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I have started on a revision of algebraic K-theory. The old version launched straight into a particular nPOV, which really just summarised the Blumberg et al paper, and did not mention any of the other ideas in the area. At present I have just put in some historical stuff, but given the importance of the subject e.g. in modern C*-algebra the page needs a lot more work.
Nice edit, Tim. By the way, I think that historical introduction should be called only historical introduction and not “Idea”. Idea is about the basic mathematical idea and not about history, so one should not trick the reader to expect what is not there. Hopefully this entry will have the true “Idea” section once, as well as separate historical and “motivation” sections.
I agree. I found the idea of algebraic K-theory is difficult to state, but the nPOV version that was there before was liable to put people off reading on. I hope to be able to write something on Volodin’s approach as that combines ideas from various other areas (e.g. in the approach via Syzygies in the work of Kapranov and Saito). That would go in a different entry.
Thanks for the historic remarks.
But can we try to find agreement on the following: I find it bad style to begin an entry with hostorical remarks. By their very nature, they tend to be long winded and arrive at the essential points only slowly.
I think an entry should start with a crisp statement of the idea. That’s the first thing one wants to know. Once one has this and is hooked, one will have the leisure to be educated about the history of the ideas.
It’s like meeting a stranger on the street. You’ll be upset if he stops you and immediately starts telling you about his family history. In fact, you will walk away and decide the guy is nuts.
On the other hand, if that same strager meets you and first helps you with some information that you were lacking, you’ll be more inclined to afterwards meet him over a beer in a pub and then hear his family history with interest.
Urs, it’s not completely obvious to me that it’s bad style to begin an entry with historical remarks. Generally speaking, historical remarks can be woven seamlessly into good exposition; consider for example the book review section of the Bulletin of the AMS, where authors are encouraged to give an account of an area for general readers, which may involve some history. It’s mainly about giving the reader hooks that he can relate his prior knowledge to.
I’m not saying we should always begin nLab entries in such fashion, and I’m certainly not saying that the Idea section is where to place them. I will say that I found Tim’s exposition gentle, congenial, and useful in its way. (And certainly not rambling and irrelevant like the nut on the street!)
The crisp statement that was there earlier is, in my humble opinion, a bit too hardcore for all but the most devoted -category theorists. If a person stumbles onto the nLab in an effort to find out what algebraic K-theory is about, he will not be enlightened by the crisp statement
From the nPOV, algebraic K-theory is an an (,1)-functor that sends stable (,1)-categories to spectra – their K-theory spectra.
I’m all for the nPOV, but I think one could lead up to that rather more gently. Doesn’t have to be through historical remarks, but it should give the Idea in a somewhat more accessible manner to start, and then lead up to the high-level description. That’s how I’d like to see more nLab articles go.
Ideally,, if we have distinct sections for idea, for motivation, for historical development, a reader chooses his own order and way through, As long as the idea is not replaced by history nor history replaced by contemporary viewpoint, people will find both useful. I have tried to do similar contemporary viewpoint in noncommutative algebraic geometry but Lieven liked more to dwell on historical papers in ring theoretic literature. So now we have a bit of both and still under development. It is wrong to discourage any of the two as both will be useful and all is under construction.
I do not mind if eventually the bit of history is put further on or abbreviated with a link to a new page in which something like that appears. However we may have to be a bit careful with the Idea sections on quite a few of the entries. An idea section should give an idea of what the entry discusses, otherwise it would not be worthy of the title Idea. It is more or less an introduction to the subject of the entry. Sometimes this should be brief and nPOVish, but for the more general wide topics such as algebraic K-theory, noncommutative algebraic geometry etc. that may be inappropriate. The ideas section then might better be an introduction to the main idea of the subject. (As there are signposts in the toc, an unwary reader who does not want the history can skip to later sections.)
One problem I have is that I cannot give a short description of what the idea of algebraic K-theory is. It is not, as the Wikipedia article say, a branch of homological algebra. I do think that a statement, however true, that algebraic K-theory ’ is an ( ∞,1)-functor that sends stable ( ∞,1)-categories to spectra – their K-theory spectra,’ gives any idea of what the subject matter of the subject deals with.
I looked at the little book ‘An introduction to Algebraic K-theory’ by John Silvester as I did not have any of the classic texts. It mentioned the Grothendieck group, the Whitehead group in simple homotopy, the Steinberg group, and Milnor’s K-theory. I am preparing notes for a mixed audience on rewriting theory, and the presentation of the Steinberg group is good as it has Stasheff polytopes amongst the syzygies. This relates to very interesting unsolved problems of Kapranov and Saito and to Volodin’s version of higher K-theory. Apart from the Grothendieck group none of this was handled by the direct statement in the old entry, yet Silvester defines K_1 as K_0 of the loops on the category of free modules (approximately) so is an early sign of a different nPOV. The links with rewriting and simple homotopy raise some interesting challenges for homotopy type theory (are homotopy equivalences able to be ’constructed’?), so again an nPOV link, and the higher syzygies of the Steinberg group are also nPOVish. There is not one nPOV involved but many, and hopefully they will interact. (I think this is the combinatorial homotopy versus model category, bottom up versus top down, false dichotomy in play here. It is a useful ’tension’ to work with.)
I do not see, at the moment, how to put these points into the entry without some historical perspective. As I said if we can give a brief summary of the history with a fuller separate ’algebraic K-theory: a historical perspective’ page as well that would be perhaps better, and certainly when I have filled in more that could/should be done. I suggest that we continue the development with that aim in mind, with some brief timeline to replace the bit I have just typed with links to other entries.
(Later: I have tried out an edit along the lines I suggested above. Please give me your reactions …. I have no doubt that you will! :-) )
Question for everyone but Zoran in particular: Should the algebraic K-theory entry mention the Operator algebra and C*-algebra versions or should they be separated out because of the different properties involved?
We need references somewhere to classics like here:
but it is not clear to me where that material should go.
They should, at least that is the way how the operator algebraists classify it (as the algebraic K-theory of operator algebras, not topological). On the other hand, there are now first versions of the model categories of operator algebras available, what will make many things cleaner in future.
now first versions of the model categories of operator algebras available, what will make many things cleaner in future.
Provided of course nothing untoward happens. There may be some divergence between the different approaches and that might be ’fun’! I will add in a link to algebraic K-theory of operator algebras and then we can build that (once it exists!)
I agree that
algebraic K-theory is an (∞,1)-functor that sends stable (∞,1)-categories to spectra
is unhelpful, but only because it says nothing about what characteristics that functor has. It would equally be true about the functor which sends every stable (∞,1)-category to the sphere spectrum, say. But if we said
algebraic K-theory is, roughly, the universal finitary Morita invariant of stable (∞,1)-categories under which all exact sequences of stable (∞,1)-categories split
then I think it would be helpful. Perhaps we should soften it somewhat for readers who are scared by (∞,1)-categories, but I have never heard a non-nPOV “Idea” of algebraic K-theory that I would find worthy of the name — it’s always “you do this funny construction”. I agree that history is not necessarily an idea.
I Alexander Rosenberg’s approach, the algebraic K-theory is defined for what he calls right exact categories (roughly, categories equipped with a precanonical singleton pretopology), what is one sided weakening of Quillen exact categories, and it is not a funny construction but a universal delta-functor like in Grothendieck. Namely, the category of all right exact categories has a dual, left ’exact’ strcuture in his sense. This class of categories is less symmetric than the stable infinity setup suggests.
@Mike: I think that the point of an Idea section is to help the non-initiate into the subject, but even to me ’stable (∞,1)-categories’ does not tell me anything really useful about the intuitions behind algebraic K-theory. It is again the top down view whilst for many the bottom up approach may give them a handle to begin the process of understanding what is going on. By this means they get to a point where the top down version becomes useful as well, and progress is often via the combination of the two ways.
What might be well worth doing is to write a bridging section, showing how to gradually change the perspective from the traditional approach to the nPOV one. That way some of the techniques from that traditional approach, which allows for detailed calculations and applications might provide insights into the newer approach.
The historical perspective is in a different entry now.
Even though we’re still discussing how to arrange the exposition (so that it can be read fruitfully by initiates and non-initiates alike), I liked what Mike wrote in #11 to the point that I incorporated it into the text.
That was a good idea. Those later parts of the entry will need some work on structuring perhaps. (I note that ’A K-theory should be given’. The ’should’ is debatable!!!! ;-) Perhaps we should avoid ’should’! especially since in this case none of is an acredited K-theorist!
@Zoran: Thanks. Certainly, the Blumberg-Gepner-Tabuada paper isn’t necessarily the only abstract context in which to describe K-theory, and not all K-theory needs to fit in that framework. But it was a good example for the point I was trying to make.
@Tim: I agree that the point of an Idea section is to help the non-initiate into the subject, and in general I am all in favor of easing into the nPOV gradually. But different things are helpful for different non-initiates, and although I heard all sorts of “introductions” to algebraic K-theory over the years, it was not until Blumberg-Gepner-Tabuada that I got an Idea of what K-theory is about that I could wrap my head around. Some of us find it much easier, at least sometimes, to start with a top-down view before getting into the messy details of the bottom-up one. Cf. John Baez’ joke about category theorists being dual to ordinary people. It’s a delicate balance, and easy to go too far in that direction (as I think we sometimes do), but one of the things I like best about the nLab is that we can be up-front about the top-down view too.
@Mike: I agree. It is very difficult to get the balance right. What I am worried about is that the very usefulness of a nPOV is obscured if it is claimed to be THE nPOV. (This is my old battle of putting more up front than at present the more combinatorial and algebraic versions of the nPOV, e.g. in rewriting, explicit formulae for homotopy limits etc. and keeping in touch with the origins of ideas, rather than the sometimes over enthusiastic model category view point (for example). )
My interest in rewriting brings in the combinatorial aspect of algebraic K-theory, and what I would hope for is a bridge to help connect the top down methods with the more concrete ideas in that area.
One thing is that the linking structure of the nLab makes it possible to present both top-down and bottom up at the same time. You can flip between the two.
Ronnie and I used to discuss methodology of maths and use the idea of analogy as an important key. Looking at two adjacent areas of maths one highly developed (and perhaps nPOVised) the other still fairly traditional. Seeing the analogies between them at a fairly basic level, you get ideas for possible developments in the second area, suggestions for possible theorems and also questions that can be asked in the ’developed’ one. (Here I am thinking specifically of the Kapranov Saito ’conjectures’, in which they ask why Stasheff polytopes are occurring in the n-syzygies of the Steinberg group (and hence in the construction of one of the main models for the K-theory spectrum of a ring). The combinatorics are interesting and also suggest links with Morse theory. The technology used links up with a lot of ideas in combinatorial group theory, as well. ) This is definitely linked with the top-down nPOV version of algebraic K-theory, but without the ’foot hills’ of that theory available the higher syzygies are very difficult to interpret.
(Added slightly later: I also wanted some place to put links into for nLab entries on some of this Kapranov Saito stuff. One point is that the structure of the classifying space of alg. K-theory (with respect to the Volodin model for K-theory) is as a homotopy colimit with the indexing categories explicitly known and very pretty and intuitive. This uses Boardman-Vogt, -homotopy coherence, etc in a very nice way, but is dotted around the literature so it is difficult to see what is missing from a ’coherent’ treatment.)
@Tim: Are you saying that a historical perspective gives a different nPOV on K-theory? Is there something nPOV about the combinatorial or arithmetic aspects?
@Mike: I am not an expert but I am trying to ’grope’ my way towards something that is a bit as you say, though to what extent ‘different’ is a big question. My reasons for believing this are that the combinatorial group theory of the Steinberg group seems to be very nicely related to Stasheff polytopes, and this is probably due to the combinatorics of total orders. Kapranov and Saito thought that they had a specific combinatorial model of the K-theory complex of a ring. This is based on Volodin’s construction, but their approach does not use results on higher generation by subgroups involving the homotopy colimit of bar constructions. In other words, a lot of good infinity category stuff plus some combinatorics yields quite small geometric models for the BGL(A)+ space. This would give a possibility not only of interpreting classical algebraic K-theory but might just be a general enough attack to allow much more general forms of such theories to be approached using the models of -types, -syzygies, etc. (Kapraonov and Saito is fun stuff but mystifying. Good material for a postgrad to take apart and put back together again with more detail, intuition and examples.)
The rewriting idea that is elementary in linear algebra would then have an nPOV aspect from the Steinberg ‘resolution’!
Okay. Is that really an nPOV on the meaning of K-theory, or an application of the nPOV on K-theory to computations?
There you are hitting the problem of the meaning of ’meaning’!
I think here it is both meaning and computational application. If you look at interpreting AKT in the case of rings, then it is a question of rewriting, hence looking for relations between the ’actions’ of rewriting equations. The higher szyzygies are the relations between relations …, but when you look at many similar situations you find the geometry of the group is reflected in the combinatorics of the syzygies. The geometry of the group is mirrored by the higher cells of a presentation of the -resolution starting with a well chosen presentation. It is as if the geometry emerges from the presentation via the resolution. There is a nice article by Loday on this sort of thing, following on from K & S to some extent but not dealing with AKT.
That sounds interesting, although I don’t really understand it. I know how to think of other sorts of (co)homology in terms of relations-between-relations; but what exactly is being “rewritten” in the case of algebraic K-theory?
I am being terribly naive in this, on the basis that my intuition seems to work and each time I follow up something I find that someone (usually Suslin) has been there 20 years ago! Think undergrad linear algebra. Sets of linear equations! You perform Gauss elimination. Oh! you rewrite the equations in another form using elementary operations and succeed in analysing them enough to decide if there is a solution or not, and finding the solutions if there are any.
The rewrites are given by elementary matrices,(which form a group, ,) and so one tries to see the relations between these matrices. There are some universal ones given by the Steinberg group relations and there may be others that depend on the particular structure of the ring being used. The Steinberg group projects onto . Milnor’s is the kernel of this epi. It therefore measures the extra relations needed to get a presentation of .
I think Volodin’s approach to higher K-theory amounts to looking for the higher relations between relations of those. (There are times when I convince myself that this is the case, and times when I convince myself that I am stupid!!! so take with possible pinches of salt.)
Hmm, okay. But Milnor K-theory is different from “ordinary” algebraic K-theory, right?
I think his is THE same but he proposed a higher form that did not have all the properties that the QUillen theory had.
Okay, right. Does your intuition about rewriting, or Volodin’s approach, extend to higher -groups, and if so which ones?
I think it was Suslin (1982-ish) who gave a proof that the Volodin definition, another of Wagoner, and two of Quillens gave the same answer. One proof seems to involve proving that Volodin’s complex is the homotopy fibre of the BGl\to BGl+ map, whilst the Volodin space seems to have nice homotopy which is very ‘rewriting friendly’. I am going to talk a bit about this aspect in Luminy when trying to introduce higher syzygies. The other key element is a very nice paper by Abels and Holz which hardly mentions K-theory but is on the interpretation of the homotopy invariants of nerves of coset covers of groups. This is very closely related.
My aim is to revisit all this and if possible do some nPOV interpretation. Another related idea is to take partial presentations of groups and eventually of theories and to adapt the methods of Abels and Holz to say what needs to be added to get more complete ones. (That is pie in the future sky but is behind this to some extent.)
Okay, I look forward to hearing what you come up with!
Some of the known stuff is being typed up for the Menagerie. I am not sure of the content yet.
I just reread the intro to Kapranov and Saito’s paper. I quote: ’the study of higher syzygies among row operations is, at least, ideally the aim of algebraic K-theory’. Interesting take on the subject with respect to the earlier discussion. :-)
I am in Luminy at the moment (excellent talks in the LI2012 meeting week 5). The cutdown version of the Menagerie that I prepared for this is available on my nLab homepage.
Mike Warren is giving an intro to HTT. Very good so far.
In case someone wants to see the schedule, here it is.
The entry algebraic K-theory was in a sad state. I have tried to bring it at least a bit more on track (it now still needs to run a bit on that track).
FIrst of all I have expanded and rewritten the Idea-section. It used to be the following, which I found was lacking mentioning of the central aspects and ended up sounding a bit like saying that its impossible to say what the idea of algebraic K-theory really is, which I don’t think is true.
This is what it used to say:
Algebraic K-theory was initially a body of theory that attempted to generalise parts of linear algebra, notably the theory of dimension of vector spaces, and determinants,to modules over arbitrary rings. It has grown into a well developed tool for studying a wide range of algebraic, geometric and even analytic situations from a variety of points of view. It is thus difficult to give a single idea of what the subject is about. It has a side that looks at the manipulation and rewriting of the elementary operations of linear algebra, but also a definite infinity category aspect. Its development uses a lot of algebraic topology, particularly homotopy theory, both stable homotopy theory and the more simplicial parts of that area, and more recently has interacted with infinity category theory in various forms.
I have now turned that into the following:
Algebraic K-theory is about natural constructions of cohomology theories/spectra from algebraic data such as commutative rings, symmetric monoidal categories and various homotopy theoretic refinements of these.
The algebraic K-theory of a commutative ring was originally defined to be the Grothendieck group of its category of projective modules. Under the relation between modules and vector bundles this is directly analogous to the basic definition of topological K-theory and hence the common term.
More generally, following the axiomatics of generalized (Eilenberg-Steenrod) cohomology any algebraic K-theory should be given by a sequence of functors from some suitable class of categories of “algebraic nature” to abelian groups, satisfying some natural conditions. Moreover, following the Brown representability theorem these groups should arise as the homotopy groups of a spectrum, the algebraic K-theory spectrum. Classical constructions producing this by combinatorial means are known as the Quillen Q-construction defined on Quillen exact categories and more generally the Waldhausen S-construction defined on Waldhausen categories.
For more on the history of the subject see (Arlettaz 04) and see at at Algebraic K-theory, a historical perspective.
There are two ways to think of the traditional algebraic K-theory of a commutative ring more conceptually: on the one hand this construction is the group completion of the direct sum symmetric monoidal-structure on the category of modules, on the other hand it is the group completion of the addition operation expressed by short exact sequences in that category. This leads to the two modern ways of expressing and viewing algebraic K-theory:
monoidal. The core of a symmetric monoidal category or more generally of a symmetric monoidal (∞,1)-category has a universal completion to an abelian ∞-group/connective spectrum optained by universally adjoining inverses to the symmetric monoidal operation. This yields the concept of algebraic K-theory of a symmetric monoidal category and more generally that of algebraic K-theory of a symmetric monoidal (∞,1)-category;
exact/stable. Analogously, inverting the addition operation expressed by the exact sequences in an abelian category or more generally in a stable (∞,1)-category yields the algebraic K-theory of a stable (∞,1)-category. Explicit ways to express this are known as the Quillen Q-construction and the Waldhausen S-construction. This turns out to be a universal construction in the context of non-commutative motives.
Here the second construction may be understood as first splitting the exact sequences and then applying the first construction to the resulting direct sum monoidal structure. Typically the first construction here contains more information but is harder to compute, and vice versa (see also MO-discussion here and here).
Both of these constructions produce a spectrum (hence representing a generalized (Eilenberg-Steenrod) cohomology theory) – called the K-theory spectrum – and the algebraic K-theory groups are the homotopy groups of that spectrum.
The classical case of the algebraic K-theory of a commutative ring is a special case of this general concept of algebraic K-theory by either forming the symmetric monoidal category and applying the abelian ∞-group-completion to that, or else forming the stable (∞,1)-category of chain complexes of -modules and applyong the Waldhausen S-construction to that. In both cases the result is a spectrum whose degree-0 homotopy group is the ordinary algebraic K-theory of as given by the Grothendieck group and whose higher homotopy groups are its higher algebraic K-theory groups.
I have then also restructured the section outline a bit, such as to be more systematic. Most of the subsections of course eventually should be filled with (more) content.
Great! This page was really in need of revision.
Thanks for the feedback.
I have further added pointers to/brief remarks on
added a pointer to another “historical account” (added it also to the page that claims to be the dedicated history page):
31 Urs, good and insightful work, though too abstract and intimidating for the starting paragraphs of the entry (people who know what stable category is have the idea certainly what algebraic K-theory is). That should be some sort of NPOV idea after the simpler down to earth idea, IMHO.
More technical remark. The idea you give is a bit too broad in a sense: cohomology for algebraic data, well say Hochschild cohomology seems to satisfy all your criteria and descriptions (except that the sample concrete means is S-construction or Q-construction). The idea from the previous one that some constructions from linear algebra are spread to general modules – number to generalized (spread-out over a space) number aka vector bundle (as Gromov put it) that is f g projective module – should also ideally not be lost from the idea section, though I do not know how to upgrade it to superabstract idea section you have now.
Zoran, the first 4 (four) paragraphs in #31 recall nothing but the classical picture.
The classical picture does not say that any cohomology theory attached to algebraic structure is an algebraic -theory.
Algebraic K-theory is about natural constructions of cohomology theories/spectra from algebraic data such as commutative rings, symmetric monoidal categories and various homotopy theoretic refinements of these.
Waldhausen construction is an advanced topic in algebraic K-theory, certainly not for an idea section. This way the Lab will die for working mathematician as the classical category community isolated itself from working mathematicians.
Why are you against the previous idea “constructions from linear algebra are spread to general modules” or so ? I think it is a greater and more central idea to K-theory than any detail of stable infinity formalism in this or that disguise.
I am not sure what you mean to add, so I cannot be against it. The fact that algebraic K-theory generalizes the construction of topological K-theory – which I suppose is what you mean by “generalizing linear algebra to modules” – is mentioned. But please feel free to add what seems missing.
I felt the Idea section which I had removed gave no idea of what algebraic K-theory is. But if you feel mine does neither, than clearly some more discussion is needed (in the entry I mean, not here in the forum).
I felt the Idea section which I had removed gave no idea of what algebraic K-theory is.
Never mind. For large part this is true that it was not that much of content, but the part which I repeated twice above made much essential sense to me. And not necessarily in your rephrase in 38 – it was not the topological K-theory which came first historically but the algebraic (not higher algebraic) – for coherent sheaves and for locally constant sheaves, in the work of Grothendieck. The fact that a number generalizes to a vector bundle, and that one generalizes further Morita invariant constructions like determinants, via K-theoretic construction, and one can go directly from the point case to the module case without a need of intermediary of topological K-theory to go on with this idea. Of course, I agree, the insight that higher algebraic theory is there is later was even historically based on wider experience of algebraic topology and homological algebra.
Side remark. I also have in mind that there are other sisters like Milnor’s K-theory, negative K-theory, and now emerging and richer Leibniz K-theory, predicted by Loday. I am not sure how to add those in unified perspective. It is also indicative that the cyclic homology of algebras, in the early work of Tsygan and Feigin has been called “additive K-theory” and it itself has few variants.
@Zoran
any good references for Leibniz K-theory?
Look into Leibniz algebra, references of Loday, in particular,
and Covez (I think he has more on his homepage) and maybe some other. The picture is now emerging but it is only very partially written.
Thanks, Zoran.
Zoran, I have added the following paragraph to the Idea-section at algebraic K-theory (I am assuming that this is what the allusion to determinants above is about)
There are canonical maps from the 0th algebraic K-theory of a ring to its Picard group and from the first algebraic K-theory group of to its group of units which are given in components by the determinant functor. This fact is sometimes used to motivate algebraic K-theory as a “generalization of linear algebra” (see e.g. this MO discussion).
added a pointer to Saito 14
added a paragraph with the actual statement of the construction as the K-theory of algebraic vector bundles:
Regard the stack of algebraic vector bundles on the gros Zariski site as taking values in symmetric monoidal (∞,1)-categories, via the direct sum of vector bundles. Then applying the K-theory of a symmetric monoidal (∞,1)-category-construction to this yields a sheaf of spectra
This is the higher algebraic K-theory spectrum, in that for a scheme, then the homotopy groups of are the higher algebraic K-theory groups of the scheme .
In this form this is due to (Bunke-Tamme 12, section 3.3). This is based on (Weibel, IV.4.8, IV.4.11.1) which characterizes algebraic K-theory via group completion of projective modules, and on (Thomason-Trobaugh 90) which shows that the resulting K-theory presheaf satisfies descent on the Zariski site.
For more on this see at differential algebraic K-theory .
turned the remarks on descent of into a standalone section algebraic K-theory – Descent and added all the references that Adeel collected in his MO comment
Made several edits at algebraic K-theory: added some constructions of K-theory of rings and schemes, some stuff about Waldhausen (oo,1)-categories, and some minor corrections.
I have added two references on algebraic K-theory of quotient stacks by reductive algebraic groups (here).
re-organized and expanded the list of references on algebraic K-theory of ring spectra. Now it reads like this:
The algebraic K-theory of ring spectra:
{#EKMM97} Anthony Elmendorf, Igor Kriz, Michael Mandell, Peter May, chapter VI of Rings, modules and algebras in stable homotopy theory, AMS Mathematical Surveys and Monographs Volume 47 (1997) (pdf)
{#BlumbergGepnerTabuada10} Andrew Blumberg, David Gepner, Gonçalo Tabuada, Section 9.5 of: A universal characterization of higher algebraic K-theory, Geom. Topol. 17 (2013) 733-838 (arXiv:1001.2282, doi:10.2140/gt.2013.17.733)
Jacob Lurie, Algebraic K-Theory of Ring Spectra, Lecture 19 of Algebraic K-Theory and Manifold Topology, 2014 (pdf)
The algebraic K-theory of specifically of suspension spectra of loop spaces (Waldhausen’s A-theory) is originally due to
added pointer to
(and also at derived category)
pointer
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