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Longer titles found: Stable homotopy theory (view), Rational homotopy theory (view), A¹ homotopy theory (view), Weak equivalence (homotopy theory) (view), Chromatic homotopy theory (view), Coherency (homotopy theory) (view), Simple homotopy theory (view), Equivariant stable homotopy theory (view)

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recent reawakening as methods have been discovered to apply motivic homotopy theory to the problems. The problem of Apollonius is an early example of an

In mathematics, the Andreotti–Frankel theorem, introduced by Aldo Andreotti and Theodore Frankel (1959), states that if V {\displaystyle V} is a smooth

American mathematician at Johns Hopkins University. Morava specialized in homotopy theory and is credited for Morava K-theory, a class of cohomology theories

and provides natural settings for intersection theory (or motivic homotopy theory) of singular algebraic varieties and cotangent complexes in deformation

contributions to algebraic topology, particularly equivariant stable homotopy theory, algebraic K-theory, and applied algebraic topology". In 2008, Carlsson

group is important because of its relation to Galois theory, étale homotopy theory, and the ring of adeles. In addition, it provides a basic tractable

localization and completion in homotopy theory, and in the study of homotopy properties of schemes (e.g. étale homotopy theory). Edwards, David A.; Hastings

professor at Amherst College. He studies, among other things, étale homotopy theory, elliptic surfaces, computer-based algebraic geometry (such as Gröbner

pro-simplicial set associated to the étale topos of a scheme is called étale homotopy theory. In good cases (if the scheme is Noetherian and geometrically unibranch)

in 1952, with a thesis titled Abstract Function Spaces and Their Homotopy Theory. All of these degrees were from UCLA where Jackson remained in the

Technology, under the supervision of Daniel Quillen, with thesis Abstract Homotopy Theory and Generalized Sheaf Cohomology. Hired by Cornell University in 1971

and Russian mathematician known for his work on algebraic topology, homotopy theory, and mathematical physics. Buchstaber's first research work was in

Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan. Real homotopy theory of Kähler manifolds. Invent. Math. 29 (1975), no. 3, 245–274. MR 0382702

at an early stage of their career. The prize is named in memory of homotopy theory pioneer J. H. C. Whitehead. More specifically, people being considered

mathematician, known for his work in algebraic topology, especially homotopy theory. Ganea left Communist Romania to settle in the United States in the

simplicial commutative ring from the point of view of homotopy theory? W. G. Dwyer and D. M. Kan. “Homotopy theory and simplicial groupoids”. In: Nederl. Akad.

Nice, and in 1995 at the University of Lille. His research deals with homotopy theory and homology of loop spaces with applications in geometry. He wrote

mathematics at the University of Manchester. Eccles specialises in homotopy theory and its applications to different topology. Eccles taught a wide variety

Pierre; Griffiths, Phillip; Morgan, John; Sullivan, Dennis (1975). "Real homotopy theory of Kähler manifolds". Inventiones Mathematicae. 29 (3): 245–274. Bibcode:1975InMat

of Edgar H. Brown, Jr. His thesis was titled On Odd Primary Stable Homotopy Theory. He did his postdoctoral training as an L.E. Dickson Instructor at

(∞, 1)-categories. Rezk, Charles (2001), "A model for the homotopy theory of homotopy theory", Transactions of the American Mathematical Society, 353 (3):

0043. S2CID 168002. Rezk, Charles (2001). "A model for the homotopy theory of homotopy theory". Transactions of the American Mathematical Society. 353 (3):

rest of his career. Cohen did influential work in several areas of homotopy theory. His thesis concerned the topology of configuration spaces, a topic

no. 3, 623–637. 2001 Michael J. Hopkins for: Nilpotence and stable homotopy theory. I. Ann. of Math. (2) 128 (1988), no. 2, 207–241. (with Ethan Devinatz

This is a timeline of bordism, a topological theory based on the concept of the boundary of a manifold. For context see timeline of manifolds. Jean Dieudonné

mathematician specializing in topology, including algebraic topology, homotopy theory, and geometric topology. She is a professor of mathematics at the University

construction the converse is also true. This can be formulated in terms of homotopy theory. Recall B G {\displaystyle BG} the classifying space for stable spherical

Professor at the University of Copenhagen. His research interests include homotopy theory, algebraic K-theory, and arithmetic algebraic geometry. Hesselholt

Technology, under the supervision of Michael Artin, (Fibrations in Étale Homotopy Theory). He was a postdoctoral instructor at Princeton University: a lecturer

rings, intersection theory, (equivariant) algebraic K-theory, motivic homotopy theory, and existence of vector bundles on singular algebraic surfaces. Around

regarding general localization theory, as they occur in the context of homotopy theory. He has also conducted research in the field of cohomology of groups

generalization of the connectivity part of the theorem from traditional homotopy theory to any other infinity-topos with an infinity-site of definition was

In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time,

Hopkins, Michael J.; Smith, Jeffrey H. (1988). "Nilpotence and stable homotopy theory. I". Annals of Mathematics. 128 (2): 207–241. doi:10.2307/1971440.

James E. Humphreys (1997, ISBN 978-0-387-90053-7) A Course in Simple-Homotopy Theory, Marshall. M. Cohen, (1973, ISBN 0-387-90056-X) Functions of One Complex

{\displaystyle {\binom {t+k}{k}}} . Johnson, Keith (2014), "Stable homotopy theory, formal group laws, and integer-valued polynomials", in Fontana, Marco;

G have been obtained by Asok, Hoyois & Wendt (2018) by means of A1 homotopy theory. Bass, H. (1973), Some problems in 'classical' algebraic K-theory.

motivic homotopy theory as well as finding a way to build algebraic varieties with a specified list of properties. From the motivic homotopy theory the proof

the Steinberg relation. In motivic cohomology, specifically motivic homotopy theory, there is a sheaf K n , A {\displaystyle K_{n,A}} representing a generalization

Riehl (2021), for "deep and foundational work in category theory and homotopy theory". Kristen Hendricks (2023), for "highly influential work on equivariant

group of integers. The homotopy hypothesis, a well-known conjecture in homotopy theory formulated by Alexander Grothendieck, states that a suitable generalization

an arbitrary integer. This was a major addendum to the existing n-Homotopy theory. After graduating from Tbilisi State University, he became a research

and pointwise multiplication are homotopic. Thus, in terms of the homotopy theory of ΩG, these maps are interchangeable. Loop groups were used to explain

arXiv:math/0608040. ISBN 978-0-691-14048-3. Simpson, Carlos (2012). Homotopy theory of higher categories. New Mathematical Monographs. Vol. 19. Cambridge:

Fellowship and in 1972 the Leroy P. Steele Prize for his paper Simplicial homotopy theory. Curtis died in Seattle, Washington on April 2, 2024, at the age of

Springer Robinson, Alan (1983), "Derived tensor products in stable homotopy theory", Topology, 22 (1): 1–18, doi:10.1016/0040-9383(83)90042-3, MR 0682056

Poincaré duality Poitou–Tate duality Pontryagin duality S-duality (homotopy theory) Schur–Weyl duality Series-parallel duality Serre duality Spanier–Whitehead

Ravenel, Douglas C. (1992), Nilpotence and Periodicity in Stable Homotopy Theory, Annals of mathematics studies, vol. 128, Princeton University Press

New York: Springer. ISBN 9780387215976. OCLC 55897585. Cohen, Marshall M. (1973) A Course in Simple-Homotopy Theory, Springer-Verlag New York v t e

1/30144. Mathew, Akhil; Meier, Lennart (2013). "Affineness and chromatic homotopy theory". Journal of Topology. 8 (2): 476–528. arXiv:1311.0514. doi:10.1112/jtopol/jtv005

algebraic topology model of a differential graded algebra used in rational homotopy theory. Neron minimal model, algebraic geometry model for abelian varieties

ISBN 0-521-81496-0 (Page 29) Goerss, P. G.; Jardine, J. F. (1999), Simplicial Homotopy Theory, Progress in Mathematics, vol. 174, Basel, Boston, Berlin: Birkhäuser

(as of 2013) (Accessed 1 April 2020) Kerodon a Stacks project inspired online textbook on categorical homotopy theory maintained by Jacob Lurie v t e

Hilton, Peter (1988), "A Brief, Subjective History of Homology and Homotopy Theory in This Century", Mathematics Magazine, 60 (5), Mathematical Association

Differential graded Lie algebra Quillen, Daniel (September 1969). "Rational homotopy theory". Annals of Mathematics. 2. 90 (2): 205–295. doi:10.2307/1970725. JSTOR 1970725

I} is an ideal of R . {\displaystyle R.} Cofibration – Concept in homotopy theory Identity function – Function that returns its argument unchanged MacLane

homotopy equivalent but not homeomorphic. This was the origin of simple homotopy theory. The use of the term geometric topology to describe these seems to

2012: Michael J. Hopkins "For his leading role in the development of homotopy theory, which has both reinvigorated algebraic topology as a central field

one publication so far, published when he was doing his PhD. "On ex-homotopy theory and generalized homotopy products". The University of Hong Kong. 1978

Grothendieck. He also collaborated with Barry Mazur to define étale homotopy theory which has become an important tool in algebraic geometry, and applied

1155/S1073792804132017, ISSN 1073-7928, MR 2037049 Quillen, Daniel (1969), "Rational homotopy theory", Annals of Mathematics, Second Series, 90 (2): 205–295, doi:10.2307/1970725

meanings: Spectrum (topology), the fundamental object of study in stable homotopy theory Spectrum of a C*-algebra, a kind of dual object Spectrum of a ring

Strecker, theorem 12.7, page 213 Riehl, Emily (2014). Categorical Homotopy Theory. New York: Cambridge University Press. p. 32. ISBN 9781139960083. OCLC 881162803

Algebraic K {\displaystyle K} -theory of spaces, concordance, and stable homotopy theory, Algebraic topology and algebraic K {\displaystyle K} -theory (Princeton

S2CID 242258794 Puppe, Dieter (1962), "On the formal structure of stable homotopy theory", Colloquium on algebraic topology, Aarhus Universitet Matematisk Institute

MR 1268594 Röndigs, Oliver; Østvær, Paul Arne (2008), "Rigidity in motivic homotopy theory", Mathematische Annalen, 341 (3): 651–675, doi:10.1007/s00208-008-0208-5

invariant problem", in Mahowald, Mark E.; Priddy, Stewart (eds.), Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997)

Grothendieck-Teichmüller Groups: Part 2: The Applications of (Rational) Homotopy Theory Methods, Mathematical Surveys and Monographs, vol. 217, American Mathematical

information and provide a diagram calculus to use in derived categories and homotopy theory. Formally, a prederivator D {\displaystyle \mathbb {D} } is a 2-functor

Proposition 5.7. Frankland, Martin (February 4, 2013). "Math 527 - Homotopy Theory – Compactly generated spaces" (PDF). Steenrod, N. E. (1 May 1967).

mathematical physics. In the 1960s he wrote fundamental papers on higher homotopy theory and homotopy algebras. He introduced A ∞ {\displaystyle A_{\infty }}

higher-dimensional groupoids, not available in group theory, with applications in homotopy theory and in group cohomology. Brown, R.; Higgins, P.J. (1981). "On the algebra

nerve of a category. Goerss, Paul; Jardine, John (2009), Simplicial Homotopy Theory, Progress in Mathematics, vol. 174, Birkhäuser Basel, ISBN 978-3-7643-6064-1

E_n-ring What is a simplicial commutative ring from the point of view of homotopy theory? What facts in commutative algebra fail miserably for simplicial commutative

computation of stable stems in The Čech centennial: a Conference on Homotopy Theory, June 22–26, 1993, pp. 299–316. MR 1320997 Mark E. Mahowald, A new

Mathematics Institutions Johns Hopkins University Thesis On stable homotopy theory and some applications (1965) Doctoral advisor C. T. C. Wall Website

the path space fibration. Mapping space Martin Frankland, Math 527 - Homotopy Theory - Fiber sequences Davis & Kirk 2001, Definition 6.14. Davis & Kirk

1016/0040-9383(73)90014-1. MR 0391071. Brown, Edgar H. (1965). "Abstract homotopy theory". Transactions of the American Mathematical Society. 119 (1): 79–85

if and only if it is fibrant in the sense of so-called "holomorphic homotopy theory". Every compact smooth manifold of dimension 2n, which has only handles

Jean-Pierre Marquis (2006) "A path to Epistemology of Mathematics: Homotopy theory", pages 239 to 260 in The Architecture of Modern Mathematics, J. Ferreiros

conjecture Mathew, Akhil; Meier, Lennart (2015). "Affineness and chromatic homotopy theory". Journal of Topology. 8 (2): 476–528. arXiv:1311.0514. doi:10.1112/jtopol/jtv005

the nLab category of presheaves at the nLab Free cocompletion at the nLab Daniel Dugger, Sheaves and Homotopy Theory, the pdf file provided by nlab.

{\displaystyle T\leftarrow P\to X\to X/G} . Jardine, John F. (2015). Local homotopy theory. Springer Monographs in Mathematics. New York: Springer-Verlag. section

object at the nLab Goerss, Paul G.; Jardine, John F. (1999). Simplicial Homotopy Theory. Progress in Mathematics. Vol. 174. Basel, Boston, Berlin: Birkhäuser

Cenkl M., Miller, H. (ed.) The Cech centennial: Proc. Conference on Homotopy Theory, Contemporary Mathematics, 181, 1995, pp. 405–418 With Elliot Gootman:

Corporation, p. 37, ISBN 9780486783475. Riehl, Emily (2014), Categorical Homotopy Theory, New Mathematical Monographs, vol. 24, Cambridge University Press,

a full professor. In 2017, Wickelgren and Jesse Kass used motivic homotopy theory to count the lines on a smooth cubic surface, generalizing known results

May (b. 1939), researcher in algebraic topology, category theory, homotopy theory, and the foundational aspects of spectra Margaret Evelyn Mauch (1897–1987)

1090/pspum/062.2/1492538, MR 1492538, S2CID 2659059 Simpson, Carlos (2012). Homotopy theory of higher categories. New Mathematical Monographs. Vol. 19. Cambridge:

Robert; Lambrechts, Pascal; Turchin, Victor; Volić, Ismar (2011). "Real homotopy theory of semi-algebraic sets". Algebraic & Geometric Topology. 11 (5): 2477–2545

Ravenel, Douglas C. (1980), "Complex cobordism and its applications to homotopy theory", Proceedings of the International Congress of Mathematicians (Helsinki

Østvær, P.A.; Röndigs, Oliver; Voevodsky, Vladimir (2007), Motivic Homotopy Theory: Lectures at a Summer School in Nordfjordeid, Norway, August 2002,

ISSN 0003-486X, JSTOR 1970299 Marshall M. Cohen (1973), "A Course in Simple-Homotopy Theory", Graduate Texts in Mathematics, Graduate Texts in Mathematics, vol

Pierre; Griffiths, Phillip; Morgan, John; Sullivan, Dennis (1975), "Real homotopy theory of Kähler manifolds", Inventiones Mathematicae, 29 (3): 245–274, Bibcode:1975InMat

Society. ISBN 0-8218-2129-6. Gromov, Mikhael (1999). "Quantitative Homotopy Theory". In Rossi, Hugo (ed.). Prospects in Mathematics: Invited Talks on

open map. Baues, Hans-Joachim; Quintero, Antonio (2001). Infinite Homotopy Theory. K-Monographs in Mathematics. Vol. 6. p. 53. ISBN 9780792369820. A

Mathematical Society, 1972. 1972 Edward B. Curtis for his paper, Simplicial homotopy theory, Advances in Mathematics, volume 6 (1971), pp. 107–209. 1972 William

Junhyong Kim. Kim has made contributions to the application of arithmetic homotopy theory to the study of Diophantine problems, especially to finiteness theorems

X)=X^{H}} . Greenlees, John; May, Peter (1995). "8. Equivariant stable homotopy theory" (PDF). In James, I.M. (ed.). Handbook of algebraic topology. Elsevier

MR 1989190. Hopkins, M and Mahowald, M., From Elliptic Curves to Homotopy Theory (1998), http://www.math.purdue.edu/research/atopology/Hopkins-Mahowald/eo2homotopy

MR 0212840 Hatcher, A.E. (1978). "Concordance spaces, higher simple-homotopy theory, and applications". Algebraic and geometric topology (Proc. Sympos

abroad, Warner was now on the margins of her Oxford-era interest of homotopy theory research, much of her knowledge obsolete due to significant advancements

homeomorphism) by invariants has motivated areas of research, such as homotopy theory, homology theory, and K-theory.[citation needed] A given set may have

module. This leads to the idea of using multiple groupoid objects in homotopy theory. More generally, the Eckmann–Hilton argument is a special case of the

functor Change of rings May, J.P.; Sigurdsson, J. (2006). Parametrized Homotopy Theory. A.M.S. p. 253. ISBN 0-8218-3922-5. Bourbaki, Nicolas (1989), Elements

(1986), "Through the looking glass: a dictionary between rational homotopy theory and local algebra", in J.-E. Roos (ed.), Algebra, algebraic topology

way responsible for much of the subtlety and complication in stable homotopy theory. An important generalisation to the non-abelian case using crossed

of Deligne's Hochschild cohomology conjecture". Recent progress in homotopy theory (Baltimore, MD, 2000). Providence, RI: Amer. Math. Soc. pp. 153–193

Texts in Mathematics 139, 1993. Cohen, Marshall M., A Course in Simple-Homotopy Theory, Springer Graduate Texts in Mathematics 10, 1973. Brody, E. J. (1960)

55P15, 55Q05 18C10, 18D05, 18G55, 55P15, 55Q05]. Dimitri Ara. On the homotopy theory of Grothendieck ∞ -groupoids. J. Pure Appl. Algebra, 217(7):1237–1278

Measure Theory Terence Tao 2011 978-0-8218-6919-2 127 Modern Classical Homotopy Theory Jeffrey Strom 2011 978-0-8218-5286-6 128 Tensors: Geometry and Applications

Cisinski, Denis-Charles; Khan, Adeel A. (2017-05-09). "Brave new motivic homotopy theory II: Homotopy invariant K-theory". arXiv:1705.03340 [math.AT]. Behrend

Chow groups. One of the motivations for higher Chow groups comes from homotopy theory. In particular, if α , β ∈ Z ∗ ( X ) {\displaystyle \alpha ,\beta \in

(2005), "La théorie de l'homotopie de Grothendieck" [Grothendieck's homotopy theory] (PDF), Astérisque, 301, MR 2200690 Denis-Charles Cisinski (2006),

Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude (2001). Rational homotopy theory. Graduate Texts in Mathematics. Vol. 205. New York: Springer-Verlag

bundle, known as the tangent bundle along the fibers of f. Using A1-homotopy theory, the Grothendieck–Riemann–Roch theorem has been extended by Navarro

1007/BF02684690, MR 0174605, S2CID 4649786 Cohen, M. A course in simple homotopy theory Graduate Text in Mathematics 10, Springer, 1973 Higman, Graham (1940)

Microsoft Research 1951 2008 Rick Jardine University of Western Ontario Homotopy theory 2008 Angus Macintyre University of Oxford Logic 1941 2008 Cameron Leigh

Inventiones Mathematicae 3 (1967), 172–178, EuDML. The case of simplicial sets is treated in Paul Goerss, Rick Jardine, Simplicial homotopy theory, Birkhäuser

homotopī-ron)" [S1-action on the cohomologic complex projection space (same homotopy theory)] (PDF). RIMS Kokyuroku (in Japanese). 319. Kyoto University: 20–25

Monographs, American Mathematical Society, (2007), Emily Riehl, Categorical Homotopy Theory, Cambridge University Press (2014) [1] Cisinski, Denis-Charles (2023)

Pontryagin, L. S. (1959). "Smooth manifolds and their applications in homotopy theory" (PDF). American Mathematical Society Translations. Series 2. 11. Translated

foundations of the usual theories of logic and connect logic with homotopy theory by introducing a Hopf structure into the generalized logic theories

families, and it is not amenable to computation. Vladimir Voevodsky, A1-homotopy theory, Proceedings of the International Congress of Mathematicians, Vol.

homology either. The homology "stabilizes" in the sense of stable homotopy theory: there is an inclusion map Sn → Sn+1, and for fixed k, the induced

Pierre Gabriel, Michel Zisman, § 1.5.4., Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer

"Introduction to the Language of Stacks and Gerbes". Retrieved 2007-05-20. Homotopy theory of presheaves of simplicial groupoids, Zhi-Ming Luo Twisted K-theory

conjecture⇐de Branges's theorem 1984 Gunnar Carlsson Segal's conjecture homotopy theory 1984 Haynes Miller Sullivan conjecture classifying spaces Miller proved

Hilton, Peter (1988), "A Brief, Subjective History of Homology and Homotopy Theory in This Century", Mathematics Magazine, 60 (5), Mathematical Association

wide-ranging techniques drawn from analytic number theory, ergodic theory, homotopy theory to address concrete problems in number theory and discover new phenomena

Daniel S.; Hopkins, Michael J. (2013). "Chern-Weil forms and abstract homotopy theory". Bulletin of the American Mathematical Society. (N.S.). 50 (3): 431–468

MR 2900173. Rourke, Colin P.; Sanderson, Brian J. (1971). "Δ-Sets I: Homotopy Theory". Quarterly Journal of Mathematics. 22 (3): 321–338. Bibcode:1971QJMat

ISSN 0273-0979. Belmont, Eva (2016-05-16). "Talbot 2016: Equivariant homotopy theory and the Kervaire invariant one problem" (PDF). math.northwestern.edu

_{2}(B(S^{-1}S)^{0})\to \pi _{1}(Ff)\to \pi _{1}(BGL(R))=GL(R)\to K_{1}(R).} From homotopy theory, we know the second term is central; i.e., π 1 ( F f ) → E ( R ) {\displaystyle

S2CID 15403121 Lev Pontryagin, Smooth manifolds and their applications in homotopy theory American Mathematical Society Translations, Ser. 2, Vol. 11, pp. 1–114

first 9 pages of [1] mention some examples of incorrect results in homotopy theory. Ilya Nikokoshev, Most interesting mathematics mistake? Kevin Buzzard

construct an analogue of the functor R f ! {\displaystyle Rf_{!}} in A1-homotopy theory. Grothendieck's relative point of view in algebraic geometry Change

Brown, R.; Sivera, R. (2009). "Algebraic colimit calculations in homotopy theory using fibred and cofibred categories". Theory and Applications of Categories

arXiv:1908.06323 [math.AT]. Jardine, J. F. (2020-10-26). "Persistent homotopy theory". arXiv:2002.10013 [math.AT]. Lesnick, Michael; Wright, Matthew (2022)

Hilton, Peter (1988), "A Brief, Subjective History of Homology and Homotopy Theory in this Century", Mathematics Magazine, 60 (5): 282–291, doi:10.1080/0025570X

physicist Emil Hilb (1882–1929), mathematician Peter Hilton (1923–2010), homotopy theory Edith Hirsch Luchins (1921–2002), mathematician Kurt Hirsch (1906–1986)

Martin C. (2008) [1968], Cohomology operations and applications in homotopy theory, Mineola, NY: Dover Publications, ISBN 978-0-486-46664-4, MR 0226634

no. 1. p. 1-2. Lyra, Carlos Benjamin de (1965). "On a conjecture in homotopy theory". Anais da Academia Brasileira de Ciências. Vol. 37, no. 2. p. 167-184

relating the groups Ω i a l m {\displaystyle \Omega _{i}^{alm}} to stable homotopy theory. The generalized Poincaré conjecture in dimension n {\displaystyle

anti-feminist activist Julie Bergner, American expert on algebraic topology, homotopy theory, and higher category theory Nicole Berline (born 1944), French researcher

if and only if it is fibrant in the sense of so-called "holomorphic homotopy theory". Meromorphic function in one-variable complex function were studied

The Kervaire invariant problem; 3) Chern-Weil theory and abstract homotopy theory. 2016 Timothy A. Gowers (University of Cambridge): Generalizations

Whitehead, Newmanry mathematician (topologist, one of the founders of homotopy theory) Bernard Willson, academic, worked in Hut 4 on Italian and Japanese

those whose negative homotopy groups vanish. (In areas related to homotopy theory, it is common to use homological conventions, as opposed to cohomological

Study, US "He defined and developed motivic cohomology and the A1-homotopy theory, provided a framework for describing many new cohomology theories for