our joint ef­forts and our com­bined know­ledge.” Like his [11], the book was writ­ten while many as­pects of the the­ory were still in flux, and it makes topos the­ory both ac­cess­ible and rel­ev­ant to the pro­ver­bi­al work­ing math­em­atician, over­all a truly as­ton­ish­ing ac­com­plish­ment.

Saun­ders’ math­em­at­ic­al activ­it­ies didn’t stop here. Find­ing good ax­ioms for sets was one of the is­sues that re­mained on his agenda un­til as late as 2000, when he gave the open­ing talk at the Como meet­ing on a theme in the do­main of al­geb­ra­ic set the­ory, an area that is still very act­ively pur­sued today. It seems that he could nev­er aban­don a sub­ject which, in his mind, had not found a sat­is­fact­ory set­ting. In [17] he con­cludes on the one hand

that the ZFC ax­io­mat­ics is a re­mark­able con­cep­tu­al tri­umph, but that the ax­iom sys­tem is far too strong for the task of ex­plain­ing the role of the ele­ment­ary no­tion of a “col­lec­tion.” It is also curi­ous that most math­em­aticians can read­ily re­cite (and use) the Peano ax­ioms for the nat­ur­al num­bers, but would be hard put to it to list all the ax­ioms of ZFC.

On the oth­er hand, he nev­er had any il­lu­sions about quick ac­cept­ance of any sys­tem that would rad­ic­ally break with tra­di­tions, as he also says in [17]:

And set the­ory without ele­ments is still un­pal­at­able to those trained from in­fancy to think of sets with ele­ments: Habit is strong, and new ideas hard to ac­cept.

Final salute

In this art­icle I have not only neg­lected to even touch upon many parts of his math­em­at­ic­al work, which in­cludes, among many oth­er things, con­tri­bu­tions to com­bin­at­or­ics, ana­lys­is and the­or­et­ic­al phys­ics, but also failed to elab­or­ate on his om­ni­pres­ence in and strong en­gage­ments with ed­it­or­i­al boards, pro­fes­sion­al so­ci­et­ies, agen­cies, and policy-mak­ing bod­ies, be it as pres­id­ent or just let­ter writer, al­ways ready to take a stand and “to do something about it” whenev­er he saw a prob­lem. At the same time he kept in touch with al­most any­body work­ing any­where in his wide ar­ray of math­em­at­ic­al in­terests, and he took the time to pro­mote his causes any­where around the world. For ex­ample, after re­marry­ing in 1987, he made a trip with his wife Osa Skot­ting to sev­er­al parts of the former So­viet Uni­on, which he vividly de­scribes in [23]. Hav­ing vis­ited Tb­il­isi, Geor­gia, he de­cided to help pub­lish George Janelidze’s work on cat­egor­ic­al Galois the­ory in the West, and thereby star­ted to in­teg­rate a highly cap­able but isol­ated group of re­search­ers in­to the in­ter­na­tion­al arena. Co­in­cid­ently, I first met George shortly af­ter­wards, at a con­fer­ence in Baku, Azerbaijan, and was happy to serve as a (some­what il­leg­al) let­ter car­ri­er for his manuscripts. On Novem­ber 3, 1987, Saun­ders found the time to send me a neatly typed let­ter, something that must have been just one of a hun­dred items on his daily agenda:

Dear Wal­ter,

Many thanks for your let­ter and the art­icle by Janelidze on “Pure Galois The­ory in Cat­egor­ies.” He is a tal­en­ted math­em­atician, and I am de­lighted to have the ac­tu­al ms — I had tried hard to re­con­struct his ideas from my notes.

I hope that your sab­bat­ic­al year goes well.

Cor­di­ally,
Saun­ders.

How he man­aged to main­tain such over­whelm­ing level of activ­ity on all ends over such a long peri­od of time truly es­capes me. Strict work at­ti­tudes and will­ing­ness to spend long hours at his desk, even throughout the night, make for ne­ces­sary but not suf­fi­cient con­di­tions, and so does Saun­ders’ math­em­at­ic­al en­dur­ance, to which Sammy Ei­len­berg al­ludes in [e2]:

One of the many qual­it­ies that I ad­mire in Saun­ders is his ca­pa­city for or­derly and in­tel­li­gent com­put­ing. Not just or­din­ary 2–3 page cal­cu­la­tions where the res­ult is known in ad­vance, but real com­pu­ta­tion­al fish­ing ex­ped­i­tions run­ning to 30–40 pages. I ad­mire this tal­ent all the more since my own ca­pa­cit­ies in this dir­ec­tion are nil; a com­pu­ta­tion run­ning more than five lines usu­ally de­feats me.

Still, where did he find the needed phys­ic­al and, per­haps more im­port­antly, in­ner strength? His fun­da­ment­ally pos­it­ive and cheer­ful nature comes to mind, and read­ers of [23] also sense how much he ap­pre­ci­ated the sup­port that he re­ceived from both his first and second wives.

Last, but not least, it helps to be born a lead­er, a role that he could as­sume in any situ­ation, and with great ease. At the 1999 cat­egory the­ory con­fer­ence in Coim­bra, Por­tugal, at which we cel­eb­rated Saun­ders’ nineti­eth birth­day, he still ap­peared strong, fol­lowed an im­mense con­fer­ence pro­gram, and took centre stage at the so­cial events. He did not skip the Wed­nes­day af­ter­noon walk on a sandy hill­side, in the burn­ing early-af­ter­noon sun­shine of Ju­ly that made every­body thirsty and ex­hausted, and he was just at his best at the won­der­ful early-even­ing re­cep­tion at a nearby castle that fol­lowed. Nobody could have writ­ten a bet­ter movie script for this scene: Our tour bus stops at the foot of the castle’s hill, Saun­ders gets out first and just loves how he is roy­ally wel­comed by the may­or of the loc­al town (who happened to be the broth­er of the con­fer­ence or­gan­izer, Manuela Sobral, and who must have been care­fully in­struc­ted on how to say the right things), and then the two of them lead the pack up­hill to the castle, as if the day had just star­ted. Also un­for­get­table the scene after the con­fer­ence din­ner two nights later, held in a grand palace in the coun­tryside. The guests were en­ter­tained for quite a while by three typ­ic­al Coim­bra fado sing­ers in their tra­di­tion­al black gowns, who were very sur­prised when, after the birth­day bash, Saun­ders joined them on stage, put on one of the gowns and, to every­body’s de­light, chimed in.

That sum­mer in Por­tugal the field that Saun­ders had helped to cre­ate and that he had nur­tured over such a long peri­od of time, cel­eb­rated its lead­er and his great ac­com­plish­ments, for the last time in his pres­ence. But cer­tainly not for the last time.

Works

[1]G. Birk­hoff and S. Mac Lane: A sur­vey of mod­ern al­gebra. Mac­mil­lan (New York), 1941. The fiftieth an­niversary of this book was com­mem­or­ated in an art­icle in Math. In­tell. 14:1 (1992). MR 0005093 Zbl 0061.​04802 book

[2]S. Ei­len­berg and S. MacLane: “Group ex­ten­sions and ho­mo­logy,” Ann. Math. (2) 43 : 4 (October 1942), pp. 757–​831. MR 0007108 Zbl 0061.​40602 article

[3]S. Ei­len­berg and S. MacLane: “Nat­ur­al iso­morph­isms in group the­ory,” Proc. Natl. Acad. Sci. U.S.A. 28 : 12 (December 1942), pp. 537–​543. See also art­icle in Gaz. Mat. 29:109–112 (1968). MR 0007421 Zbl 0061.​09203 article

[4]S. Ei­len­berg and S. MacLane: “Gen­er­al the­ory of nat­ur­al equi­val­ences,” Trans. Am. Math. Soc. 58 : 2 (September 1945), pp. 231–​294. MR 0013131 Zbl 0061.​09204 article

[5]S. MacLane: “Groups, cat­egor­ies and du­al­ity,” Proc. Natl. Acad. Sci. U.S.A. 34 : 6 (June 1948), pp. 263–​267. MR 0025464 Zbl 0031.​24702 article

[6]S. MacLane: “Du­al­ity for groups,” Bull. Am. Math. Soc. 56 : 6 (1950), pp. 485–​516. MR 0049192 Zbl 0045.​29905 article

[7]S. Mac Lane: Ho­mo­logy. Die Grundlehren der math­em­at­ischen Wis­senschaften 114. Aca­dem­ic Press (New York), 1963. In Ger­man. A Rus­si­an trans­la­tion was pub­lished in 1966. MR 0156879 Zbl 0133.​26502 book

[8]S. Mac Lane: “Nat­ur­al as­so­ci­ativ­ity and com­mut­ativ­ity,” Rice Univ. Stud. 49 : 4 (1963), pp. 28–​46. MR 0170925 Zbl 0244.​18008 article

[9]S. MacLane: “Cat­egor­ic­al al­gebra,” Bull. Am. Math. Soc. 71 : 1 (1965), pp. 40–​106. MR 0171826 Zbl 0161.​01601 article

[10]S. Mac Lane: “Found­a­tions for cat­egor­ies and sets,” pp. 146–​164 in Cat­egory the­ory, ho­mo­logy the­ory and their ap­plic­a­tions (Bat­telle Me­mori­al In­sti­tute, Seattle, WA, 24 June–19 Ju­ly 1968), vol. II. Edi­ted by P. J. Hilton. Lec­ture Notes in Math­em­at­ics 92. Spring­er (Ber­lin), 1969. MR 0242919 Zbl 0218.​18003 incollection

[11]S. Mac Lane: Cat­egor­ies for the work­ing math­em­atician. Gradu­ate Texts in Math­em­at­ics 5. Spring­er (New York), 1971. A Ger­man trans­la­tion was pub­lished as Kat­egori­en: Be­griffss­prache und math­em­at­ische The­or­ie (1972). MR 0354798 Zbl 0232.​18001 book

[12]S. Mac Lane: “To­po­logy and lo­gic as a source of al­gebra,” Bull. Am. Math. Soc. 82 : 1 (January 1976), pp. 1–​40. Re­tir­ing Pres­id­en­tial ad­dress. MR 0414648 Zbl 0324.​55001 article

[13]S. Mac Lane: Se­lec­ted pa­pers. Edi­ted by I. Ka­plansky. Spring­er (New York), 1979. With con­tri­bu­tions by Al­fred Put­nam, Ro­ger Lyn­don, Ka­plansky, Samuel Ei­len­berg and Max Kelly. MR 544841 Zbl 0459.​01024 book

[14]S. Mac Lane: “A late re­turn to a thes­is in lo­gic,” pp. 63–​66 in S. Mac Lane: Se­lec­ted pa­pers. Edi­ted by I. Ka­plansky. Spring­er (New York), 1979. incollection

[15]S. Mac Lane: Proof, truth and con­fu­sion, 1982. The 1982 Ry­er­son Lec­ture, Uni­versity of Chica­go (1982). misc

[16]S. Mac Lane: Math­em­at­ics, form and func­tion. Spring­er (New York), 1986. MR 816347 Zbl 0675.​00001 book

[17]S. Mac Lane: “Con­cepts and cat­egor­ies in per­spect­ive,” pp. 323–​365 in A cen­tury of math­em­at­ics in Amer­ica, part I. Edi­ted by P. Duren, R. A. As­key, and U. C. Merzbach. His­tory of Math­em­at­ics 1. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1988. An ad­dendum to this art­icle was pub­lished in His­tory of Math­em­at­ics, part III (1989). MR 1003181 Zbl 0665.​01002 incollection

[18]S. Mac Lane and G. Birk­hoff: Al­gebra, 3rd edition. Chelsea (New York), 1988. Re­pub­lic­a­tion of the {1967 ori­gin­al|MR:0214415}. MR 941522 Zbl 0641.​12001 book

[19]S. Mac Lane and I. Mo­er­dijk: Sheaves in geo­metry and lo­gic: A first in­tro­duc­tion to topos the­ory, re­prin­ted edition. Uni­versitext. Spring­er (New York), 1994. MR 1300636 book

[20]S. Mac Lane: “Math­em­at­ics at Göt­tin­gen un­der the Nazis,” No­tices Am. Math. Soc. 42 : 10 (1995), pp. 1134–​1138. MR 1350011 Zbl 1042.​01536 article

[21]S. Mac Lane: “Cat­egor­ies in geo­metry, al­gebra and lo­gic,” Math. Ja­pon. 42 : 1 (1995), pp. 169–​178. MR 1344646 Zbl 0831.​18001 article

[22]S. Mac Lane: “Samuel Ei­len­berg and cat­egor­ies,” pp. 127–​131 in Cat­egory the­ory 1999 (Coim­bra, Por­tugal, 13–17 Ju­ly 1999), published as J. Pure Ap­pl. Al­gebra 168 : 2–​3. Issue edi­ted by J. Adamek, P. T. John­stone, and M. Sobral. 2002. MR 1887153 Zbl 0992.​18001 incollection

[23]S. Mac Lane: Saun­ders Mac Lane: A math­em­at­ic­al auto­bi­o­graphy. A K Peters (Welles­ley, MA), 2005. With a pre­face by Dav­id Eis­en­bud. MR 2141000 Zbl 1089.​01010 book