What is Category Theory in mathematics? Johns Hopkins' Dr. Emily Riehl explains

17 Dec 2020 JOHNS HOPKINS UNIVERSITY

The President's Frontier Award was established with a $2.5 million donation from trustee Louis J. Forster. Forster helped design the award to support exceptional scholars among the Johns Hopkins faculty who are on the cusp of transforming their fields. The award recognizes one person each year with $250,000 in funding for their work. This PFA lecture features 2020 recipient Dr. Emily Riehl presenting on Category Theory. Thinking categorically can help serve as a guide post as you trek the grand scheme of (mathematical) things. As written in Quanta Magazine in a 2020 interview with Riehl, “category theory and its next-generation version, higher category theory, are central to many fields of math, from algebraic geometry to mathematical physics. In those areas, Riehl said, ‘I think it would be impossible to describe the kind of basic objects of study without categorical language.’” Join us for this fantastic event honoring the brilliant work of Dr. Riehl, opened by Provost Sunil Kumar and Dr. John Toscano. Denis Wirtz, Vice Provost for Research, provides closing remarks. To learn more, visit jhu.edu/hopkinsathome

Transcript

0:19

>>Welcome everyone and thank you for joining us  for the sixth President's Frontier Award lecture.  

0:27

Before we go to the lecture itself, I'd  like to say a little bit about the award.   The President's Frontier Award was initially  established with 1.25 million dollar donation  

0:38

from trustee Louis J Forster, who is now  the Chair of the Board of Trustees, who is a  

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1982 Arts and Sciences graduate, and a 1983  SAIS graduate. The President's Frontier Award  

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has been awarded six times so far, and it will  be expanded for another five years thanks to  

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Mr. Forster's continued commitment  with another 1.25 million dollar gift.  

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The award is designed with the goal of  supporting exceptional scholars among the  

1:14

Johns Hopkins faculty who are on the cusp of  transforming their fields. So the aspiration  

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isn't merely transforming their careers, but  their fields themselves. And I'm delighted to  

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say that there is no doubt in my mind that  all six of the winners so far easily meet  

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this very high bar. And so I could not think of a  better award to celebrate the born innovation and  

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pursuit of excellence that Hopkins is known for  than the Presidential Frontier Award. As Provost  

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I'm contractually forbidden from commenting on  the actual work of somebody. And even though I  

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know a little bit of Matt, to attempt to speak  on category theory, even for people who know a  

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lot of math is a dangerous proposition. However, I  can't stop myself from saying this simply because  

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it's, for me, it's one of the most beautiful parts  of mathematics. Sometimes by asking the question,  

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how would this problem be solved if it wasn't  as special as it is? What is even more general?  

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The answer becomes easier. In my own field,  probability, a great breakthrough was when  

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somebody realized the best way to think of a  probability is simply the length of an intro.  

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From that abstraction, suddenly a lot of ideas  simplified. And category theory represents  

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one of the logical extremes of that kind of  reasoning and therefore is one of the most  

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elegant ways of saying, if I  free myself of special structure,  

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will the more general I get make the problem  easier and more elegant? And I think Emily  

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represents the highest form of that art and so  I'm delighted that she is the sixth President's  

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Frontier Award lecture winner. So with that, let  me turn to John Toscano, who is the Interim Dean  

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of the Krieger School of Arts and Sciences to  introduce our award winner. Thank you again.   

3:34

>>Great. Thanks, Sunil, appreciate that. When  I, when I first learned last winter that Emily  

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Riehl would receive this year's President's  Frontier Award I was obviously delighted, but  

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not at all surprised. We knew Emily was a star  from the moment she arrived at Hopkins in 2015.  

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Emily has been taking the mathematics world by  storm, as her colleagues say. She studies category  

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theory and you heard a bit about that from Sunil.  Math Department Chair, David Savitt describes  

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category theory as providing a framework in  which outwardly different ideas can be viewed  

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as examples of a single more general concept,  which can then be specialized to new concept  

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context to obtain fresh insight. Although Emily  is less than 10 years out from earning her PhD,  

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she's already widely recognized and admired  for her ability, seemingly effortless to use  

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this framework, to build bridges, to other  disciplines, bringing her colleagues and   students along with her. She's built a reputation  for consolidating the approach of category theory,  

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simplifying the literature in streamlining  contact complex proofs. David also described  

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Emily as a true force of nature. In her department  for her ability to attract visitors and post-docs,  

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offering sought after seminars and graduate  classes, and pursuing a formidable research   program, all at the same time. In addition, while  writing books is rare for any mathematician,  

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let alone one in her early stage  of career, Emily's written three,   including an introduction to category theory, a  book that prepares graduate students for research  

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and indeed has a fourth on the way. Emily's  books, along with her numerous journal articles  

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have earned her richly deserved praise. She's a  groundbreaking mathematician who has already made  

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monumentous and lasting changes to the field.  She is also a highly collaborative convener and  

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organizer in the academic community. For example,  she hosts the end court end category cafe,  

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which is a group blog on math, physics, and  philosophy. She founded the Kan Extension Seminar,  

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which is a graduate reading course in algebraic  topology, and also organized a seminar program,  

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Higher Categories and Categorification for  the Mathematical Society Research Institute.  

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She's doing no less than changing the culture  of the field. Emily received her PhD in 2011  

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from the University of Chicago and spent  four years as an NSF and Benjamin Peirce  

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Postdoctoral Fellow at Harvard before joining  us here at Johns Hopkins where she was promoted  

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to Associate Professor in 2019. She has been  recognized with numerous prestigious awards,  

6:27

residencies, and invited lectures, mentored  graduate students and post-docs and won awards  

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for her undergraduate teaching. It's easy to see  why the selection committee unanimously chose to  

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recommend Emily to receive this year's President's  Frontier Award. Emily, because this award was  

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designed, as Sunil mentioned for exceptional  scholars on the cusp of transforming their fields,  

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it's tailor made for you. Congratulations. The  virtual stage is now yours. >>Great. Thanks so  

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much to Provost Sumar and Dean Toscano. That's  really quite an introduction I'm very honored.  

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So this award was announced in January and quite  a lot has changed in the world since then. So I  

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wanted to start by acknowledging that I certainly  knew in January how lucky I was to, I mean,  

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not only to have found a tenure track position,  but to have found a tenure track position at   Johns Hopkins. When I came here in 2015, I didn't  really know my departmental colleagues very well,  

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we'd only just met a few months before, but  already, one of my colleagues, Steve Wilson  

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threw a party to welcome me in the spring before  I officially arrived. And put me up in his  

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apartment when I was looking for a place to stay  in Baltimore. He's the one who then nominated me  

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for this award, you know, completely, without  my knowledge, several years later. And I was   humbled to even have been considered among the  finalists much less to actually receive it. I know  

7:54

that a lot of people are really suffering right  now. It's, it's really been a terrible year. And  

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you know, there's something that feels a bit  incongruous at times and sort of persisting with,   you know, academic inquiry in the face of all  of the present and urgent suffering. I do,  

8:12

I guess also think that there's something hopeful  in a basic science research. I mean, certainly   there was some hope that sustained Katalin Kariko  and Andrew Weissmann in the 1990s when they  

8:22

were following up on this crazy idea of an MRNA  based vaccine. And you know, it was years of  

8:30

sort of patient development of expertise in  my colleagues Lauren Gardner and Ensheng Dong  

8:35

on the epidemiological modeling that allowed them  to then so quickly stand up the Johns Hopkins   coronavirus dashboard that's now been such as key  policy tool used worldwide. And thinking towards  

8:48

the future, I'm hopeful that my colleagues at  the Agora Institute and elsewhere around the   University can help us dream up a better,  maybe more functional political system and  

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some policies that can sort of test or sort of,  sort of survive the gauntlet that we have today.  

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Anyway, I just wanted to say thanks to everyone  at Johns Hopkins, I, I feel very fortunate   to be at an institution that is you know,  protecting us in many ways and allowing us to  

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continue this very long ranging  inquiry, even at such a turbulent time.  

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What I'm going to invite you to explore with  me today is the realm of abstract mathematics,  

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which is where I like to spend my time. What  I'm going to describe is a joint project   with a longtime collaborator Dominic Verity.  And even before I arrived at Johns Hopkins,  

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we were working on redeveloping the foundations of  infinite dimensional category theory. So these are  

9:49

infinite dimensional categories are  homes for a sort of very complex   objects that arise in mathematical  physics or in mathematics in general. And  

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we've been working on a sort of new  approach to develop the basic theory   to kind of rigorously prove all the  theorems from, from the very beginning,  

10:07

so they can be used by other mathematicians. And  specifically what I want to tell you about today   is a new result that was proven in 2020,  and very much with the help of the frontier  

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award. And what it says is that there is a formal  language that can be used to encode statements  

10:24

about infinity categories and every statement  that is written in this language is then invariant   under change of model. Okay. So what's going to be  challenging about this is I realized that none of  

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those words make any sense. So mathematicians have  developed this very precise language. It sounds  

10:39

like English on the surface, but we've redefined  all the words to mean something very particular.   And it's, it's kind of a lingua franca around sort  of spoken common among mathematicians all around  

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the world and completely opaque to everybody  else. So I'm going to explore all of the ideas  

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of a formal language of models of invariants under  change of model by starting with something that I  

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hope is a bit more familiar to us which are  the natural numbers. So when we think about  

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the natural numbers, so, so what I'm referring to  are that the counting numbers, I like to include   zero, but if you'd rather not, it doesn't  really make a big difference. So zero one,  

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two, three, four, and the question is sort of  what are they really? So there's some sense in  

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which this is kind of a philosophical question,  but so a way that a mathematician might approach  

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defining the real numbers or sort of saying  what they are is by describing some of their   properties. So we know we can add numbers, we can  multiply, we can, there's certain properties of  

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that addition and multiplication. We can take any  natural number and factor it uniquely as a product  

11:47

of primes. But the problem with the description  or definition of the natural numbers along those  

11:52

lines would be that it would be very long. You  know, we humans have been studying the national   numbers for millennia and there's, there's quite  a lot that we can say about them. So in the  

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late 19th century Giuseppe Peano formulated some  axioms, they're commonly referred to as the Peano  

12:11

postulates in a paper, erythematous, Arithmetices  principia, nova methodo exposita. And so these  

12:19

are some true facts about the natural numbers and  it's a, it's a sort of a shortlist of true facts   and so I'll sort of say what they are. So one  of them is that there's a natural number zero,  

12:30

and then every natural number has a successor  that's also a natural number. The successor  

12:35

is meant to be like the next one in the  sequence, so the successor of zero is one   successor of one is two, the successor of two is  three, and so on and so forth. So zero is not the  

12:45

successor of any natural number, meaning that this  sequence doesn't wrap around to the beginning,   it just kind of carries on. And moreover, no two  natural numbers have the same successor. So every  

12:54

new natural number that you define as a successor  of a previous one is a, is a distinct number. So  

13:00

we can generate infinitely many natural numbers  in this way. And then the final axiom, this is the  

13:07

complicated one, it's also sort of where the real  power lies. If you're familiar with the principle   of mathematical induction, this is secretly  justifying that. So it says that any set that  

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contains the number zero and then is closed under  the successor operation, meaning it contains the  

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successor of any natural number, it contains that  necessarily contains all of the natural numbers.  

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Okay. So these are some facts that mathematicians  accept about the natural numbers. We sort of   believe in our bones that these are true. I've  reformulated these postulates in, so formal  

13:44

languages are going to be a theme later on, so  I've rewritten five axioms, zero is a natural   number, every natural number has a successor  and so on in this formal language. So there's a  

13:55

formal syntax that mathematicians can use to  write sentences. And then you can ask a computer,  

14:02

whether the sentence is a logically well-formed  mathematical sentence or not. So that's the,  

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this idea of a formal language. This is sort  of how I would teach a computer about the Peano  

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postulates and the computer would say, okay, these  sentences make sense at least. They're logically   reasonable. So another mathematician, Dedekind,  was studying the, or formulated a very similar  

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series of axioms at around the same time in a  paper, Was sind und was sollen die Zahlen. So  

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what numbers are and what they should be. Which  has this starts with this very beautiful quote  

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in German, so in science, nothing capable  of proof should be accepted without proof.   And what's so exciting about the Peano postulates,  what's , why we can think of the natural numbers  

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as being defined somehow by these Peano postulates  is all of the other properties that we know about   them follow from these axioms. So using just these  sort of five simple statements about the natural  

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numbers, it's possible to define addition,  and then prove that addition is communitive,  

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that A plus B is equal to B plus A, and to prove  that it's associative, that you can add any finite  

15:15

collection of natural numbers together in any  order and the result will always be the same.   You can similarly define multiplication, prove  the distributive property of multiplication over  

15:24

addition. And then prove more sophisticated  statements in number theory, starting from  

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what's called the fundamental theorem of  arithmetic, which is the property that any   natural number can be factored uniquely as a  product of primes. So there's a sense in which  

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if we're asking what the natural numbers are the,   the Peano postulates give an answer to that  somehow. If it's a list that we can write down and  

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not too much time that somehow generates all of  this complex mathematical structure. Number theory  

15:56

is very much still an active research area today.  Okay. But mathematicians don't like sort of,  

16:07

you know, founding every area of mathematics  on its own set of axioms. There's, there is a  

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desire at around the same time around the  turn of the 20th century to come up with   a common foundation for all mathematics. So a  list of axioms, like these Peano axioms that  

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characterize the national numbers, but that we  can use to talk about all kinds of mathematical   structures. So not just numbers, but geometric  structures, analytics structures as well. And  

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the accepted common foundation of mathematics that  we implicitly refer to today is something called  

16:36

set theory. And so there's a secondary question,  which is, can we reason about the natural numbers   in set theory? Can we use some, maybe  more primitive axioms, axioms about sets  

16:47

to begin to talk about the natural numbers? And  so there, it is possible to construct the natural  

16:55

numbers formally in the system of set theory.  And in fact, there are different ways to give   a construction. So, so set theory includes,  so what we're trying to do is we're trying to  

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model that's, that's the word the mathematician  would use, to sort of model or define, introduce  

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the natural number to zero, one, two, three,  and so on as sets. Because for mathematicians,   somehow sets are more primitive than numbers.  So, so how do we do this? So one of the axioms  

17:24

of set theory tells us that there was an empty  set. So this is the, it's written as a zero   with a line through it. So we can associate  the natural number zero with the empty set.  

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So then in set theory, you're allowed to take a  collection of sets and form a new set with those  

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sets as elements. This is one of the things  I don't actually like about the axioms of set   theory, the elements of sets in formal set theory  are themselves sets, which is very confusing, but  

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this is just how the world works according to  the axioms of set theory. So once we have the  

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empty set as a set, we can use that as an element  and form another set. This is the set containing   the empty set, and that's how we're going to  think of the number one, as a set. And so now,  

18:07

if we wanted to find the number two using Von  Neumann's construction, what we do is we take   the two sets that we've mentioned so far, there's  the empty set that doesn't have any elements,  

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and then there's the set that has the empty set as  its element. And I can form now a two element set,  

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including the empty set and the set containing  the empty set, and that's the number two.   And then we define the number three using the  three sets that I've just mentioned. So we have  

18:27

the empty set, the set containing the empty set  and the set who has two elements, one of which is   the empty set and the other is the set containing  the empty set. That's the definition of three  

18:35

in Von Neumann's natural numbers and four is  then the set that contains the empty set, the   set of the empty set, the set of the empty set,  plus the set of the empty set, and then the set  

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that has all of those three sets as its element.  Okay, I'm going to stop there or I'm going to  

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run out of time. But in parallel, Zermelo  proposed a different construction. It starts  

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out the same we're identifying the empty set with  the number zero, and we're identifying the set,   containing the empty set for the number one. And  then to form the number two what I'm going to  

19:03

do is build a set, which has a single  element and that element is the   set that corresponds to one. So this is the set  of the set of the empty set. And then three is  

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the set of the set of a set of the empty set and  four is the set of the set of the set of the set   of the empty set, and so on and so forth.  Okay, so these are complicated constructions,  

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but they're valid within the rules of set  theory with the accepted axioms of set theory.   And when I collect all these sets together, I  have two different models of the natural members.  

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I have defined that the set of natural numbers  in these two different ways, and you can check   that they satisfy the Peano postulates. So this  gives us a construction of the natural numbers  

19:40

in set theory. Okay. But what we need to observe  about these constructions is they're not the  

19:45

same. They're, the reason is the sort of way  the successors formed is different in the  

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two different constructions. With the Von Neumann  construction of the naturals and this are Zermelo   construction that natural members, and they're  not literally equal. Which seems weird because  

20:00

we think of the natural number as being some  sort of unified concept. And now we have two  

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different models of this one thing. So that's  okay because mathematicians have a more refined  

20:14

notion of equality and Dedekind proved in this  paper...that something that's now referred to as  

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Dedekind's Categoricity Theorem. It says that  if you have any triple given by a set, and so  

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that's like the set of all the natural numbers,  an element, zero and a successor function, then as  

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long as that data satisfies the Peano postulates,  so the five axioms that I mentioned previously,  

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then those triples are isomorphic. So I'll, I'll  say more later on about what I mean by isomorphic,  

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but its etymological roots are same shape. So  these are the same, in some sense, not equal,  

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not literally equal, it's a more sophisticated  sense of equality. And a corollary is that I mean,  

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from the point of view of mathematics, having two  objects that are isomorphic is awesome because   it means that all kind of meaningful properties  of one also hold true of the other. So  

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a consequence of this isomorphism between these  two constructions of the natural numbers is that  

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all number theoretic properties of the Zermelo  naturals hold equally for the Von Neumann   naturals. So each is kind of, can be used to prove  a theorem and then if that theorem will be true  

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of the other model as well. Okay. But there's  one subtlety here that was pointed out by Paul  

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Benacerraf in 1965 paper, what numbers could  not be, sort of referencing to Dedekind's, what  

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the numbers should be. And he asks a question,  is three an element of 17? So this is a really  

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wild question. I mean, on the one sense it's like  total nonsense. I mean, it doesn't make any sense   for a number to be an element of another number.  I mean, that's just not what numbers are about.  

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I mean, it's, you know, there's, there's sort of  no way to answer that question because it doesn't  

22:06

make any sense. But on the other hand, if we're  thinking about these numbers as something that   are constructed formally in set theory, this  question makes perfect sense. It's something  

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that you could write in the formal language, this  language that a computer would recognize and say,   yes, this is a mathematically valid question. You  can write this question in the formal language of  

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set theory. And moreover the answer to this  question depends on which model you're using.  

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So the way that the Von Neumann natural numbers  are constructed, the set 17 contains 17 elements  

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and one of those is the element three.  So three is in fact, an element of 17.   Whereas in the Zermelo construction of the natural  numbers, the set 17 contained a single element,  

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namely the element 16. So three would not be an  element there. So it's true for the Von Neumann   natural numbers, but false for the Zermelo natural  numbers, but also totally nonsense. So this is  

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a question that we're going to have to leave  open, which will I'll address again at the end.   The question is can we sort of restrict our formal  language? So there's a common formal language,  

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first-order logic that mathematicians use to  write very rigorous sentences that allows us in  

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the case of the natural numbers to ask nonsense  questions like is three an element of 17. So a   question that I'll leave open for now is whether  we can restrict that language so that it only  

23:24

includes meaningful statements, mathematically  meaningful statements about the natural numbers.  

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So just to summarize the story so far, that if  we're reading about the natural numbers in the   classical foundations of mathematics, we need a  model. So something like this set of Von Neumann  

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natural numbers or Zermelo natural numbers  that are built within the axioms of set theory.  

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So the problem with these models is they're  not unique. We have two different constructions   and in, you know, in fact there are many, many  more. But there's a sense in which they are  

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the same. So mathematicians have a  more sophisticated notion of sameness,   not equality, but something called isomorphism  that I'll tell you more about very soon.  

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But a problem remains, is there a way to restrict  the formal language of mathematics include these   nonsense statements, like is three, an element  of, of 17? And the idea that's going to appear  

24:11

in the solution is that we need some sort of more  precise taxonomy of mathematical objects so that  

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we don't ask whether X is equal to Y, if X is  a natural number and why is a triangle? I mean,  

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that's, that's kind of the problem with using set  theory as a foundation for mathematics is that   everything in set theory is a set. That's sort  of the only sort of stuff we have to build more  

24:32

sophisticated mathematical objects. So you could  absolutely ask, is this number the same as this   triangle, even though that doesn't make sense  to sort of intuitive humans. And the, what we're  

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going to bring in to help us solve this problem to  kind of come up with this taxonomy of mathematical  

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objects is something called category theory, which  is the area of mathematics where I work. Okay. One  

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of the things that we can do in category theory  is we can define what an isomorphism actually   means. So I'm going to give you the definition of  this idea of seeing shape of mathematical objects.  

25:03

So it's something you can define in any category.  So I have to tell you, firstly, what a category   is. So this is a mathematical definition. It's  kind of bread and butter for our math majors, but  

25:13

maybe less familiar for the rest of us. So what is  a category? It's something that consists of some  

25:18

collection of objects. You can think of these as  being sets if you'd like, or as being some sort of   data types. So they're objects of some specified  kind, but they really can be whatever you want.  

25:28

And then we also specify some arrows between  the objects. So an arrow should have a name   telling you sort of who it is. And it also has a  specified source object and a target object. So  

25:38

we're drawing an arrow is pointing from one thing  to another. And this you can think of as like a   function or maybe it's a algorithm that converts  a thing of type X into a thing of type Y. And then  

25:51

on top of that, we have two additional structures.  So each object necessarily has an identity arrow,  

25:57

which you can think of as the process that  does nothing, that sort of leaves the state   unchanged. And then whenever I have an arrow from  X to Y, like F, and an arrow from Y to Z, like G,  

26:08

there needs to be a specified composite arrow,  which you can think of as, you know, running two   processes in sequence or composing a function with  another. And this would be an arrow from X to Z.  

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Okay. So this is a very, very abstract definition  as a Provost Sunil said in the introduction.  

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This is all sorts of mathematical objects  assemble into categories. They're kind of   all over the place. And what's cool about them  is you can define a notion of isomorphism at this  

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level of generality. So between two objects in any  category, no matter what that category is. So what  

26:43

is an isomorphism? So you, the data consists  of two different objects in a common category.   So A and B have to be in the same category.  And then an isomorphism consists of an arrow,  

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one pointing from A to B and one pointing from  B to A, so that when I compose the arrows, if I  

27:00

compose them one way around, I get an endo arrow  of A, and that needs to be this identity arrow.  

27:06

And similarly, if I compose them the  other way around, I get an endo arrow of B   and that needs to be the identity arrow again.  So the idea of an isomorphism is F gives us some  

27:14

way of translating from A to B and G gives us  some way of translating from B back to A, and  

27:20

this condition about the composition says that no  information is lost in those translations. You can   pass from A to B and then back to A, and  you've come back to exactly where you started,  

27:31

right? So again, what's, what's powerful about  this definition of isomorphism is it can now be   deployed all across mathematics for any type of  mathematical object, whatever you want. You can  

27:41

ask are two objects of that type isomorphic  or not? Are they same in this sense? And so  

27:46

in particular, we've just defined this notion  of category. So categories are now themselves  

27:51

mathematical objects, and we can ask, what is an  isomorphism then between categories? What does it   mean for two categories to be isomorphic? And so  there's an answer for that. But before we think  

28:02

about it, let's think about an example. So there's  a category of matrices defined as follows. So it's  

28:09

objects are just natural numbers, so like zero,  one, two, three, four. So we have natural numbers,  

28:15

that's the objects. And then an arrow from  the number N to the number of M is an M by N  

28:21

matrix. So an arrow from three to four would be  a four by three matrix of real numbers. So it's  

28:26

got 12 different real numbers in an array.  And then we have a composition and identity.  

28:33

And then there's a similar category of vector  spaces. So the objects here are written sort  

28:39

of RK, RM, RN. These are RN is an N dimensional  Euclidean space. So this is the collection of all  

28:48

points in N dimensional space. So if I have an  ordered list of N real numbers, that defines a   point in RN. So these are sort of much bigger  somehow as mathematical objects than just  

28:58

the numbers themselves. So these are the  objects now, these Euclidean spaces, you   can think of a point, a line, a plane, these are  examples of these Euclidean spaces. And an arrow  

29:07

now from RN to RM, from N dimensional space to  M dimensional space is a function that takes an  

29:13

N dimensional point to an M dimensional point and  then satisfies this property of linearity which is  

29:19

probably unfamiliar to many of you, but most  of our students at Johns Hopkins learn about   linear transformations at some point in a  linear algebra course. Okay. So it turns out,  

29:31

so it makes sense to ask whether these two  categories are isomorphic because they are both   objects of a category of categories. They're not  in fact isomorphic. However, they are same in a  

29:42

more refined sense of sameness, something called  equivalents. And a consequences of this is that  

29:49

all category theoretic properties of the category  of matrices also hold for the category of vector   spaces. So a way I like to think about this  equivalence is we actually teach two different  

29:59

linear algebra courses at Johns Hopkins. In one  of the courses we spend most of the time in the  

30:04

category of matrices learning about a row echelon  form and matrix operations and so on. And in the  

30:09

other course, we spend most of our time in the  category of vector spaces. I think you may be  

30:15

more abstractly about inner products and so on  and so forth, but somehow those are the same   course. So both the matrix course and the vector  space based course are teaching the same subject  

30:23

of linear algebra. And the justification  for that is that this category of matrices   is equivalent, so the same in the sense of  categories to the category of vector spaces.  

30:35

But again, these categories are not isomorphic,  so they don't satisfy this reduced property of   sameness. And that means you can ask questions  that are kind of nonsense from the point of  

30:46

view of category theory, but which will have  different answers depending on which category   you're thinking about. So here you can ask whether  your category has countably infinitely, many  

30:55

objects. Something that's a bit wild is that's  actually the smallest notion of infinity, that's   countable infinity. There are other larger notions  of infinity. And in this case, the category of  

31:04

matrices does have countably infinitely, many  objects where the category of vector spaces has   many, many, many more objects than that. So  somehow these categories can be equivalent  

31:13

with having different numbers of objects, which  is a bit of a strange thing to think about.  

31:21

Okay. So I said that the categories of  vector spaces and matrices are equivalent,   not isomorphic, and what does that actually  mean? Well, like isomorphism was a concept  

31:30

you can define in any category, equivalence is a  concept you can define in a 2-category. So really  

31:36

categories themselves live as objects in some  two dimensional category, and that's where this  

31:41

notion of equivalence can be understood. But if  I defined a 2-category for you, then we could  

31:46

ask the same question, what is the notion of,  what's the correct notion of equivalence between   2-categories? And that's something that you  can define, but you need a third dimension. You  

31:55

define that enemy three-dimensional category.  And three-dimensional categories also have a   notion of equivalence that's correctly defined in  a four-dimensional category and four-dimensional  

32:03

categories, have a notion of equivalence to find  in a five-dimensional category, five categories of   notion equivalents in a six-dimensional category.  And you can see this is never going to stop until  

32:11

we get to infinity. So these are actually the sort  of categories that I spend most of my time with,  

32:17

infinite dimensional categories. And let me  introduce them to you now. So an infinity  

32:23

category has, like an ordinary category, it has  some objects and some, one arrows between them.  

32:30

I say one, cause I'm thinking of an arrow as being  something that lives in dimension one. I mean,   if you like, the picture of an arrow is a sort  of one dimensional path from X to Y. But now we  

32:40

have an addition, these two-dimensional arrows. So  one arrow had a source object and a target object,  

32:45

a two-dimensional arrow, alpha here, has a source  one arrow, which is F and a target one arrow,  

32:51

which is G, and there's a requirement that F and G  need to be parallel arrows. So they have a common   source object X and a target object Y. And then  we have three-dimensional arrows between parallel  

33:02

two-dimensional arrows. So here are alpha and  beta are two dimensional arrows from F to G.   And so we can have...as a three-dimensional arrow  from alpha to beta. And think of it as inhabiting  

33:14

this sort of three-dimensional ball. So X and  Y are sort of objects here and F and G define  

33:19

paths between them and alpha and beta inhabit  sort of the front and back faces of the ball  

33:24

and the interior is really where, so it's somehow  a three-dimensional object and this continues in  

33:30

all higher dimensions. So the pictures are sort  of harder to draw. And like an ordinary category,  

33:36

these infinite dimensional categories need  to have identities and composition, but it's,   it's a little bit more subtle. Now these are,  composition is, is not an operation. I don't  

33:46

have a well-defined composite of any two arrows.  Instead, it's something that's witnessed. So  

33:52

for any arrow from X to Y and arrow from Y to  Z, you can recognize when an arrow from X to Z  

33:57

is a composite of those arrows, but it's not  a unique property anymore. And the reason that  

34:05

this definition is as complicated it is, is  it's reflecting what we see in the examples  

34:10

when we meet in the real world. This is actually  the point in the talk that comes the closest to   the real world, so let me take you there. So any  space, so space is something that a term that a  

34:21

mathematician would recognize, so any space has an  associated, infinite dimensional category called  

34:26

the fundamental infinity groupoid of that space.  So let me describe a space for you so we can have  

34:32

a common visual metaphor. Let's say you're baking  donuts, but you're doing it badly. So you've,  

34:38

you've got all your donuts and you put them in  a tray, a baker's dozen donuts, but then you put   them too close and so when you put it in the oven,  I realize this isn't how you normally bake donuts,  

34:47

but let's just do this anyway, you put it in the  oven and they all kind of run together. They all   sort of stick together. And the space I actually  want to imagine is the glaze on this sort of sheet  

34:57

of congealed together donuts in the oven. You can  imagine you're an ant sort of walking around this  

35:04

space. This, a mathematician would call this the  sort of oriented surface of genus g or genus 13,  

35:11

where 13 refers to the baker's dozen donuts  that you congeal altogether. Okay. So what's  

35:17

the infinity category here? So the objects of the  infinity category are the points, the positions,  

35:22

the ant could be on walking around the glaze  of this congealed together sheet of donuts.   The one -dimensional arrows are the paths that an  ant could take between two different points. So  

35:32

the two-dimensional arrows are something called  homotopies. These are notions of paths between   paths. I don't know how to describe this in terms  of ants, but it's like a continuous deformation  

35:40

from one path to another. And the reason I like  this space is you see that some paths are sort  

35:47

of deformable and continuously deformable and  others aren't. You have to respect the holes in,  

35:53

you know, an ant can't sort of teleport across the  hole in a donut. It's got to kind of walk around   sort of one side or the other. And in a way,  these continuous paths, there are sort of paths  

36:03

between paths, between paths, between paths in all  dimensions and that's the notion of an infinity   category here. So these are the sort of infinity,  these are the sorts of categories that might  

36:13

collaborate in Dom and I's study. And again, if  we want to reason about them rigorously using the  

36:19

common foundations of mathematics, sort of this  lingua franca that all mathematicians use, we need   models. So we need a way to construct instances  of these, construct sort of rigorous mathematical  

36:32

versions of these infinite dimensional categories  within set theory. Now, these are sort of much,   much, much, much bigger objects. So they live  in a much, much larger set theoretical universe.  

36:42

And what's distinctive about the way that Dom and  I think about them is we're sort of zoomed out  

36:49

an extra category level. So, so it's kind of  classical at this point, mathematicians like  

36:56

to think about the infinity category of spaces or  the infinity category of spectra, or there's an  

37:02

infinity category of homotopic coherent diagrams,  which was first imagined here at Johns Hopkins by  

37:08

Michael Boardman. So these are sort of specific  instances of infinity categories. And what Dom   and I think about instead is something that  we call an infinity cosmos. It's the universe,  

37:17

the infinite dimensional categorical universe  in which these infinity categories live as  

37:23

object. So, I'm drawing here in orange  are these specific infinity categories,  

37:29

which are themselves, you know, very, very large  mathematical objects. And we're thinking about   them as themselves being objects in a much larger  thing, the infinity cosmos of quasi categories,  

37:39

which is one of the models of infinity categories  or the infinity cosmos of Segal categories. So  

37:45

the right visual metaphor is the sort of Men  in Black outro, which takes the power of 10  

37:51

video where you, you know, start with a couple  in Millennium Park in Chicago and then sort of   zoom out. And then eventually you see the United  States and then you see the Earth and that you see  

38:00

the solar system, and then you see the Milky  Way. And then at the very end, you, you see that,   that whole thing lives inside a marble that some  alien is playing with. So these infinity cosmos or  

38:10

the marbles. So they're, they're small somehow  because we have different models that we're   thinking about at once, and we're sort of juggling  between them, but they're also very big because  

38:17

they contain the whole universe within them.  Okay. So, what Dom and I have done is that we have  

38:25

proven that the theory of infinity categories,  so theorems that mathematicians want to use about   infinity categories can be developed in any model  and are independent of the features of the model.  

38:35

And the way we use that is we've observed that  when you zoom way, way out, and you're thinking  

38:40

at the level of an infinity cosmos, all of the  models kind of look the same. So there are common  

38:46

properties of the infinity cosmos of infinity  categories in one model and the infinity cosmos  

38:51

of infinity categories in another model. And if  we prove a theorem starting from those common   properties, then it applies, it's one of these  very abstract results that then applies in all  

38:59

sorts of specific contexts. Usefully though,  our theorem is kind of general enough to  

39:07

also encompass analytically proven theorems.  I do refer to the coordinates of a specific  

39:12

model. But, we still have this problem of these  sort of evil statements that are not variant under  

39:19

equivalence between infinity categories or under  change of models. So these sort of questions that  

39:25

somehow are beside the point that are asking about  features of the model, as opposed to features of   the infinity category. And there's a question  about how to exclude statements like that.  

39:36

And so, we are developing a formal language  for infinity categories that are based on a  

39:43

formal language for category theory developed  by Michael Makkai. And the idea is he's looking,  

39:51

using a sort of familiar first-order logic, so  the familiar formal language of mathematics,  

39:56

but then imposing some additional restrictions. So  one of the restrictions is that equality cannot be  

40:03

sort of used everywhere in a formal sentence in  the language of categories. It's permissible to  

40:09

ask whether two parallel arrows are equal, but  it's not permissible to ask whether two objects   are equal. And then there are also restrictions  involving the quantifiers. So this upside down A  

40:18

is, reads to a mathematician as for all, and  the backwards E reads to mathematicians as there  

40:24

exists, and there are also restrictions there.  So Dominic and I have developed an analogous  

40:31

signature. So these are describing the structure  of the variables in a formal language for infinity  

40:36

categories. You can see it's considerably  more complicated than the signature for   ordinary categories, but infinity categories  are somehow infinitely more complicated than  

40:45

ordinary categories, and the fact that this is  sort of finitely more complicated, is I think a  

40:51

better than one could have hoped. Okay. So there's  considerably more to say about that, but I think  

40:58

at this point, I'll leave that to the references.  So Dom and I are very close to the end of a book,  

41:05

Elements of Infinity Category Theory, that I hope  will appear in 2021. It's also currently available  

41:10

on the web. I should say this, the intended  audience for this is sort of graduate level  

41:16

mathematicians, but there's some pretty pictures  and maybe you want to check it out. I also   just want to say thank you to everyone who's  helped me here in so many different ways. So.   

41:27

>>Fantastic, Dr. Riehl, amazing lecture. If  that's okay with you, I'd like to ask you  

41:36

questions that were posed through the website or  through chat. And the first question is from Dr.  

41:45

Sonja Armstrong, who's asking, why do you include  zero in the set of natural numbers? >>It's not,  

41:53

right, this is always a point of controversy. So  the set of natural numbers with zero is isomorphic  

42:00

to the set of natural numbers without zero. So  from that notion of sameness, it doesn't actually  

42:05

matter whether you include zero or not. So it's  really kind of a matter of personal preference.  

42:13

I guess I like to have zero there because it's  nice when you're adding numbers to have a number   you can add that doesn't change anything, but if  you'd like to take it out I won't object too much   

42:25

>>Also from Dr. Armstrong, are you redefining  the fundamental theorem of arithmetic since the  

42:32

theorem refers to prime and composite numbers?  How you do, do you define prime and composites?   

42:39

>>Right. So, great. So the fundamental theorem of  arithmetic is the result about the natural numbers  

42:46

that says that any natural number can be expressed  uniquely as a product of primes. And I guess here,  

42:53

I should maybe say any non-zero natural numbers.  So that might be related to the question about   zeros, but so it is certainly possible to define  a prime number in the sort of formal language for  

43:06

mathematical, the formal language for natural  numbers that we've been discussing. So a prime is  

43:12

a number that has no proper factors. So if  your number P is equal to a product, A times B  

43:20

of other natural numbers, then either A is equal  to one or B is equal to one. And that is certainly  

43:26

a sentence that you can write down. It's negation  then sort of the formal compliment of that  

43:32

sentence defines the composite numbers. So those  are certainly things that you can start to prove  

43:37

theorems about beginning with the Peano axioms.   >>Thank you. What do you see as the advantages  

43:47

of category theory over set theory as a  foundation for mathematics? >>Right. So  

43:56

I think it's useful to have a foundation for  mathematics that's a little closer to mathematical  

44:02

practice. So sort of famously when mathematicians  in the early 20th century, we're trying to  

44:11

write very, very formal versions of proofs of, you  know, sort of basic results. Two plus two equals  

44:16

four, that all mathematicians accept are true.  It, it took, you know, hundreds of pages. This is,  

44:22

this is just really, really, really difficult,  and it's also kind of completely unreadable.  

44:28

So more recently as a mathematics gets  very, very complicated, there has been  

44:35

increased interest in using computer proof  assistance. So what a computer is doing actually   is it's, it's not at this stage helping you come  up with new ideas or coming up with new theorems,  

44:45

but it's checking that the proof that you've  claimed you've written is actually correct. So,  

44:53

so the idea is that you know, this is very cutting  edge today in, in, in limited use, but the idea is  

44:58

that in the future that a mathematician might be  able to formally verify all of the little lemmas,  

45:05

all of the building block results in their papers.  And so when you write a paper and share it with  

45:10

other mathematicians, you could also share this  computer file that checks that everything that   you've claimed is true is in fact, correct.  So there's a sense in which this should be  

45:20

possible because there are these formal languages  that you can teach a computer to recognize   whether a statement is valid and you can also  teach a computer, the basic rules of inference  

45:31

in mathematical proof. So that if A is true and A  implies B is true, then B as true, that's called  

45:37

modus ponens, it's one of the logical principles.  That's kind of the building block of mathematical   proofs. But for this to be practicable,  so for a mathematician who has, you know,  

45:47

is also teaching and is also advising students  and, you know, you know, has limited time to  

45:54

be able to write every single step of every  single logical argument in a computer, you  

46:00

really need the formal foundations to be  much closer to mathematical practice. You   need this sort of sentences. You would teach  a computer to be close to the way that you're  

46:07

thinking about these objects already. And set  theory just really is not that. So one of the  

46:13

most basic concepts in mathematics is the notion  of a function between sets and the way that   functions are thought about or formalized in set  theory, is somewhat complicated. It's a formalized  

46:23

as a graph or as it's relation. And you know, it's  just, it's just not the way they're mathematicians  

46:29

tend to think about functions. So there are  other formal systems that are more categorical  

46:35

such as type theory, for instance, that I think  are closer to mathematical practice. And this is   one of the things that I'm very interested in as  well. >>A couple of questions about your career,  

46:47

your scholarship, you as a person...of  course one is from Andrea was asking,  

47:00

how did you, Dr. Riehl, find your way  into this field of mathematics? >>I mean,  

47:11

it's kind of a very human story. I, you know, I  loved mathematics as a child and loved it even  

47:17

more when I started to learn about proof-based  mathematics and, you know, kind of how creative   it is and how you know, how much writing is  involved. And before I started my PhD, I went to  

47:28

Cambridge, England for a year and took a category  theory course there. And it was kind of love at  

47:33

first sight. It felt to me like, this is the  style of proof that I liked the best. This is   the way I like to think about mathematical  objects. And so that was that. >>Terrific.  

47:45

Also these question, right, from...what is a  common misconception people have about your  

47:52

research? Either experts or non-experts.   >>So there's this there's this sort of  

47:59

classical phrase, abstract nonsense to  describe proofs and category theory. And  

48:05

some people view it as a pejorative, other people  embrace it. So I guess you know, when the first  

48:12

paper in category theory was written in the 1940s,  it was simultaneously viewed as very profound  

48:19

and completely meaningless. And I, I think I  think that could still apply. I don't know.   

48:28

>>So I think we all have a same question and  that's okay too, you know but from Jenny,  

48:35

how does one apply category three  through so-called real life problems?   

48:41

>>So actually applied category theory is a new and  growing field within category theory. And they are  

48:49

thinking about modeling sort of compositional  frameworks, whether these are like Petri nets or  

48:56

you know, some sort of computer processes  where you can run things in parallel,   or you can run them in sequence, or you  can do some sort of combination of them  

49:03

or quantum systems. It's an extremely active  research area. One of the applications is also  

49:10

if say you've written a computer program, and  you want to verify that it functions as intended,   you know, that its implementation matches with  its intended design. So that can be expressed as a  

49:21

functor. In other words, an arrow in the language  of category theory. There are also applications   to mathematical physics of the kind of infinite  dimensional categories that I was mentioning. So  

49:34

there's a notion of a quantum field theory which  what it means mathematically is it's a functor, so  

49:40

again, an arrow between these infinite dimensional  categories. So, so yeah, the applications are  

49:47

there though a lot of mathematicians are motivated  primarily by kind of aesthetics and, you know,  

49:52

you want to know the answer because you want to  know the answer and that would describe myself.   

50:00

>>Thank you from...do you  think this also....all of this   could someday be considered basic enough in  ordinary syllabus for high school students or  

50:11

this is necessary necessarily advanced and only  of interest to specialists? >>Great. That's a  

50:17

good question. I don't anticipate teaching this  to high school students ever, but I do dream about  

50:22

teaching it to mathematics undergraduates. So  the things that we teach our undergraduates today  

50:28

are were so cutting edge, you know, a hundred  years ago, 150 years ago, these were, you know,  

50:34

were to blown the minds of mathematicians just  a few centuries or a few decades before. So,  

50:40

so our undergraduates today are handling extremely  sophisticated mathematical concepts. This work  

50:45

that I was talking about today is not well enough  understood yet, it's complicated and challenging  

50:51

to explain to a graduate student, but I hope that  in a hundred years' time, we understand things  

50:57

in you know, we, we really we understand to the  point of being able to simplify and, you know,  

51:02

perhaps have improved our foundation system,  which I think at present is really holding us   back and our teaching this sort of things to  advanced undergraduates yeah. >>From...why or  

51:13

how is functional programming related to category  theory? >>Right. So I'm not a computer scientist,  

51:23

but there is kind of sophisticated operations  that you can have on data structures that can be  

51:32

expressed in a categorical way. So a program  you know, has, the way a program runs might be  

51:43

affected by that current state, a machine is in  and might leave that state that the machine is in  

51:48

altered. So they, we have these sort of  interactive input output and side effects   and if you're understanding programs in that  way, then understanding what it means to  

51:57

combine programs to compose programs is quite  complicated. But the language of category theory  

52:02

can express that. So the people who designed  programming languages like Haskell are aware of  

52:09

category theory and are implementing categorical  concepts in that programming language and the way  

52:16

that people study programming languages  can be informed by categorical thinking.   

52:25

>>I guess I have a question also about   what seems to be your acute awareness of the  historical context. So who have been contributors  

52:35

to your field. And I was wondering, where was  that happy balance between knowing enough of  

52:42

what's been done, but maybe at some point may  diminish or...the original because you think  

52:49

maybe not, maybe at this point you see a lot of  open space in front of you or and versus almost  

52:56

being happily innocent or ignorant  of all those previous advances as,  

53:02

as a way to be feeling liberated in your, in your  pursuits. I don't know if you think about that.   

53:10

>>I do. I don't have an answer for that. I  mean, one, one thing that we're, we're lucky in   mathematics is that we're, we're really in a very  long conversation in a way that our colleagues in  

53:19

the sciences aren't. You know, we Euclid has this  beautiful proof that there are infinitely many   primes that's over 2000 years old and it's still  as true today as it was then, and as useful to us  

53:29

as it's always been. So I think you can't,  I mean, you can't ignore the history. But  

53:36

you're right that just because somebody thought  about something in one way 50 years ago, doesn't  

53:42

necessarily mean that's the most productive way  to think about it today. And I think that's one  

53:48

of the exciting things about doing research  in a teaching institution is, you know,  

53:54

there are new people coming through all the time  with new ideas and new interests and already  

54:00

you know, infinite dimensional category theory is  more popular among kind of younger people in my   field than older people in my field. And so you  know, interests shift and times, and there are,  

54:10

my students are more interested in type theory.  And I think you know, being part of this structure  

54:17

where we can sort of challenge these hierarchies  and have sort of established senior people and  

54:24

ambitious junior people together in the  same discussion, I think is really helpful   in driving the science forward. >>I think it  was said before, I think by, by John, that  

54:37

you've made extra efforts, especially for, to  really make your field more accessible. I think  

54:43

this lecture, you just give epitomizes this, this,  this, you know, you, you could be still trying to  

54:48

impress and make contribution when vis-a-vis your  peers, but you you've seen more than that. An  

54:55

opportunity really to bring in kind of the next  generation of mathematicians into the field and  

55:01

your, your texts I think have helped  tremendously. They're super highly cited where,  

55:07

you know...especially in mathematics. So Emily,  if I may, if I may I'd like to thank you for a  

55:16

wonderful lecture and thank all attendees who  gave great feedback, great questions, I think.  

55:25

I hope also that you've inspired many to, to  talk about math as a really a wonderful pursuit  

55:35

and academic and, and really almost,  I think you touched on, on philosophy  

55:40

to some extent as well. So thank you  again. And I like to mention that  

55:48

the recipient of our next Presidential Frontier  Award will be announced early next year. So  

55:55

checkout who have been our past recipients besides  Professor Riehl. It's quite a cohort. They are  

56:03

really creating a huge and wonderful solid  foundation to for students and peers at Hopkins.  

56:12

So again, thank you so very much Professor  Riehl for a wonderful lecture and  

56:18

one congratulations again for receiving the  Presidential Frontier Award. >>Thanks so much.