#adjoint functor

[bubbloquacious]

Do you have any recommendations for texts for a first reading of category theory? :))

So this is a super common question and I think it's a little hard to give a good answer to! I'll tell you what I did.

Back in the summer of the beautiful year 2019 I had just finished my first year of university mathematics education. I don't remember where I had even heard of the subject, but the first math book I ever self-studied out of was Mac Lane's Categories for the Working Mathematician (the second edition, specifically). It's a good book, I think! But I only really appreciated it on my reread (i.e. my more-than-the-first-three-chapters read) last year. For one, I had not done a lick of topology by this point, and I really do recommend knowing basic topology and basic abstract algebra before getting into categories; it gives you a nice two-pronged approach to view every construction.

So I washed out of that one pretty quickly. I learned the basic definitions of categories, functors, and natural transformations. I could work out what it meant for a product to be the limit of a discrete diagram (super neat when you just learned what a group was five months ago!), but if you asked me to give an example of a pair of adjoint functors that wasn't in the text I would've been stumped (natural transformations between hom-functors? what?).

Over the course of the next semester I got familiar with the basics of topology (and I made this blog!). Now I could see why a lot of Mac Lane's examples were so nice. Compositions of homotopic maps are homotopic, so you can take a quotient category the same way you take a quotient group. The fundamental group is so useful exactly because it is functorial. A map into a product space is continuous precisely when its composition with the projections is continuous. And so on. I sampled some more books. I read the first few chapters of Awodey's Category Theory and of Adámek, Herrlich, and Strecker's Abstract and Concrete Categories. Each offers a slightly different perspective, but every time I would wash out after the first several chapters.

But it was exactly rereading all of these different perspectives that made me grok what it was all about. How to work with these structures. I had limits and colimits pretty much down by this point. Ah, it's not a coincidence that the least upper bound and greatest lower bound are categorical limits, they're expressing the same idea. I realized on my own that not only is the discrete space functor to topological spaces left adjoint to the forgetful functor to sets, the indiscrete (or chaotic, or trivial) space functor is right adjoint. The abelianization of a group or the completion of a metric space are left adjoint to the inclusion functors of abelian groups and complete metric spaces, because a map from a group/metric space into an abelian group/complete metric space determines a unique map (in fact it factors through this map) out of the abelianization/completion. Even though the abelianization makes the group smaller, and the completion makes the space bigger, they're expressing the same relationship.

This went on for a while. I read Bartosz Milewski's blog on Category Theory for Programmers, but I didn't know Haskell so that didn't get me very far. I read a lot of nLab pages (I still do). I read a small bit of Borceux's Handbook of Categorical Algebra, which is very nicely written if you can get your hands on it. I spent a summer in a reading group for Tai-Danae Bradley's Topology: A Categorical Approach (this one's probably a good pick for you in particular!). Like two years ago I played the Natural Number Game (which they changed recently apparently!), which got me really into Lean and proof assistants in general. Playing around with Lean gave me a great appreciation for what category theory brings to the table for logic and computational mathematics. Reading Dan Marsden's blog on monads got me really into monad theory, which is just super cool and I love it very much. I had a brief fling with bicategories, mostly because of the monads. Around this time I made a bunch of explanatory posts on the things I was learning about, and that was just terrifically helpful for my understanding. I read Mac Lane and Eilenberg's original paper introducing categories, General Theory of Natural Equivalences, and it's fascinating to see the perspective that it takes. Last year I got really into sheaf theory by reading Mac Lane and Moerdijk's Sheaves in Geometry and Logic, which finally made me understand the Yoneda Lemma.

These last few examples are what I would call pretty intermediate, as far as categories go. Somehow by reading all these different sources (and progressing in my studies in other mathematical subjects!) I had gathered up enough experience to understand the higher-level structures built on top of the basics. I still don't understand Kan extensions (though I tried reading Riehl's Category Theory in Context to understand it) or fibrations, or model categories, or co/end calculus, but I'm sure I will once the need arises.

So that's my advice! Read as many different introductions to the subject as you can, in as many different contexts as you have the experience for! The power and beauty of category lies in the bridging over the gaps between these contexts. Any of the books I mentioned I would recommend, but really the most important rule of category theory is to have fun and be yourself :-)

#math #adventures in cat theory #this went kinda long huh lol #i really like your posts panda i hope this is helpful <3 #i have pdfs of everything i mentioned also so hit me up if you can't find one

[measurementcat]

I am attempting to learn “real algebraic geometry” (schemes) and it is the most brutal. The density of new concepts is like, 5 times as high as any previous math class I’ve been in. It took us about a month to define what a scheme even is (using locally ringed spaces). I’m somewhat hoping the difficulty / density of new concepts thins out from here, but I’m not sure if it will or not. 

I do have this wondering about, learning about sheafs and categories and so on so far doesn’t feel super directly related to what schemes actually are (this is a bit silly, I suppose, since a scheme is “mostly” “just a sheaf” (plus the Zariski topology)). But I can’t help but wonder whether I’m feeling this overwhelmed because I’ve done the equivalent of learning assembly before learning C, instead of learning C or better Python and then opening it up under the hood. I do expect we’ll be using a lot of category theory and some sheaf theory though. The first big proof we’ve done so far was the proof that the global sections functor is adjoint to the Spec functor, which both is a theorem of category theory as well as using a lot of category theory in the proof. 

It is definitely the first time a math class has made me wonder if I should make flashcards to memorize things though. 

#mathblr #math #algebraic geometry #anki

[endofunktor]

So you can forget that a unital ring is unital.

Denote Ring the category of unital rings and Rng the category of (general) rings. Then this should constitute a forgetful functor

S : Ring → Rng.

Can this functor be adjoint? What does it yield?

#category theory #ring theory #mathblr

[theohonohan]

Cocktails and Adjunctions

Loosely inspired by Hans Peter Luhn's Cocktail Oracle, this is a brief, and relatively concrete, exploration of adjunctions (the category theory concept) in the setting of relations.

It is straightforward to consider a mapping from cocktail recipes to ingredients as a relation. A given recipe contains zero or more ingredients. (Let's assume that some conceptual cocktails exist which have no physical ingredients). This relation can be thought of as taking the name of a cocktail and mapping it to a set of ingredients. Because it's a relation (not a function) it is unlikely to have an inverse: the same set of ingredients could map to multiple cocktails.

Category theory puts forward the concept of an adjunction, which is a kind of best-effort analogue of an inverse. A left adjunction is analogous to a left inverse, and a right adjunction is analogous to a right inverse.

Concretely, if we use our recipe relation to map a cocktail name to a set of ingredients, we can then use the left adjoint of the recipe relation to map that set of ingredients back to a set of recipes. This new set of recipes will include any cocktail that includes any one of the ingredients. This is an example of the "exists" functor—it is expansive (or inclusive) in its attempt to figure out, from a set of ingredients, what cocktail was originally asked for. Of course, the set of ingredients that we pass to the left adjoint might not come directly from a cocktail recipe. It could just reflect the contents of a drinks cabinet. The "exists" functor performs best in this case when the set only includes ingredients that are unique to a particular cocktail, and worst when an ingredient that is used in many cocktails is present.

On the other side, we can use the right adjoint, the "forall" functor, to map a set of ingredients to a set of cocktail names. Very simply, this set of cocktail names will include only the cocktails that can be made using those ingredients. The right adjoint is restrictive (or strict).

Every adjunction gives rise to a monad. So what monads can we derive from this situation? Well, the first one arises from composing our recipe map with its left adjoint. It maps a set of ingredients to a set of recipes, and that set of recipes is then mapped back to a set of ingredients. The effect of this is to complete the set of ingredients so that any cocktail that might be "suggested" by the initial set of ingredients can now be mixed. The fact that a monad is a kind of generation of a closure operator is clear here: what we've calculated is a kind of closure of the set of ingredients, pulling in extra ingredients where needed so that any cocktail which uses any one of the initial set of ingredients can now be made.

The second monad we can derive comes from our recipe mapping and its right adjoint. We compose the right adjoint with the recipe map. The result of this is to take a set of cocktail names, find all of the ingredients necessary, and then find all of the cocktails that can be made using only ingredients from that set. So this monad is a kind of closure of the cocktail "menu": if, given a menu listing cocktails A, B and C, we must have the ingredients to also make D and E, then this monad maps our menu "A, B, C" to "A, B, C, D, E": it is a completion of the menu based on all of the ingredients we have to hand.

In a sense, the adjoints attempt to reconstruct a set of cocktail names based on set of ingredients. I think of this as a detective problem: in a room, you come across a drinks cabinet with some half-full bottles in it, and you want to know what cocktails the inhabitant was making. It isn't possible to give a definitive correct answer, in general. If you think some bottles might be missing, you'll need to use the left adjoint (the "exists" functor) to get a best guess. If you suspect that there are some extraneous bottles, the right adjoint (the "forall" functor) might help in ignoring them.

Trying to draw inferences in this way is logical but maybe not that impressive. Like category theory in general, though, it's an absolutely principled and mechanical method.

I learned some of this by asking a StackExchange question.

[mathsuggestions]

Try to understand adjoint functors one more time.

#math #suggestion

[vimagubikoga]

Advanced linear algebra book pdf

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            Course notes for a 12-week advanced linear algebra course based on this textbook (just 10 weekly PDFs are provided, as it takes the author 12 weeks to get through this material due to midterms, snow days, and course projects): - Week 1 ( blank PDF, annotated PDF) - Week 2 ( blank PDF, annotated PDF) - Week 3 ( blank PDF, annotated PDF) The ten years since the first edition have seen the proliferation of linear algebra courses throughout the country and have afforded one of the authors the opportunity to teach the basic material examples, which are usually presented in introductory linear algebra texts with more abstract de nitions and constructions typical for advanced books. Another speci c of the book is that it is not written by or for an alge-braist. So, I tried to emphasize the topics that are important for analysis, geometry, probability, etc., and did not Featuring updates and revisions throughout, Advanced Linear Algebra, Second Edition: Contains new chapters covering sesquilinear forms, linear groups and groups of isometries, matrices, and three important applications of linear algebra. Adds sections on normed vector spaces, orthogonal spaces over perfect fields of characteristic two, and Harvard Mathematics Department : Home page 8CONTENTS Preface This is a book on linear algebra and matrix theory. While it is self contained, it will work best for those who have already had some exposure to linear algebra. It is also assumed that the reader has had calculus. Some optional topics require more analysis than this, however. Linear Algebra Igor Yanovsky, 2005 7 1.6 Linear Maps and Subspaces L: V ! W is a linear map over F. The kernel or nullspace of L is ker(L) = N(L) = fx 2 V: L(x) = 0gThe image or range of L is im(L) = R(L) = L(V) = fL(x) 2 W: x 2 Vg Lemma. ker(L) is a subspace of V and im(L) is a subspace of W.Proof. Assume that fi1;fi2 2 Fand that x1;x2 2 ker(L), then L(fi1x1 + fi2x2) = fi1L(x1)+fi2L(x2 These linear algebra lecture notes are designed to be presented as twenty ve, fty minute lectures suitable for sophomores likely to use the material for applications but still requiring a solid foundation in this fundamental branch of mathematics. The main idea of the course is to emphasize the concepts of vector spaces and linear transformations as mathematical structures that can be used to 2 Advanced Algebra - Pareigis 5.1. Representable functors 46 5.2. The Yoneda Lemma 49 5.3. Adjoint functors 51 5.4. Universal problems 52 6. Limits and Colimits, Products and Equalizers 55 6.1. Limits of diagrams 55 6.2. Colimits of diagrams 57 6.3. Completeness 58 6.4. Adjoint functors and limits 59 7. The Morita Theorems 60 8. Simple and In this sense, our collection of algebra books in PDF format will be very useful for your studies and research. The first algebra treatise was published in 820 A.D. by the astronomer Al-Khwarizmi. Therefore, its origin was placed in the Arabic culture. Math 4377/6308 { Advanced Linear Algebra I Vaughn Climenhaga December 3, 2013. 2 The primary text for this course is \Linear Algebra and its Applications", second edition, by Peter D. Lax (hereinafter referred to as [Lax]). The lectures will follow the presentation in this book, and many of the homework exercises will be taken from it. You may occasionally nd it helpful to have access to other Math 4377/6308 { Advanced Linear Algebra I Vaughn Climenhaga December 3, 2013. 2 The primary text for this course is \Linear Algebra and its Applications", second edition, by Peter D. Lax (hereinafter referred to as [Lax]). The lectures will follow the presentation in this book, and many of the homework exercises will be taken from it. You may occasionally nd it helpful to have access to other

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            Course notes for a 12-week advanced linear algebra course based on this textbook (just 10 weekly PDFs are provided, as it takes the author 12 weeks to get through this material due to midterms, snow days, and course projects): - Week 1 ( blank PDF, annotated PDF) - Week 2 ( blank PDF, annotated PDF) - Week 3 ( blank PDF, annotated PDF) The ten years since the first edition have seen the proliferation of linear algebra courses throughout the country and have afforded one of the authors the opportunity to teach the basic material examples, which are usually presented in introductory linear algebra texts with more abstract de nitions and constructions typical for advanced books. Another speci c of the book is that it is not written by or for an alge-braist. So, I tried to emphasize the topics that are important for analysis, geometry, probability, etc., and did not Featuring updates and revisions throughout, Advanced Linear Algebra, Second Edition: Contains new chapters covering sesquilinear forms, linear groups and groups of isometries, matrices, and three important applications of linear algebra. Adds sections on normed vector spaces, orthogonal spaces over perfect fields of characteristic two, and Harvard Mathematics Department : Home page 8CONTENTS Preface This is a book on linear algebra and matrix theory. While it is self contained, it will work best for those who have already had some exposure to linear algebra. It is also assumed that the reader has had calculus. Some optional topics require more analysis than this, however. Linear Algebra Igor Yanovsky, 2005 7 1.6 Linear Maps and Subspaces L: V ! W is a linear map over F. The kernel or nullspace of L is ker(L) = N(L) = fx 2 V: L(x) = 0gThe image or range of L is im(L) = R(L) = L(V) = fL(x) 2 W: x 2 Vg Lemma. ker(L) is a subspace of V and im(L) is a subspace of W.Proof. Assume that fi1;fi2 2 Fand that x1;x2 2 ker(L), then L(fi1x1 + fi2x2) = fi1L(x1)+fi2L(x2 These linear algebra lecture notes are designed to be presented as twenty ve, fty minute lectures suitable for sophomores likely to use the material for applications but still requiring a solid foundation in this fundamental branch of mathematics. The main idea of the course is to emphasize the concepts of vector spaces and linear transformations as mathematical structures that can be used to 2 Advanced Algebra - Pareigis 5.1. Representable functors 46 5.2. The Yoneda Lemma 49 5.3. Adjoint functors 51 5.4. Universal problems 52 6. Limits and Colimits, Products and Equalizers 55 6.1. Limits of diagrams 55 6.2. Colimits of diagrams 57 6.3. Completeness 58 6.4. Adjoint functors and limits 59 7. The Morita Theorems 60 8. Simple and In this sense, our collection of algebra books in PDF format will be very useful for your studies and research. The first algebra treatise was published in 820 A.D. by the astronomer Al-Khwarizmi. Therefore, its origin was placed in the Arabic culture. Math 4377/6308 { Advanced Linear Algebra I Vaughn Climenhaga December 3, 2013. 2 The primary text for this course is \Linear Algebra and its Applications", second edition, by Peter D. Lax (hereinafter referred to as [Lax]). The lectures will follow the presentation in this book, and many of the homework exercises will be taken from it. You may occasionally nd it helpful to have access to other Math 4377/6308 { Advanced Linear Algebra I Vaughn Climenhaga December 3, 2013. 2 The primary text for this course is \Linear Algebra and its Applications", second edition, by Peter D. Lax (hereinafter referred to as [Lax]). The lectures will follow the presentation in this book, and many of the homework exercises will be taken from it. You may occasionally nd it helpful to have access to other

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[sharans-math-science-dump]

Here is a picture of me trying to think with pen and paper, based on the tantalizing crumbs left behind by Lawvere in Conceptual Mathematics, on exactly what functors are. Conceptual Mathematics is nominally a book for those with no background in mathematics, but since it treats topics that are very advanced and very new, there can sometimes be oddities in the way the material is presented. Aside from a few notes scattered in the front, which is what this is based on, most of the presentation of key topics like functors and presheaves are concentrated in the last few chapters, where difficulty zooms up hyperbolically and frankly I’m very lost and I need to re-read those segments. Here I attempt to classify and poke around with the basic functors presented in the chapter on endomorphisms, trying to figure out what sort of structure functors have. Very interesting bits of thinking, and also probably very pretentious, but of little help when left- and right-adjoint functors jumped out at me later. Ugh. Also, some notes on retractions and injections

#mathblr #mathematics #math #studyblr

[isomorphismes]

There are no theorems in category theory.

Emily Riehl, Category Theory In Context

Mathematicians often tell her this; hence the book.

If I had to summarise her views in one sentence, it would be:

Everything is an adjunction.

I also like the division these mathematicians are making to her: essentially, a theorem is anything that solves Feynman’s challenge: by a series of clear, unsurprising steps, one arrives at an unexpected conclusion.

Examples for me include:

17 possible tessellations

6 ways to foliate a surface

27 lines on a cubic

1, 1, 1, 1, 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, 16, 16, 523264, 24, 8, 4 ways to link any-dimensional spheres.

the existence of sporadic groups

surprising rep-theory consequences of Young diagrams, Ferrers sequences, and so on (you could say the strangeness of integer partitions is really to blame here…)

59 icosahedra

8 geometric layouts

Books which are bristling with mathematical ideas of this kind include Montesinos on tessellations, Geometry and the Imagination (the original one), and Coxeter’s book on polyhedra (start with Baez on A-D-E if you want to follow my path). Moonshine and anything by Thurston or his students, I’ve found similarly flush with shockng content—quite different to what I thought mathematics would be like. (I had pictured something more like a formal logic book: row by row of symbols. But instead, the deeper I got into mathematics, the fewer the symbols and the more the surnames thanking the person who came up with some good idea.)

Note that a theorem is different here to some geometry — as in The Geometry of Schemes. The word geometry used in that sense, I feel, is to have a comprehensive enough vision of a subject to say how it “looks” — but the word theorem means the result is surprising or unintuitive.

This definition of a theorem, to me, presents a useful challenge to annoying pop-psychology that today lurks under the headings of Bayesianism, cognitive _______, behavioural econ/finance, and so on.

Following Buliga and Thurston to understand the nature of mathematical progress, within mathematics at least (where it’s clearer than elsewhere whether you understand something or not—compare to economic theory for example), there is a clear delination of what’s obvious and what’s not.

What is definitely not the case in mathematics, is that every logical or computable consequence of a set of definitions is computed and known immediately when the definitions are stated! You can look at a (particularly a good) mathematical exposition as walking you through the steps of which shifts in perspective you need to take to understand a conclusion. For example start with some group, then consider it as a topological object with a cohomology to get the centraliser. Or in Fourier analysis: re-present line-elements on a series of widening circles. Use hyperbolic geometry to learn about integers. Use stable commutator length (geometry) to learn about groups. Or read about Teichmüller stuff and mapping class groups because it’s the confluence of three rivers.

Sometimes mathematical explanations require fortitude (Gromov’s "energy") and sometimes a shift in perspective (Gromov’s (neg)"entropy").

This view of theorems should be contrasted to the disease of generalisation in mathematical culture. Citing two real-life grad students and a tenured professor in logic (one philosophical, one mathematical, the professor in computer science):

I like your distinction between hemi-toposes, demi-toposes, and semi-toposes

I care about hyper-reals, sur-reals, para-consistency, and so on

Abstract thought — like mathematicians do — is the best kind of thought.

(twitter.com/replicakill, the author of twitter.com/logicians, ragged on David Lewis by saying “What do mathematicians like?” “What do mathematicians think?” —— And Corey Mohler has done a wonderful job of mocking Platonism, which is how I guess the thirst for over-generalisation reaches non-mathematicians.)

Paul Halmos knew that cool examples beat generalisations for generalisation’s sake, as did V. I. Arnol’d. And it seems that the people a Harvard mathematician spends her time with make reasonable demands of a mathematical idea as well. It shouldn’t just contain previous theories; it should surprise. In Buliga’s Blake/Reynolds dispute, Blake wins hands down.

#category theory #theorems #J P May #J. P. May #Emily Riehl #topos theory #toposes #topoi #mathematics #maths #math #adjunctions #adjoint #adjoint functor #functors #Daniel Kan #Kan extensions #tensors #tensor product #⊗ #1958 #algebra #analytic philosophy #logicians #logic #William Blake #Joshua Reynolds

[twocubes]

the span thing is getting more and more complicated

like. I have now a bit of evidence that my notion of generalized polycategory is "correct" (it has extremely weak requirements and very classical examples that don't work within the paradigm of... the paper I corrected i guess...) but to get it to work i had to use a notion of "unbiased lax bicategory" which iirc was implicit in a different paper on multicategories but not spelled out

and, a lot of the theory of internal profunctors seems to generalize... somewhat.

like, ok, an arrow f : A → B can be made into a span B ← A = A and a span A = A → B which are left and right adjoint in the bicategory of spans, right, and every span X ← F → Y factors as a left followed by a right adjoint. Similarly, a functor F: C → D can be made into a profunctor F*: C ⊸ D and a profunctor F°: D ⊸ C which are adjoint and if you have a profunctor C ← P → D (C and D are categories this time) then there is a category P' and functors F : P' → C and G : P' → D such that P factors as like, G*F°, or w/e

Ok, skipping over how, given arbitrary monads S and T (and some other data), i can make an (unbiased lax) bicategory of spans SX ← F → TY, which stands in relation to generalized polycategories as the category of spans stands in relation to internal categories, right

Now, given a Kleisli arrow A → TB, you can make that into a span SA ← A → TB by using the unit for S, right; in fact for... reasons... arbitrary spans SX ← F → TY will "laxly factor" as a composite of SX ← F → TF and SF ← F → TY. But arrows of those types... aren't usually adjoints. Which is unfortunate.

Now, some... incomplete work... seems to suggest that something like this should extend to factoring poly-profunctors between polycategories. Or at least I... think I can get the polycategory P' to exist; the trouble is that it factors as a pair of "kinda functors" that like, send the objects of P' to S- and T-collections of objects of C and D (respectively). (I call these "kinda" functors, because there are also "proper" functors (which incidentally can themselves be lifted to both those kinds of "kinda" functors.))

Or at least that's how it looks right now... the coherences for these things are, complicated to work through, so i haven't nailed everything down yet.

Anyways this is unfortunate because the limit-preservation properties of adjoints are like, an important part of how you characterize geometric morphisms among profunctors so... either i have to find some other trick to characterize right-torsors or "right torsors are such that tensoring with them on the right is cartesian" is the wrong definition...

And the only way to tell if that last one is the case is to keep reading the Elephant.

So maybe I'll get back on that tomorrow...

Anyways. The point is: the theory seems to... "work", like I've gotten partial results, stuff seems to lead to other stuff, but also i keep having to develop new notions and the definitions for a lot of these things are kind of complicated...

[eatsleepmath]

So I realized that part of a post I wrote this time last year

about non-adjoint equivalences between categories was wrong, because I didn’t make myself check that an “obvious-looking” unit map was actually a natural transformation.

Anyways, I spent last night re-convincing myself that non-adjoint equivalences still exist.

I ended up checking the following (identifying a group with its delooping, i.e. the one-object category carrying that group as the automorphism group of the unique object):

Any equivalence \(F : H \leftrightarrows H : G\) of a group with itself comprises two automorphisms \(F, G\), such that \(F G\) and \(G F\) are inner. The unit and counit are the group elements \(g_{\rho}\) such that \(GF(k) = g_{\rho} k g_{\rho}^{-1}\) and  \(g_{\sigma}\) such that \(FG(k) = g_{\sigma}^{-1} k g_{\sigma}\) for any \(k \in H\).

Any equivalence of \(H\) with itself where \(F\) and \(G\) are themselves also inner is an adjoint equivalence.

If \(H\) has trivial center, then any equivalence of \(H\) with itself is an adjoint equivalence.

To obtain a non-adjoint equivalence, we therefore need a group \(H\) with nontrivial center and nontrivial outer automorphisms, such that we can pick two whose products are inner.

So take \(H = K\) the Klein 4-group. This is a product of abelian groups, so abelian, so is its own center. In fact, it’s \(\mathbb{Z}/2 \mathbb{Z} \times \mathbb{Z} / 2 \mathbb{Z}\), so let \(F = G\) the automorphism which interchanges coordinates. \(FG = GF = \operatorname{id}_{K}\), which is given by conjugation by any element.

One of the triangle equalities stipulates that \(F(g_{\rho}) = g_{\sigma}^{-1}\). We can pick \(g_{\rho}\) and \(g_{\sigma}\) to break this. For example, let \(g_{\rho}\) be \((1,1)\) and let \(g_{\sigma}\) be \((0,1)\).

This is an illuminating special case of the fact that, given a non-adjoint equivalence, you can always replace its unit with another unit that makes the equivalence adjoint, without any real loss.

#math #adjoint functors #category theory

[notarealwelder]

...i'm guessing that three exercises ten pages of exercises and a couple pages of handwaving is only a surface scratch of adjoint functors

But finally I can at least see how some of them work

...is it that no-one can be told about adjoints, one can only do enough exercises to see for themself, is that why no explanation I read before made sense

(beside the formal definition, which was definitive and useful for computations, but not enlightening)

[bubbloquacious]

So the fact that the underlying sets of products and coproducts of topological spaces are themselves the products and coproducts of the underlying sets is a consequence of the fact that the forgetful functor from the category of topological spaces and continuous maps to the category of sets and functions has both a left and right adjoint (namely the discrete space functor and the indiscrete space functor), so it preserves both limits and colimits (compare this with the fact that the underlying set of a direct sum of abelian groups is not generally the disjoint union of their underlying sets).

But it also goes the other way! The discrete space functor has a right adjoint (namely the forgetful functor), so any disjoint union of discrete spaces is discrete. Products of discrete spaces can be non-discrete, so it does not have a left adjoint. The indiscrete space functor has a left adjoint, so products of indiscrete spaces are indiscrete. Disjoint unions of indiscrete spaces are almost never indiscrete, so it does not have a right adjoint.

#math #topology #adventures in cat theory

[thetwomeatmeal]

Just gave my partner this cute category theory problem:

Define four functors  Sets -> Spaces -> Sets -> Spaces -> Sets such that each one is left adjoint to the one before. The second one is the forgetful functor, and the others are all things you’ve seen before.

#mathema #category theory #topology

[mathematicalmemer]

To be honest, I was mostly thinking adjoint functors for that first one. (Now, is there a mathematical object called an 'estate'?)

Honestly, I don’t know anything about adjoint functors, but I’ll take your word for it that it could have been simpler ;P

I was considering going for replacing “estate” with “a state”, but I’m thinking that might be a bit of a push aha

[alexyar]

if you have a forgetful functor but you don’t have an adjoint free functor, then did you really have a forgetful functor

#midnight research with alex

[oregontrailmixtape]

2019/06/05

Reading:

Wikipedia, “Ramification group”

in the lower numbering, the \(s\)-th ramification subgroup of \(\mathrm{Gal}(\mathbb{Q}_p(\mu_{p^n})/\mathbb{Q}_p)\) is \(\mathrm{Gal}(\mathbb{Q}_p(\mu_{p^n})/\mathbb{Q}_p(\mu_{p^e}))\), where \(e=\lfloor\log_ps\rfloor+1\)

the lower numbering is compatible with subgroups, while the upper numbering is compatible with quotient groups and subgroups 

Notes:

using the Grothendieck–Lefschetz trace formula for \(\mathrm{Bun}_{\mathrm{SL}_n}\), one can show that \(\lvert\zeta_{X_{\mathbb{F}_{q^n}}}(-m)\rvert\sim(q^{3(2m+1)(g-1)})^n\) as a function of \(n\), where \(X\) is a geometrically connected smooth proper curve over \(\mathbb{F}_q\) of genus \(g\), and \(\zeta\) denotes the Hasse–Weil zeta function

by the adjoint functor theorem, the functor \(M\mapsto\mathbb{Z}[M]\) from absorbing monoids to \(\Lambda\)-rings has a right adjoint \(R\). by comparing with the forgetful-Witt adjunction to rings, we see \(R\) takes \(W(A)\) to the absorbing monoid underlying \(A\) for any ring \(A\)