a call for examples

a call for examples

[Emily Riehl]

Hi all,

I am writing in hopes that I might pick the collective brain of the categories list. This spring, I will be teaching an undergraduate-level category theory course, entitled “Category theory in context”:

http://www.math.harvard.edu/~eriehl/161

It has two aims: 

(i) To provide a thorough “Cambridge-style” introduction to the basic concepts of category theory: representability, (co)limits, adjunctions, and monads.

(ii) To revisit as many topics as possible from the typical undergraduate curriculum, using category theory as a guide to deeper understanding.

For example, when I was an undergraduate, I could never remember whether the axioms for a group action required the elements of the group to act via *automorphisms*. But after learning what might be called the first lemma in category theory -- that functors preserve isomorphisms -- I never worried about this point again.

Over the past few months I have been collecting examples that I might use in the course, with the focus on topics that are the most “sociologically important” (to quote Tom Leinster’s talk at CT2014) and also the most illustrative of the categorical concept in question. (After all, aim (i) is to help my students internalize the categorical way of thinking!)

Here are a few of my favorites:

* The Brouwer fixed point theorem, proving that any continuous endomorphism of the disk admits a fixed point, admits a slick proof using the functoriality of the fundamental group functor pi_1 : Top_* —> Gp. Assuming the contrapositive, you can define a continuous retraction of the inclusion S^1 —> D^2. Applying pi_1 leads to the contradiction 1=0.

* The inverse image of a function f : A —> B, regarded as a functor f* : P(B) —> P(A) between the posets of subsets of its codomain and domain, admits both adjoints and thus preserves both intersections and unions. By contrast, the direct image, a left adjoint, preserves only unions.

* Any discrete group G can be regarded as a one-object groupoid in which case a covariant Set-valued functor is just a G-set. The unique represented functor is the G-set G, with its translation (left multiplication) action. By contrast, a *representable* functor X, not yet equipped with the natural ($G$-equivariant) isomorphism $G \cong X$ defining the representation, is a $G$-torsor. I learned this from John Baez’s this week’s finds:

http://math.ucr.edu/home/baez/torsors.html

My favorite example is still the one that John uses: n-dimensional affine space is most naturally a R^n-torsor.

* The universal property defining the tensor product V @ W as the initial vector space receiving a bilinear map 

@ : V x W —> V @ W

can be used to extract its construction. The projection to the quotient V @ W —> V @ W/<v @ w> by the vector space spanned by the image of the bilinear map @ must restrict along @ to the zero bilinear map, as of course does the zero map. Thus V @ W must be isomorphic to the span of the vectors v @ w, modulo the bilinearity relations.

* By the existence of discrete and indiscrete spaces, all of the limits and colimits one meets in point-set topology -- products, gluings, quotients, subspaces -- are given by topologizing the (co)limits of the underlying sets. Of course this contradicts our experience with the constructions of colimits in algebra.

* On that topic, the construction of the tensor product of commutative rings or the free product of groups can be understood as special cases of the general construction of coproducts in an EM-category admitting coequalizers.

I would be very grateful to hear about other favorite examples which illustrate or are clarified by the categorical way of thinking. My view of what might be accessible to undergraduates is relatively expansive, particularly in the less-obviously-categorical areas of mathematics such as analysis.

I have also posted this query to the n-Category Cafe and am hoping to collect examples there as well:

https://golem.ph.utexas.edu/category/2014/12/a_call_for_examples.html

Best wishes to all for a happy and productive new year.

Emily Riehl

--
Benjamin Peirce & NSF Postdoctoral Fellow
Department of Mathematics, Harvard University
www.math.harvard.edu/~eriehl



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: a call for examples

[Richard Garner]

> * Any discrete group G can be regarded as a one-object groupoid in which
> case a covariant Set-valued functor is just a G-set. The unique
> represented functor is the G-set G, with its translation (left
> multiplication) action. By contrast, a *representable* functor X, not yet
> equipped with the natural ($G$-equivariant) isomorphism $G \cong X$
> defining the representation, is a $G$-torsor.

Related to this is the use of what is essentially the Yoneda lemma in
normalisation proofs. For example Britton's lemma (giving normal forms
for words in an HNN extension) proceeds by taking the set of "normal
forms" and showing that it is a torsor for the group G at issue. In
other words one takes a presheaf on G and shows that it is
representable, and so is isomorphic to G itself. Another argument of a
similar Yoneda kind is the usual proof of the Poincare--Birkhoff--Witt
theorem (giving a basis for the universal enveloping algebra of a Lie
algebra). In fact the technique is very widely applicable (and very
widely applied); for instance you could prove the normal form theorem
for the simplicial category using it. A related (but more general)
technique is what is sometimes called "normalization by evaluation" by
computer scientists; Peter Dybjer has some articles on the relation to
the Yoneda lemma. But that is maybe straying a bit far from
undergraduate mathematics.

Richard


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: a call for examples

[Ross Street]

On 2 Jan 2015, at 4:47 am, Ronnie Brown <ronnie.profbrown@btinternet.com> wrote:

> Just to put a slightly different emphasis, I like to present categories
> and groupoids as good examples of structures having the dual roles of
>  (i) algebraic structures in their own right, and also
> (ii) of value for talking about mathematical structures.

Yes, I absolutely agree (I might even replace `algebraic’ in (i) by `mathematical’).

Another aspect (perhaps already mentioned) is that (monoidal) categories allow
one branch of mathematics to inspire another and, indeed, talk to each other.
I have in mind the recognition that concepts like dual finite dimensional vector
space and trace of a linear endomorphism inspire application to knot theory;
the talking is done by strong monoidal functors in explaining (in their own
way) link invariants.    

Ross
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: a call for examples

[Ronnie Brown]

Dear Emily and all,

Just to put a slightly different emphasis, I like to present categories
and groupoids as good examples of structures having the dual roles of
   (i) algebraic structures in their own right, and also
(ii) of value  for talking about mathematical structures.

The advantage of (i) is the move to the interest in *partial *algebraic
structures, and so allow an easy move to higher dimensional algebra, as
the study of partial algebraic structures whose operations have domains
defined by geometric conditions, and also gets people used to the
transition from groups to groupoids.

Thus the groupoid object {\cal I},   which has two objects 0,1 and one
non identity morphism \iota: 0 \to 1, is very easy to understand, but
when one identifies 0 and 1 one gets the integers! Hence the fundamental
group of the circle is the integers! This process of identification is
best understood in the context of saying the functor
                        Ob: Groupoids \to Sets
is a bifibration. Groupoids are  an example of an algebraic structure
with structure in levels 0,1 and the functor Ob forgets the top level.
This idea is of course relevant to other constructions in homotopy
theory, since in homotopy theory identifications in low dimensions have
higher dimensional homotopical implications.  Our book "Nonabelian
algebraic topology: ....."  has an Appendix in fibrations of categories
because the notion crops up in terms of earlier results, often connected
with excision.   That book does not use monads: it would be interesting
to know of expository improvements by using that notion.

Analogously one can define the natural numbers in terms of a pushout of
categories, using the category *2*, and identifying 0 and 1.

The notion of fibration of groupoids is also useful algebraically, and
for modelling homotopy theory, including exact sequences, particularly
at the bottom end. See for example a recent article arXiv:1207.6404
<http://arxiv.org/abs/1207.6404>

This notion is used in my book "Topology and groupoids"  to construct
operations on certain homotopy sets, generalising the usual change of
base point. The point is that groupoids are used in that book as  a
combinatorial and calculational tool, not just for theoretical reasons.

Again, covering space theory is treated there using covering morphisms
of groupoids, which are equivalent to actions of groupoids on sets.

While writing, I mention

http://education.lms.ac.uk/2014/12/alexander-grothendieck-some-recollections/


Best regards

Ronnie





On 28/12/2014 21:52, Emily Riehl wrote:

Hi all,

I am writing in hopes that I might pick the collective brain of the 
categories list. This spring, I will be teaching an undergraduate-level 
category theory course, entitled ???Category theory in context???:

http://www.math.harvard.edu/~eriehl/161

It has two aims:

(i) To provide a thorough ???Cambridge-style??? introduction to the basic 
concepts of category theory: representability, (co)limits, adjunctions, 
and monads.

(ii) To revisit as many topics as possible from the typical undergraduate 
curriculum, using category theory as a guide to deeper understanding.

For example, when I was an undergraduate, I could never remember whether 
the axioms for a group action required the elements of the group to act 
via *automorphisms*. But after learning what might be called the first 
lemma in category theory -- that functors preserve isomorphisms -- I never 
worried about this point again.

Over the past few months I have been collecting examples that I might use 
in the course, with the focus on topics that are the most 
???sociologically important??? (to quote Tom Leinster???s talk at CT2014) 
and also the most illustrative of the categorical concept in question. 
(After all, aim (i) is to help my students internalize the categorical way 
of thinking!)

Here are a few of my favorites:

* The Brouwer fixed point theorem, proving that any continuous
   endomorphism of the disk admits a fixed point, admits a slick proof
   using the functoriality of the fundamental group functor pi_1 : Top_*
   ???> Gp. Assuming the contrapositive, you can define a continuous
   retraction of the inclusion S^1 ???> D^2. Applying pi_1 leads to the
   contradiction 1=0.

* The inverse image of a function f : A ???> B, regarded as a functor f* :
   P(B) ???> P(A) between the posets of subsets of its codomain and domain,
   admits both adjoints and thus preserves both intersections and unions.
   By contrast, the direct image, a left adjoint, preserves only unions.

* Any discrete group G can be regarded as a one-object groupoid in which
   case a covariant Set-valued functor is just a G-set. The unique
   represented functor is the G-set G, with its translation (left
   multiplication) action. By contrast, a *representable* functor X, not
   yet equipped with the natural ($G$-equivariant) isomorphism $G \cong X$
   defining the representation, is a $G$-torsor. I learned this from John
   Baez???s this week???s finds:

http://math.ucr.edu/home/baez/torsors.html

My favorite example is still the one that John uses: n-dimensional affine 
space is most naturally a R^n-torsor.

* The universal property defining the tensor product V @ W as the initial
   vector space receiving a bilinear map

@ : V x W ???> V @ W

can be used to extract its construction. The projection to the quotient V 
@ W ???> V @ W/<v @ w> by the vector space spanned by the image of the 
bilinear map @ must restrict along @ to the zero bilinear map, as of 
course does the zero map. Thus V @ W must be isomorphic to the span of the 
vectors v @ w, modulo the bilinearity relations.

* By the existence of discrete and indiscrete spaces, all of the limits
   and colimits one meets in point-set topology -- products, gluings,
   quotients, subspaces -- are given by topologizing the (co)limits of the
   underlying sets. Of course this contradicts our experience with the
   constructions of colimits in algebra.

* On that topic, the construction of the tensor product of commutative
   rings or the free product of groups can be understood as special cases
   of the general construction of coproducts in an EM-category admitting
   coequalizers.

I would be very grateful to hear about other favorite examples which 
illustrate or are clarified by the categorical way of thinking. My view of 
what might be accessible to undergraduates is relatively expansive, 
particularly in the less-obviously-categorical areas of mathematics such 
as analysis.

I have also posted this query to the n-Category Cafe and am hoping to 
collect examples there as well:

https://golem.ph.utexas.edu/category/2014/12/a_call_for_examples.html

Best wishes to all for a happy and productive new year.

Emily Riehl

--
Benjamin Peirce & NSF Postdoctoral Fellow
Department of Mathematics, Harvard University
www.math.harvard.edu/~eriehl


-- 




[For admin and other information see: http://www.mta.ca/~cat-dist/ ]