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is a way to have versions of theorems 'with parameters', i.e. allowing for continuous variation, for which the 'frozen' version reduces the parameters to
166:. This set-theoretic language is too naĂŻve to fit the required context, certainly, from algebraic geometry. It combines, though, with the use of the
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in foundational matters. Assuming that we don't have a commitment to one 'set theory' (all topoi are in some sense equally set theories for some
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from about 1956 is usually cited as the key moment for the introduction of this circle of ideas. The more classical types of
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105:; from a technical point of view base change becomes a major issue for the whole approach (see for example
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situations, with a rough meaning of taking for consideration families of 'objects' explicitly depending on
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224:) it is possible to state everything relative to some given set theory that acts as a base topos.
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34:, as the basic field of study, rather than a single such object. It is named after
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142:. The 'fiber' terminology is significant: the underlying heuristic is that
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to replace the 'point' idea with that of treating an object, such as
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38:, who made extensive use of it in treating foundational aspects of
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is a fixed object. This idea is made formal in the idea of the
42:. Outside that field, it has been influential particularly on
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In other applications, this way of thinking has been used in
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In the usual formulation, the point of view treats, not
158:, which described by fibers is for each point of
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150:is a family of fibers, one for each 'point' of
101:. To move from one slice to another requires a
154:; the fiber product is then the family on
112:A base change 'along' a given morphism
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20:Grothendieck's relative point of view
232:This article uses terminology from
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183:Grothendieck–Riemann–Roch theorem
189:are recovered in the case where
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193:is a single point (i.e. the
26:applied in certain abstract
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259:Encyclopedia of Mathematics
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134:, producing an object over
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162:the fiber at its image in
130:is typically given by the
216:, to clarify the role of
107:Beck–Chevalley conditions
241:Fiber product of schemes
197:in the working category
174:, as 'as good as' the
36:Alexander Grothendieck
16:Mathematical heuristic
176:representable functor
222:intuitionistic logic
187:Riemann–Roch theorem
60:of a given category
40:algebraic geometry
48:categorical logic
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201:). Using other
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138:from one over
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93:of objects of
91:slice category
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281:Scheme theory
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195:final object
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178:it sets up.
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168:Yoneda lemma
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28:mathematical
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103:base change
275:Categories
247:References
218:set theory
32:parameters
264:EMS Press
207:constants
66:morphisms
24:heuristic
228:See also
97:'above'
266:, 2001
55:objects
85:where
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146:over
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181:The
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203:S
199:C
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164:S
160:T
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152:S
148:S
144:X
140:S
136:T
125:S
121:T
117:g
99:S
95:C
87:S
80:S
76:X
72:f
62:C
58:X
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