Geometry.Net - Pure_And_Applied_Math: Grothendieck Topology

Grothendieck Topology - Wikipedia
grothendieck topology. From Wikipedia, the free encyclopedia. A categorytogether with a grothendieck topology on it is called a site.
http://www.wikipedia.org/wiki/Grothendieck_topology

Extractions: Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help From Wikipedia, the free encyclopedia. A Grothendieck topology is a structure defined on an arbitrary category C which allows the definition of sheaves on C , and with that the definition of general cohomology theories. A category together with a Grothendieck topology on it is called a site . This tool is mainly used in algebraic geometry , for instance to define . Note that a Grothendieck topology is not a topology in the classical sense. The motivating example is the following: start with a topological space X and consider the sheaf of all continuous real-valued functions defined on X . This associates to every open set U in X the set F U ) of real-valued continuous functions defined on U . Whenver U is a subset of V , we have a "restriction map" from F V ) to F U ). If we interpret the topological space
The Primitive Topology Of A Scheme - Walker (ResearchIndex)
Abstract We define a grothendieck topology on the category of schemes whose associated sheaf theory coincides in many
http://citeseer.nj.nec.com/walker98primitive.html

Extractions: Abstract: We define a Grothendieck topology on the category of schemes whose associated sheaf theory coincides in many cases with the Zariski topology. We also give some indications of possible advantages this new topology has over the Zariski topology. Key Words: K-theory, local-global ring, primitive criterion, pretheory. 1 Introduction A ring satisfies the "primitive criterion" if every polynomial whose coefficients generate the ring as an ideal represents a unit i.e., takes on a unit value... (Update)
Lambda Definability With Sums Via Grothendieck Logical Relations - Fiore, Simpso
j 1 ; w k g 2 S This definition appears to be related to the notion of a grothendieck topology (Fiore and Simpson, 1999).
http://citeseer.nj.nec.com/fiore99lambda.html

Extractions: Abstract: . We introduce a notion of Grothendieck logical relation and use it to characterise the definability of morphisms in stable bicartesian closed categories by terms of the simply-typed lambda calculus with finite products and finite sums. Our techniques are based on concepts from topos theory, however our exposition is elementary. Introduction The use of logical relations as a tool for characterising the -definable elements in a model of the simply-typed -calculus originated in the work of... (Update)
Abstract:001108bm
Suppose J is a grothendieck topology on C which is generated by the subcanonicalpretopology J' for which a family (C i D) is in J' if and only if the
http://www.maths.usyd.edu.au:8000/u/stevel/auscat/abstracts/001108bm.html

Extractions: Let C be a category with pullbacks and let E be a pullback-stable class of morphisms of C which is closed under composition with the isomorphisms. E defines a pseudofunctor from C op to Cat , also denoted by E , which sends an object C to E C , the full subcategory of the slice category C C consisting of arrows in E with codomain C . We may then consider the category of E -descent morphisms in C as defined in G. Janelidze and W. Tholen, Facets of Descent I, Applied Categorical Structures 2, 1994:1-37 Suppose J is a Grothendieck topology on C which is generated by the subcanonical pretopology J' for which a family (C i ->D) is in J' if and only if: the coproduct C of the C i exists, it is universal and disjoint, and the induced morphism C>D is both a universal regular epimorphism and an E -descent morphism. Using the Yoneda embedding Y: C op ->Sh( C ,J) we prove several results related to E -effective descent morphisms, Galois objects, and torsors in C . As an application, we get the following two theorems.
Points And Co-points In Formal Topology
an hint of grothendieck topology we can introduce a relation
http://www.math.unipd.it/~silvio/papers/FormalTopology/PointsCoPoints.pdf
Week68
symmetries. Then there are *really* highpowered things like topoi ofsheaves on a category equipped with a grothendieck topology .
http://math.ucr.edu/home/baez/week68.html

Extractions: Okay, now the time has come to speak of many things: of topoi, glueballs, communication between branches in the many-worlds interpretation of quantum theory, knots, and quantum gravity. 1) Robert Goldblatt, Topoi, the Categorial Analysis of Logic, Studies in logic and the foundations of mathematics vol. 98, North-Holland, New York, 1984. If you've ever been interested in logic, you've got to read this book. Unless you learn a bit about topoi, you are really missing lots of the fun. The basic idea is simple and profound: abstract the basic concepts of set theory, so as to define the notion of a "topos", a kind of universe like the world of classical logic and set theory, but far more general! For example, there are "intuitionistic" topoi in which Brouwer reigns supreme - that is, you can't do proof by contradiction, you can't use the axiom of choice, etc.. There is also the "effective topos" of Hyland in which Turing reigns supreme - for example, the only functions are the effectively computable ones. There is also a "finitary" topos in which all sets are finite. So there are topoi to satisfy various sorts of ascetic mathematicians who want a stripped-down, minimal form of mathematics. However, there are also topoi for the folks who want a mathematical universe with lots of horsepower and all the options! There are topoi in which everything is a function of time: the membership of sets, the truth-values of propositions, and so on all depend on time. There are topoi in which everything has a particular group of symmetries. Then there are *really* high-powered things like topoi of sheaves on a category equipped with a Grothendieck topology....
Ultrapowers As Sheaves On A Category Of Ultrafilters
3.1. The grothendieck topology on U
http://www.math.uu.se/~jonase/papers/ultra.pdf
The Primitive Topology Of A Scheme, By Mark E. Walker
We define a grothendieck topology on the category of schemes whose associatedsheaf theory coincides in many cases with that of the Zariski topology.
http://www.math.uiuc.edu/K-theory/0214/
A S HEAF-THEORETIC VIEW OF LOOP SP A C ES
grothendieck topology. The importance of both stacks and simplicial sheaves alone should
http://emis.impa.br/journals/TAC/volumes/8/n19/n19.pdf
Relative Cycles And Chow Sheaves, By Andrei Suslin And Vladimir Voevodsky
a presheaf on the category of Noetherian schemes over S. Moreover this presheafturns out to be a sheaf in a grothendieck topology called the cdhtopology.
http://www.math.uiuc.edu/K-theory/0035/

Extractions: For a scheme X of finite type over a Noetherian scheme S we define a group of relative equidimensional cycles. We show that it is contravariantly functorial with respect to base change and thus provides a presheaf on the category of Noetherian schemes over S. Moreover this presheaf turns out to be a sheaf in a Grothendieck topology called the cdh-topology. The main goal of the paper is to study these Chow sheaves. In the particular case of X being a projective variety over a field of characteristic zero the Chow sheaf of effective cycles of dimension d is representable by the corresponding Chow variety. Even in this case though it turns out to be more convenient to work with sheaves than with varieties. In particular we construct certain short exact sequenecs of Chow sheaves which in the case of varieties over a field lead to localization long exact sequences in algebraic cycle homology and which do not have any obvious analog for Chow varieties. 0035.bib
Research Topics
Fujiwara, et al, gives a remedy for such a difficulty, changing virtually the topologicaltexture of spaces by means of grothendieck topology (in Fujiwara's
http://www.kusm.kyoto-u.ac.jp/~kato/Research/topics.html

Extractions: Fake Projective Plane - This is a compact complex algebraic surface of general type having the same betti numbers as the projective plane. Such a surface was first imagined in connection with Severi's conjecture, which expects that the projective plane would be characterized only by its topological type, and, at first, it was completely unknown as to whether such a surface really exists or not. Since then, although the first knowledge of it was anything but substantial, the fake projective plane has attracted mathematician's curiosity even after S.-T. Yau's affirmative solution to Severi's conjecture because of the fact that, as well as being looking like a mysterious of the projective plane, it satisfies the equality in the famous Miyaoka-Yau inequality. The first example of such surfaces was discovered by Mumford in 1979, well-known nowadays as Mumford's fake projective plane . Amazing is not only his discovery itself but also the way of construction by means of dyadic uniformization , which realizes the surface in question as a discrete fixed-point free quotient of a certain symmetric domain in rigid analysis. The last fact can be seen in an interesting parallelism with the fact that any possible fake projective plane, being on the Miyaoka-Yau critical line, should be realized as a discrete fixed-point free quotient of the complex unit-ball by the procedure usually referred to as a uniformization in complex analysis.
CS 59/93
It is proven that a class of finite automata defines a grothendieck topology andthe conditions are developed when a set of states of an automation determines
http://www.cs.ioc.ee/~bibi/resrep/cs/cs59.html

Extractions: Author(s): KALJULAID, Uno, MERISTE, Merik, PENJAM, Jaan. Title: Algebraic theory of tape-controlled attributed automata.28p. Language: ABSTRACT. Compositional theory of tape-controlled attributed automata is considered together with related developments in formal languages theory and algebra. It is proven that a class of finite automata defines a Grothendieck topology and the conditions are developed when a set of states of an automation determines a sheaf of sets of objects in the induced topological category. These two results are expected to be used in the proof that the induced fiber product of a Grothendieck topology is suitable for decomposition of tape- controlled attributed automata.
Math.wesleyan.edu/~mhovey/archive/letter119
We prove the general theorem that internal equivalences of presheaves of groupoidswith respect to a grothendieck topology on Aff give rise to equivalences of
http://math.wesleyan.edu/~mhovey/archive/letter119

Extractions: There are 7 new papers this time. This is a good time to remind you that people decide whether to download your paper based on your abstract. It is therefore crucial that there be an abstract and that it be readable by humans. It is not enough to just e-mail Clarence a dvi file; you must also e-mail him an abstract, under separate cover, with minimal TeX symbols. Mark Hovey New papers appearing on hopf between 5/16/01 and 6/1/01 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Kitchloo/BrKi Classifying spaces of Kac-Moody groups Carles Broto and Nitu Kitchloo broto@mat.uab.es nitu@math.nwu.edu We study the structure of classifying spaces of Kac-Moody groups from a homotopy theoretic point of view. They behave in many respects as in the compact Lie group case. The mod p cohomology algebra is noetherian and Lannes' T-functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p-subgroups. Also, spaces of maps from classifying spaces of finite p-groups to classifying spaces of Kac-Moody groups are described in terms of classifying spaces of centralizers while the classifying space of a Kac-Moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p-subgroups, up to p-completion. We show that these properties are common to a larger class of groups, also including parabolic subgroups of Kac-Moody groups, and centralizers of finite p-subgroups. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Christensen-Hovey/relative (This is the final version, to appear in Math Proc Camb Phil Soc) Quillen model structures for relative homological algebra. by J. Daniel Christensen and Mark Hovey Univ. of Western Ontario Wesleyan University London, ON Middletown, CT jdc@julian.uwo.ca hovey@member.ams.org AMS classification: Primary 18E30; Secondary 18G35, 55U35, 18G25, 55U15 Submitted. 28 pages. An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category A is exactly the information needed to do homological algebra in A. The main result is that, under weak hypotheses, the category of chain complexes of objects of A has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the "pure derived category" of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and cohomology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/hopfalgebroids Morita theory for Hopf algebroids and presheaves of groupoids Mark Hovey Wesleyan University Middletown, CT mhovey@wesleyan.edu 5/17/01 AMS classification nos: 14L05, 14L15, 16W30, 18F20, 18G15, 55N22 Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff, the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of presheaves of groupoids with respect to a Grothendieck topology on Aff give rise to equivalences of categories of sheaves in that topology. We then show using faithfully flat descent that an internal equivalence in the flat topology gives rise to an equivalence of categories of quasi-coherent sheaves. The corresponding statement for Hopf algebroids is that weakly equivalent Hopf algebroids have equivalent categories of comodules. We apply this to formal group laws, where we get considerable generalizations of the Miller-Ravenel change of rings theorems in algebraic topology. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lazarev/ainf Author: Andrey Lazarev Title: Spaces of multiplicative maps between highly structured ring spectra. We uncover a somewhat unexpected connection between spaces of multiplicative maps between A-infinity ring spectra and topological Hochschild cohomology. As a consequence we show that such spaces become infinite loop spaces after looping only once. We also prove that any multiplicative cohomology operation in complex cobordisms theory MU canonically lifts to an A-infinity map MU>MU. This implies, in particular, that the Brown-Peterson spectrum BP splits off MU as an A-infinity ring spectrum. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Lazarev/tower Towers of MU-algebras and the generalized Hopkins-Miller theorem Author: A.Lazarev Department of Mathematics, Univ. of Bristol, Bristol, BS8 1TW, UK. email A.Lazarev@bristol.ac.uk AMS classification number 55N22 Our results are of three types. First we describe a general procedure of adjoining polynomial variables to A-infinity-ring spectra whose coefficient rings satisfy certain restrictions. A host of examples of such spectra is provided by killing a regular ideal in the coefficient ring of MU, the complex cobordism spectrum. Second, we show that the algebraic procedure of adjoining roots of unity carries over in the topological context for such spectra. Third, we use the developed technology to compute the homotopy types of spaces of strictly multiplicative maps between suitable K(n)-localizations of such spectra. This generalizes the famous Hopkins-Miller theorem and gives strengthened versions of various splitting theorems. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mitchell/localb The algebraic K-theory spectrum of a 2-adic local field by Stephen A. Mitchell mitchell@math.washington.edu (There was no abstract with this paper, so I made one up. If you don't like it, Steve, send in one!) A local field F of characteristic is a finite extension of the L-adic rationals of finite degree d, where L is a prime. When L is odd, Dwyer and the author determined the homotopy type of the etale K-theory spectrum of F, but their methods fail when L=2 and -1 is not a square in F. The purpose of this paper is to study this remaining case. The recent work on the Lichtenbaum-Quillen conjecture at 2 by Rognes and Weibel allows the author to get from the etale K-theory of F to the 2-adic completion of the algebraic K-theory of F. The result essentially says that, rather than a splitting as you get in the odd primary case, there is some room for a few non-trivial extensions (which are completely determined). This is a generalization of Rognes' calculation of the 2-adic K-theory of the 2-adic rationals. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/YauD/catcocat Title: Clapp-Puppe Type Lusternik-Schnirelmann (Co)category in a Model Category Donald Yau AMS Classification: Primary 55M30; Secondary 55P30, 55U35 math.AT/0104267 Department of Mathematics MIT, 2-230 77 Massachusetts Avenue Cambridge, MA 02139 USA donald@math.mit.edu We introduce Clapp-Puppe type generalized Lusternik-Schnirelmann (co)category in a Quillen model category. We establish some of their basic properties and give various characterizations of them. As the first application of these characterizations, we show that our generalized (co)category is invariant under Quillen modelization equivalences. In particular, generalized (co)category of spaces and simplicial sets coincide. Another application of these characterizations is to define and study rational cocategory. Various other applications are also given.
Academic Bibliography For Willaert, Luc
1995. Van Oystaeyen F., Willaert L.. grothendieck topology, coherentsheaves and Serre's theorem for schematic algebras. - In Journal
http://lib.ua.ac.be/AB/a10529.html

Extractions: Go to the STARTSCREEN The author belongs to: Years: Vancliff M. Van Rompay K. Willaert L. Some quantum P3s with finitely many points. - In: Communications in algebra , 26:4(1998), p. 1193-1207 Van Oystaeyen F. Willaert L. Examples and quantum sections of schematic algebras. - In: Journal of pure and applied algebra , 120(1997), p. 195-211 Verschoren A. Willaert L. Noncommutative algebraic geometry: from pi-algebras to quantum groups. - In: Bulletin of the Belgian Mathematical Society Simon Stevin , 4(1997), p. 557-588 Van Oystaeyen Fred Willaert Luc Cohomology of schematic algebras. - In: Journal of algebra , 185:1(1996), p. 74-84 Van Oystaeyen Freddy Willaert Luc The quantum site of a schematic algebra. - In: Communications in algebra , 24:1(1996), p. 209-222 Van Oystaeyen Fred Willaert Luc Valuations on extensions of Weyl skew fields. - In: Journal of algebra , 183:2(1996), p. 359-364 Van Oystaeyen F. Willaert L. Grothendieck topology, coherent sheaves and Serre's theorem for schematic algebras. - In: Journal of pure and applied algebra , 104(1995), p. 109-122
Overview
The main idea of the paper is that relationships between systems can be expressedby a suitable grothendieck topology on the category of systems.
http://www.mpi-sb.mpg.de/~sofronie/abstracts.html
Www.lehigh.edu/~dmd1/h1017.txt
Italy, vezzosi@dm.unibo.it Included gzipped .ps file ABSTRACT For a (semi)modelcategory M, we define a notion of a ''homotopy'' grothendieck topology on M
http://www.lehigh.edu/~dmd1/h1017.txt

Extractions: Subject: new Hopf listings Date: 17 Oct 2001 14:33:14 -0400 From: Mark Hovey < Internet Explorer) to the URL listed. The general Hopf archive URL is http://hopf.math.purdue.edu There are links to Purdue seminars, and other math related things on this page as well. The largest archive of math preprints is at http://xxx.lanl.gov There is an algebraic topology section in this archive. The most useful way to browse it or submit papers to it is via the front end developed by Greg Kuperberg: http://front.math.ucdavis.edu To get the announcements of new papers in the algebraic topology section at xxx, send e-mail to math@xxx.lanl.gov with subject line "subscribe" (without quotes), and with the body of the message "add AT" (without quotes). You can also access Hopf through ftp. Ftp to hopf.math.purdue.edu, and login as ftp. Then cd to pub. Files are organized by author name, so papers by me are in pub/Hovey. If you want to download a file using ftp, you must type binary before you type get . To put a paper of yours on the archive, cd to /pub/incoming. Transfer the dvi file using binary, by first typing binary
Www.lehigh.edu/~dmd1/h117
IL 60208 rezk@math.nwu.edu November 3, 1998 We show that homotopy pullbacks of sheavesof simplicial sets over a grothendieck topology distribute over homotopy
http://www.lehigh.edu/~dmd1/h117

Extractions: Subject: new Hopf listings From: Mark Hovey before you type get . To put a paper of yours on the archive, cd to /pub/incoming. Transfer the dvi file using binary, by first typing binary then put You should also transfer an abstract as well. Clarence has explicit instructions for the form of this abstract: see http://hopf.math.purdue.edu/pub/submissions.html. In particular, your abstract is meant to be read by humans, so should be as readable as possible. I reserve the right to edit unreadable abstracts. You should then e-mail Clarence at wilker@math.purdue.edu telling him what you have uploaded. I am solely responsible for these messages-don't send complaints about them to Clarence. Thanks to Clarence for creating and maintaining the archive. - End of forwarded message -
Www.risc.uni-linz.ac.at/research/category/risc/catlist/ortho-topos
with respect to cones, generalising that of orthogonality with respect to maps andthe sheaf condition for a cover in a grothendieck topology 1. We say that
http://www.risc.uni-linz.ac.at/research/category/risc/catlist/ortho-topos

Extractions: Date: Thu, 14 Mar 1996 17:11:37 -0400 (AST) Subject: Orthogonality and toposes question. Date: Thu, 14 Mar 1996 18:35:00 GMT From: Marcelo Fiore Below I consider a notion of orthogonality with respect to cones, generalising that of orthogonality with respect to maps and the sheaf condition for a cover in a Grothendieck topology: 1. We say that an object K is orthogonal to a cone D > C whenever for every cone D > K there exists a unique C > K such that (D > C > K) = (D > K). 2. For a category K and a class J of cones in K we define O(K,J) as the full subcategory of K consisting of all those objects orthogonal to every cone in J. My question is: Let A be a small category and write Psh A for the topos of presheaves on A. Is there a characterisation of the classes J of cones in A for which O(Psh A,J) is a topos? Comments and pointers to relevant literature are welcome. Many thanks, Marcelo. Date: Thu, 14 Mar 1996 23:08:58 -0400 (AST) Subject: Re: Orthogonality and toposes question. Date: Thu, 14 Mar 1996 19:21:43 -0500 (EST) From: Dan Christensen With regard to Marcelo Fiore's question about orthogonality with respect to cones: together with Gabriele Castellini and George Strecker I have studied some aspects of this phenomenon for discrete cones in Regular Closure Operators Applied Categorical Structures 2 (1994), 219244 Kluwer Academic Publishers Section 5 of this article might be of interest. Best regards, J"urgen Koslowski
Www.risc.uni-linz.ac.at/research/category/risc/catlist/topos-useful
C be the corresponding category of elements (object = element q of Q/Z, morphisms(q,f) q q+f for f in Q+) and generate a grothendieck topology on C
http://www.risc.uni-linz.ac.at/research/category/risc/catlist/topos-useful

Extractions: Date: Thu, 19 Nov 1998 11:53:12 +0000 From: s.vickers@doc.ic.ac.uk (Steven Vickers) Subject: categories: Has this topos ever been found useful? After seeing Eric Goubault's work on directed homotopy, where he defines a notion of "locally partially ordered space", I found myself considering the following topos. Does anyone know if it has been used, for instance in algebraic geometry or algebraic topology? What I am trying to capture by the definition is the process of taking the lower semicontinuous reals in [0,1] (classically, [0,1] with its Scott topology, so that the numerical order becomes the specification order) and identifying and 1. Then the specialization from to 1 becomes a non-trivial endomorphism e of 0, so we are forced to consider the modified space as a topos, not a locale. In addition, an extra point will spring into existence, namely the (filtered) colimit of > > > ... e e e The topos is given by a site as follows. Let Q+ be the additive monoid of non-negative rationals and Q/Z the rationals modulo the integers. Q+ acts on Q/Z by addition. Let C be the corresponding category of elements (object = element [q] of Q/Z, morphisms ([q],f): [q] -> [q+f] for f in Q+) and generate a Grothendieck topology on C^op by letting any object [q] of C be cocovered by all the morphisms ([q],f) for f > 0. Let E be the topos corresponding to this site on C^op. Its points are flat presheaves F on C with the condition that if x is in F([q]) then x = ([q],f)y for some f > 0, y in [q+f]. The points arising from the wrapped-round [0,1] are like half-infinite helices (F([q]) = N for all q), and the new point is like an infinite helix (F([q]) = Z for all q). Steve Vickers.
Br.crashed.net/~loner/sheaves/topos1.txt
respects. For one thing, for this grothendieck topology, a sheafis a functor which can be collated over each such cover. 1706
http://br.crashed.net/~loner/sheaves/topos1.txt

Extractions: I was asked to do this talk by ^LoNeR^. If you get anything out of this, you may thank him. [16:57] * [-K-] raises an eyebrow at R[[x]] [16:57] Ok. It is always a pleasure for me to introduce one of our speakers. All the more so today since R[[x]] aka [Sic] is one of the founders of the channel. [16:58] Many of you probably attended nerdy's talk on sheaves and topoi. I will attempt to make contact with that talk where I am able. Algebraic geometry is not my strong suit. [16:58] The title is: Sets and topoi... why Lawvere is a god [16:58] First, as is my custom, let us have a short history lesson on sheaves. My apologies to the algebraic geometers present, for the development will be informal and lacking in detail. The object is to provide motivation for the beginner, and to illustrate the deep connections between geometry and logic. [16:59] The notion of a sheaf originated in algebraic topology, although it has been suggested that it goes as far back as studies in the 19th century on the analytic continuation of functions. [16:59] The idea in algebraic topology is that we want a cohomology with variable coefficients, for example, varying under the action of the fundamental group, ala Steenrod. [16:59]

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