European Mathematical Society - 18 Category theory, homological algebra
https://euro-math-soc.eu/msc/18-category-theory-homological-algebra
enGreat Circle of Mysteries
https://euro-math-soc.eu/review/great-circle-mysteries
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The role of mathematics in nature and why in sciences it is so effective in catching the laws that govern the phenomena that we observe, has been the subject of speculations and conjectures throughout human history. As more recently we were able to demystify some of the elementary building blocks that make up life and we also got some insight into the processes by which our brain functions, some circle is being closed. Indeed, it is with the help of mathematics that scientists were able to unravel the mysteries of life, and to model our brain while on the other hand mathematics is an abstract construction of the brain and the brain/mind defines the identity of a living organism. This vague circularity starts shimmering faintly through but it is still largely mystical and far from being understood.</p>
<p>
In 2011 the <em>Fondation Cartier pour l'art contemporain</em> organized an exhibition <em>A Beautiful Elsewhere</em> in Paris. Part of the exposition was a <em>Library of Mysteries</em>. This work was realized by David Lynch in collaboration with rock icon Patti Smith and the geometer Misha Gromov. Using quotes from great scientists, Lynch visualized the mysteries of time, space, matter, life, knowledge, and mathematics, presented subsequently as</p>
<p>
</p>
<ul>
<li>
the mystery of physical laws</li>
<li>
the mystery of life</li>
<li>
the mystery of the mind</li>
<li>
the mysterey of mathematics</li>
</ul>
<p>
</p>
<p>
An excellent report on this exhibition by Michael Harris was published in the <a href="http://www.ams.org/notices/201206/rtx120600822p.pdf" target="_blank">Notices of the AMS 2012 vol 59, no.6, p.822-826</a>.</p>
<p>
</p>
<p>
Gromov was asked to prepare this book as a "naive mathematician's version" of what Lynch had done in Paris, and this is how this book came about. If you know some of the films by David Lynch in which he creates his unearthly mysteries, then you know that they can take some shocking bends that lift the spectator out of reality wondering what had just happened. A sense of humour is not completely absent. It seems like an impossible task to translate this atmosphere into a book format. I was not able to attend the 2011 exhibition in Paris, but I know some of the work of David Lynch and with Harris' detailed report, I can imagine what it must have been like, and I assume that Gromov succeeds well in keeping the original ideas. After all he knows them very well because of his involvement in the 2011 project. This being said, you may not expect to read a linearly structured straightforward book. It uses different colours for (parts of) sentences and different fonts to give a precise meaning to words. Some would classify it as a painting with words and ideas or even poetry. It is full of quotes and ideas and it is seasoned with this this particular slight humorous undertone. Every section, and almost every sentence is an invitation to contemplate about its deeper meaning and the consequences. Some explanation is required, and a lot of these extras are included, not only in the text but also with several footnotes that can be found on each and every page. There are many illustrations, most are borrowed from the web, but their role is more to "give some air" to the text and make it lighter to work your way through. The hope is that the reader will play with the ideas and opinions and whatever is in between those two, so that he or she will experience the <em>Beautiful Elsewhere</em> called Mathematics.</p>
<p>
It is difficult to summarize the content because it is multi-branched like a fractal tree that floats on a stream of conciousness. A first part of some seventy pages is mainly a discussion of quotes that reflect ideas of scientists of all times who gave an opinion on science in general, on numbers, laws (physics and others), truth, life, evolution, the brain, and the mind. All these are elements in the chain that eventually will form the circle. The conclusion is that mysteries remain. We do not know much about how space/time/matter/energy is transformed into life/brain, how the latter brings about what we can classify as mind/thought. It seems that we cannot conceive these relations but by using mathematics. This closes a circle because we therefore need to understand how mathematics can be the result of this brain/mind/thought complex and mathematics is precisely the instrument that tells us something about space/time/matter/energy. Only in this last relation we know something as shown by the results of physicists. The mathematics(?) to describe the other connections/transformations/mechanisms are still unknown. This book wants to be a first attempt to throw a hook at these missing mathematics.</p>
<p>
The second part is called <em>Memorandum ergo</em> that one wants to complete spontaneously with the missing <em>sum</em>. The ideas/opinions/conjectures in the remaining 130 pages are analysing the finer structure of the components in the volatile circle of mysteries sketched above. For example, there is definitely a difference between brain and mind. The term "ergo" appears for the first time in the conjecture that there is something like an ergo-brain that is not accessible by introspection and that is responsible for unconscious thoughts. It contains structural patterns that we can recognize for example in natural language. The idea is that this ergo-brain is an instance of a wider ergo-system that hopefully can be analysed using mathematics and that eventually will also shed light on mathematics itself. This is the ergo-project and it requires to investigate all the components of this very complex system. The ergo-brain is alert to the unexpected and is bored by the ordinary, repetitive impulses. It is alert to the signals that are "interesting", not the ones that are "obvious" or "logical". It is responsible for a child learning a language or to read and write, or even walk. How do we learn things? It is not sufficient to understand the electrochemical system at the level of brain cells to explain how we attach a meaning to signals that we receive when seeing or hearing something. A strong ergo-brain is probably also responsible for children being gifted for mathematics or music or chess. It is responsible for goal-free learning, meaning that it is <em>not</em> the result of evolution which defines behaviour as maximizing the chance of survival. Evolution has a big hand in forming our ego-mind. The ego-mind is rational and intelligent. It plays by the rules and has common sense, while the ergo-brain wants to be free and wanders around always looking for the new and interesting. A cave-man with a super-ergo-brain would probably not survive, but it made Ramanujan fill up his notebooks with remarkable formulas. The ergo-system is responsible for our agility with our language, for finding pleasure in playful and "useless" activity like solving sudokus, for getting bright ideas, for progress in science and mathematics. The problem is that the ego-mind has no access to the ergo-system. Direct observation is impossible. Moreover it is not logical in the usual sense but requires some ergo-logic to deal with it.</p>
<p>
All the elements that play a role in the whole process are analysed in the book: how external signals arrive in the brain, how language has to be analysed, how do we recognize structure, etc. This allows to formulate some principles (16 rules of the ergo-learner) of how we learn through the ergo-system. The trailing part of the book is more technical. It describes in terms of category theory how the ergo-system can analyse language and give meaning to words and sentences. More generally it has to recognize structure and units, classify them through partitioning and clustering and identify connections and relations between units. This is only a first attempt to formalise how an ergo-system can make sense of a text being read or being heard. Although formal in a framework of categories and functors, the description is still more qualitative than quantitative,</p>
<p>
The fact that the text is more or less written as a freewheeling stream of consciousness to, in the end, arrive at some result that is still somewhat fuzzy, is a perfect illustration of how our ergo-brain works. This is how it gives meaning to observations and thus how it is feeding the knowledge of our ego-mind. Reading the book is a strange experience that will certainly keep your ergo-brain on alert since what you read is "interesting" and certainly not boring like a standard text is. The book unfolds its ideas following an ergo-logic and therefore should be read by an ergo-brain.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This book is a spur of the exhibition <em>A Beautiful Elsewhere</em> organized in Paris 2011 where the author collaborated with the filmmaker David Lynch and rock icon Patti Smith to evoke the mysteries of life (including the brain, mind, and language), all existing in a physical world (with its time, space, matter, and energy), and how mathematics (a product of our brain) can be useful to, not only explain the physics, but also to explain how life, brain and mind can originate in this physical world. Category theory is used to make a first attempt to catch the mechanisms used by our ergo-system (an assumed autonomous system, well separated from our conscious ego-mind) can make sense of language.<br />
</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/misha-gromov" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Misha Gromov</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer-nature-birkh%C3%A4user-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer Nature / Birkhäuser</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-53048-2 (hbk), 978-3-319-53049-9 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">90.09 € (hbk); 71.39 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">208</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.springer.com/gp/book/9783319530482" title="Link to web page">https://www.springer.com/gp/book/9783319530482</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/18-category-theory-homological-algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18 Category theory, homological algebra</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/18-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18-02</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/18d35" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18D35</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/18-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18-03</a></li></ul></span>Mon, 17 Sep 2018 05:25:32 +0000Adhemar Bultheel48686 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/great-circle-mysteries#commentsCakes, Custard + Category Theory
https://euro-math-soc.eu/review/cakes-custard-category-theory
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Eugenia Cheng is a senior mathematics lecturer at Sheffield University (UK) whose domain is higher-dimensional category theory. She has gained some popularity from her YouTube videos where she mixes her love for cooking and for mathematics to show the analogy between both and to show that knowledge of one can help understanding the other. This is exactly what she also wants to achieve in this book as the subtitle promises: <em>Easy recipes for understanding complex maths</em>. She is also active as a pianist, but that is not used much in this context. This illustrates that she is a very enthusiast communicative and talkative ambassador for the popularization of mathematics. She definitely wants to convince people that it is not mathematics that is difficult, but that it is life that is complicated and mathematics is just there to simplify it and make it much easier to solve problems.</p>
<p>
In this book she engages in the task to explain what category theory is to mathematical lay people, which is certainly not an easy or obvious choice. I doubt that a mathematical illiterate after reading the book will be able to tell you what category theory really is. But they will have gotten at least a vague idea. Fortunately, Cheng starts from scratch and is meandering along many other topics along the way. In fact, there are two parts: the first explains what mathematics is about, and the second part explains what category theory is: the mathematics of mathematics. The two parts are not much different. Cheng is following the mathematical river of concepts flowing to its estuary of understanding. She also tells about the many brooks, streamlets, and bourns that feed it. The recipes that she starts each chapter with, are not really essential in my opinion. Of course cookery programs are currently very popular and it is a kind of a opening sentence to start a discussion about something that really matters. The recipes sum up the ingredients and give a brief description of the method, and you will get some ideas of how to deal with certain allergies in your cooking, but I believe you should know something about cooking if you want to really use them since not many details are given. More or less the same holds for the mathematics. The most elementary topics of mathematics are explained, but it is advisable that you know a bit of mathematics to keep apace with Cheng. You do learn that the concept of a number is not that obvious, you learn about logic, what a proof is, how one arrives at an axiomatic system by repeatedly asking `why?', you learn about complex numbers, and a group, about the unsuccessful attempts to prove the fifth axiom of Euclidean geometry, you are convinced that distance is not always the same as a Euclidean distance, and you are introduced to topology. That's a whole lot if you only have secondary school mathematics in you backpack, and certainly if it has been a while since you needed it. All this is wrapped up in much story telling featuring Fermat, Poincaré, and Riemann, and a lot of foody and cookery stuff. And this is just the mathematics part.</p>
<p>
In the category part, relations (morphisms) represented as arrows connecting objects become important. The example of genetic and mathematical family ties (e.g. the Erdős number), are examples. It is all about structures and removing as much as possible to keep the simplest skeleton. Some of the properties of the mappings are explained and simple examples are given, but a clear and strict axiomatic definition is not really given. However you learn about what it can mean to say that structures are `the same', what a monoid or a universal property is, and even what a colimit is. And again the wrapping consists of many stories e.g. about Nelson's last message to his fleet before the battle of Trafalgar, the three domes of St. Paul's Cathedral and Battenberg cakes. I find the discussion in the concluding chapter about truth most interesting. It is about different gradations or meanings of `truth' depending on (1) what we know, (2) what we understand, and (3) what we believe. The most `secure' truth is what is in the intersection of the three.</p>
<p>
The enthusiasm of Cheng is contagious, and she knows how to take the reader along on her hiking tour (not really a stroll in the park). Do not expect that after reading the book you will be ready to start reading current research in category theory. Even the reader that is a mathematician may be somewhat confused because it is too different from the top-down axiomatic and much less verbose books that he probably is more used to. But I do not think professional mathematicians are the first targets that Cheng had in mind when writing this book. Nevertheless, it is entertaining reading stuff that the professional and the non-professional will appreciate.</p>
<p>
To avoid some confusion, let me finally point out that this book is published in UK by Profile Books, but that the same book is available in the US under a different title <em>How to bake pi: an edible exploration of the mathematics of mathematics</em> published by Basic Books.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a brave attempt to show us that mathematics is there to make our lives easy and not the other way around. Cheng does not use applied mathematics to convince the reader, but instead explains the layman what category theory, her own research field, is about, and how it simplifies structures to their bare minimum, so that the proof of a certain property still holds.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/eugenia-cheng" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Eugenia Cheng</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/profile-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">profile books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-781-25287-1 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 12.99</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">302</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Algebra</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.profilebooks.com/isbn/9781781252871/" title="Link to web page">http://www.profilebooks.com/isbn/9781781252871/</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/18-category-theory-homological-algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18 Category theory, homological algebra</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/18-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li></ul></span>Sat, 20 Jun 2015 07:33:51 +0000Adhemar Bultheel46268 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/cakes-custard-category-theory#commentsCategory Theory
https://euro-math-soc.eu/review/category-theory
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book is intended as a text book on category theory not only for students of mathematics but also, as the author says in the preface, "for researchers and students in computer science, logic, cognitive sciences, philosophy and students in other fields that now make use of it". Very few mathematical prerequisites are expected of the reader. Illustrative examples are concentrated on various aspects of posets and monoids (e.g. the poset as a category on one side and the category of posets and their monotone maps on the other side, and analogously for monoids). No example is given from topology (even in the paragraph "Stone duality" no topology is mentioned).<br />
The author has been giving courses on category theory at Carnegie Mellon University over the last ten years. The lecture course, based on the material in this book, consists of two 90 minute lectures every week for fifteen weeks. The author himself says that “the selection of material was easy. There is a standard core that must be included: categories, functors, natural transformations, equivalence, limits and colimits, functor categories, representables, Yoneda's lemma, adjoints, and monads. That nearly fills a course. The only 'optional' topic included here is Cartesian closed categories and the lambda-calculus, which is a must for computer scientists, logicians and linguists.” The book is written with the pedagogical mastership of a skilled teacher trying to help the reader as much as possible. This excellent textbook can be recommended to everybody who would like to learn the basis of category theory.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">vtr</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/s-awodey" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">s. awodey</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/clarendon-press-oxford-oxford-logic-guides-49" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">clarendon press, oxford: oxford logic guides 49</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">0-19-856861-4</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 65</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/18-category-theory-homological-algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18 Category theory, homological algebra</a></li></ul></span>Sat, 22 Oct 2011 17:03:18 +0000Anonymous39978 at https://euro-math-soc.euRational Representations, The Steenrod Algebra and Functor Homology
https://euro-math-soc.eu/review/rational-representations-steenrod-algebra-and-functor-homology
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>These notes represent a sequel to a series of lectures delivered in Nantes, December 12-15, 2001 for Société Mathématique de France’s “État de la Recherche” session. They contain the following five articles: T. Pirashvili: Introduction to functor homology; E. M. Friedlander: Lectures on the cohomology of finite group schemes; L. Schwartz: Algèbre de Steenrod, modules instables et foncteur polynomiaux; L. Schwartz: L’algèbre de Steenrod et topologie and V. Franjou & T. Pirashvili: Stable K-theory is bifunctor homology (after A. Scorichenko). The introduction (written by V. Franjou) together with the first paper by T. Pirashvili introduce a reader into the domain of problems under consideration and make him/her familiar with necessary notions. Using polynomial functors introduced in the first article, E. M. Friedlander investigates in his article the cohomology of finite group schemes. The main result states that the cohomology of a finite group scheme is finitely generated. The first article by L. Schwartz presents a completely algebraic description of the Steenrod algebra (over any prime) and deals with unstable modules over this algebra. Again the polynomial functors play an important role here. The second article by L. Schwartz is devoted to the role of the Steenrod algebra in the topological framework. It is only a short note. The last article by V. Franjou & T. Pirashvili brings a result due to A. Scorichenko (with a proof) which shows that stable K-theory is functor homology. The whole book represents a nice introduction to the circle of problems described above. The articles contain the classical results as well as the most recent ones. They are all very well written, and I think that everybody who desires to understand them and is ready to devote a necessary effort, will finally understand them. It is a relatively short but excellent book.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jiva</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/v-franjou" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">v. franjou</a></li><li class="vocabulary-links field-item odd"><a href="/author/e-m-friedlander" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">e. m. friedlander</a></li><li class="vocabulary-links field-item even"><a href="/author/t-pirashvili" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">t. pirashvili</a></li><li class="vocabulary-links field-item odd"><a href="/author/l-schwartz" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">l. schwartz</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/soci%C3%A9t%C3%A9-math%C3%A9matique-de-france-paris-panoramas-et-syntheses-no-16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">société mathématique de france, paris: panoramas et syntheses, no. 16</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2003</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">2-85629-159-7</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">EUR 25</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/18-category-theory-homological-algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18 Category theory, homological algebra</a></li></ul></span>Wed, 28 Sep 2011 14:10:54 +0000Anonymous39712 at https://euro-math-soc.euVirtual Topology and Functor Geometry
https://euro-math-soc.eu/review/virtual-topology-and-functor-geometry
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book has a special character. Its main theme is to describe development of new branches of non-commutative geometry on a different level of realizations, ranging from areas already fully developed to many different suggestions for possible future investigations. In particular, a lot of attention in the book is concentrated on a formulation of a suitable version of non-commutative topology and sheaves in this situation. A standard version of non-commutative geometry consists of an associative algebra, which is a generalization of the commutative algebra of functions on an ordinary space. It is a pointless geometry, which makes the formulation of a topology seemingly hopeless. The author has earlier developed a version of scheme theory over a non-commutative algebra based on module theory and quasi-coherent sheaves. The language used in the book is that of category theory (summarized briefly in Chapter 1). In the book, the author discusses possibilities of extending it to a more general setting, using a non-commutative version of lattices as a tool. This is contained in Chapter 2, ending with the two representative examples of the theory (the lattice of torsion theories and the lattice of closed linear subspaces of Hilbert space). Chapter 3 includes the use of a general notion of a quotient representation for a description of a relation between non-commutative affine spaces and non-commutative projective spaces. A dynamical version of topology and sheaf theory is introduced in the last chapter. A very particular feature of the book is a formulation of numerous suggestions for research projects throughout the book. Some of them are more accessible but often they are quite advanced. On the whole, the book is very inspiring and worth reading.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">vs</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/f-van-oystaeyen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">f. van oystaeyen</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/chapman-hallcrc" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">chapman & hall/crc</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2007</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-4200-6056-0</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 99.95</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/18-category-theory-homological-algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18 Category theory, homological algebra</a></li></ul></span>Wed, 15 Jun 2011 12:44:59 +0000Anonymous39509 at https://euro-math-soc.euFoncteurs en Grassmanniennes filtration de Krull et cohomologie des foncteurs
https://euro-math-soc.eu/review/foncteurs-en-grassmanniennes-filtration-de-krull-et-cohomologie-des-foncteurs
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Let V denote a vector space over a finite field k and let Gr(V) be the Grassmannian of subspaces in V. Basic objects of study in this book are various categories of functors (called grassmannian functor categories) from the category of couples (V,W), where W belongs to Gr(V), to the category ₣ of vector spaces over k. The book contains a study of finite objects in these categories and their homological properties. General vanishing properties are proved, together with an application of grassmannian functor categories to the Krull filtration of the category of functors of the category ₣. Special attention is given to the case of the basic field k with two elements (in this case it is possible to prove Noetherian properties of studied functors).</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">vs</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/djament" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">a. djament</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/soci%C3%A9t%C3%A9-math%C3%A9matique-de-france" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">société mathématique de france</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2007</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-2-85629-248-8</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">EUR 37</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/18-category-theory-homological-algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18 Category theory, homological algebra</a></li></ul></span>Wed, 15 Jun 2011 11:49:21 +0000Anonymous39475 at https://euro-math-soc.euLa théorie de l'homotopie de Grothendieck
https://euro-math-soc.eu/review/la-th%C3%A9orie-de-lhomotopie-de-grothendieck
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The aim of the book by G. Maltsiniotis is to give an introduction to some of the key ideas of A. Grothendieck's unpublished treatise on abstract homotopy theory via category theory (“Pursuing Stacks”). The fundamental concept is that of a “test category” A, whose main property is that the localization of the presheaf category Ã with respect to weak equivalences is naturally equivalent to the usual homotopy category (a typical example is the category of simplices/cubes, for which Ã is the category of simplicial/cubical sets). Furthermore, one can consider more general homotopy categories by replacing the class of weak equivalences by an arbitrary “fundamental localize” (a class of functors that satisfies, among other things, Quillen's Theorem A). The exposition of these concepts and of their basic properties is very clear.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jnek</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/g-maltsiniotis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">g. maltsiniotis</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/soci%C3%A9t%C3%A9-math%C3%A9matique-de-france" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">société mathématique de france</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2005</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">2-85629-181-3</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">EUR 26</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/18-category-theory-homological-algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18 Category theory, homological algebra</a></li></ul></span>Wed, 08 Jun 2011 12:10:35 +0000Anonymous39410 at https://euro-math-soc.euSets for Mathematics
https://euro-math-soc.eu/review/sets-mathematics
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The book introduces set theory as an algebra of mappings. This approach translates usual notions (sums, products, axiom of choice and others) to the language of mappings. The category S of abstract sets and mappings is defined using several axioms (S is a category; S has all finite limits and colimits; for any objects X and Y in S, there is a power X to Y; representation of truth values; S is Boolean; S is two-valued; axiom of choice). The category S of abstract sets and arbitrary mappings is a topos that is two-valued with an infinite object and the axiom of choice. This abstract approach includes all known situations in one simple frame. The material presented in the book is illustrated with many useful exercises. The book is suitable for advanced undergraduates or those beginning graduate studies. It gives a well-founded basis for the study of mathematics. The book consists of 10 chapters devoted to abstract sets and mappings; sums, monomorphisms, finite inverse limits; colimits, epimorphisms and the axiom of choice; mapping sets and exponentials; consequences and uses of exponentials; power sets; variable sets and models of additional variation; together with three appendices (logic, maximal principles, definitions, etc.). Many diagrams illustrate the book. It is an excellent book for those mathematicians who wish to study foundations of mathematics in an axiomatic form based on an algebraic approach.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">ppy</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/f-w-lawvere" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">f. w. lawvere</a></li><li class="vocabulary-links field-item odd"><a href="/author/r-rosebrugh" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">r. rosebrugh</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2003</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">0-521-01060-8</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£19,95</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/18-category-theory-homological-algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18 Category theory, homological algebra</a></li></ul></span>Mon, 23 May 2011 20:55:51 +0000Anonymous39155 at https://euro-math-soc.euHomological and Homotopical Aspects of Torsion Theories
https://euro-math-soc.eu/review/homological-and-homotopical-aspects-torsion-theories
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Torsion pairs (originally called ‘torsion theories’) entered module theory in the 60s through the works of Dickson and Gabriel, and were later developed into a powerful tool, e.g. in the monographs of Stenström and Golan. In the 80s, analogues of torsion pairs, called t-structures, were used in the seminal work of Beilinson, Bernstein and Deligne on triangulated categories. At about the same time, tilting theory emerged providing important examples of torsion pairs both in module categories and, later, in bounded derived categories of modules. Yet another source of torsion pairs, this time in stable module categories, came from (co)tilting theory via covariantly and contravariantly finite subcategories, notably in the works of Auslander and Reiten. Moreover, torsion pairs in stable categories were later shown to be closely related to complete cotorsion pairs in the original Abelian categories, and the latter to closed model structures in the sense of Quillen (e.g. in the works of Hovey). Thus it has become clear that torsion pairs tie together a number of important areas of contemporary algebra, topology and geometry. </p>
<p>The Beligiannis-Reiten memoir not only provides a comprehensive treatment of these ties but also finds remarkable generalisations, clarifications and extensions of the results mentioned above. The point is that the authors work in the general setting of pretriangulated categories, which includes both the Abelian and the triangulated setting as special cases. The core of the memoir consists of proving general correspondence theorems (some of which were mentioned above). Moreover, the authors use torsion pairs to develop universal cohomology theories generalising the Tate-Vogel (co)homology theory. There are also a number of concrete applications presented, notably to Gorenstein and Cohen-Macaulay categories (generalizing the classical Gorenstein and Cohen-Macaulay rings). The memoir covers an important area of contemporary pure mathematics; it is particularly recommended to anyone interested in modern representation theory, homological algebra or algebraic topology.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jtrl</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/beligiannis-i-reiten" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">a. beligiannis. i. reiten</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/american-mathematical-society" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">american mathematical society</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2007</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-8218-3996-6 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 72</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/18-category-theory-homological-algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18 Category theory, homological algebra</a></li></ul></span>Sat, 14 May 2011 11:54:20 +0000Anonymous39055 at https://euro-math-soc.eu