Math 429, Quiz 5.
Your name:
1. Let A be a connected subspace of space X and A Y A. Prove that Y is
connected.
2. Let X be a topological space with two points a and b and only three open
sets , cfw_a, and cfw_a, b. Show that X path connected.
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CONTINUOUSLY VARIABLE SETS;
ALGEBRAIC GEOMETRY = GEOMETRIC LOGIC
F. William LAWVERE
University of Perugia, Perugia, Italy
The (elementary) theory of topoi, the fundamentals of which were
outlined in Prof. Mac Lanes talk at this colloquium, (see also [6, 1
Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 124.
TAKING CATEGORIES SERIOUSLY
F. WILLIAM LAWVERE
Abstract. The relation between teaching and research is partly embodied in simple
general concepts which can guide the elaboration of e
Reprints in Theory and Applications of Categories, No. 1, 2002, pp. 137.
METRIC SPACES, GENERALIZED LOGIC, AND CLOSED
CATEGORIES
F. WILLIAM LAWVERE
Author Commentary:
Enriched Categories in the Logic of Geometry and Analysis
Because parts of the following
Variable Quantities and
Variable Structures in Topoi
F. WILLIAM
LAWVERE
In memory of my eldest son, William
Nevin.
I have organized this chapter into three sections as follows:
1. The conceptual basis for topoi in mathematical experience with
variable set
STPITE 'CFITEGUFIIES, CLOSED CATEGORIES; MID. THE EKISTENCE
SEMIcfw_CONTINUOUS ENTRUFY FUNCTIONS
Ef'
F. HILLIAM LAHVERE
IMA PrEpPirIt Sarias I BE
Jun 3; 3984
INSTITUTE FOFI MATHEMATICS AND ITS APPLICATIONS
UNWEHSITV cfw_IF MINNESOTA
514 Vincent Hall
20
VARIABLE SETS ETENDU
AND
VARIABLE
STRUCTURE iN TOPOi
by
F. William
Lawvere
Notes by Steven Landsburg
of Lecture,s
and Conversations
Spring
Department of Mathematic s
1975, The University
of Chicago
The University of Chicago
I.
Introduction
and Historical
FUNCTDRIRL REMFHRKS ON THE GENERL CONCEPT OF CHUS
Ev
F.H. LWUERE
II'III. Fraprint Series cfw_I 8?
dub; 1984
INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS
UNIVERSITY 'DF MINNESOTA
51 Vlnneni Hall
206 Church Slret SE.
Minneapolis, Minnesota 55455 Functo
Reprints in Theory and Applications of Categories, No. 16, 2006, pp. 116.
ADJOINTNESS IN FOUNDATIONS
F. WILLIAM LAWVERE
Authors commentary
In this article we see how already in 1967 category theory had made explicit a number of
conceptual advances that we
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Actes, Congrs intern, math., 1970. Tome 1, p. 329 334.
QUANTIFIERS AND SHEAVES
by F. W. LAWVERE
The unity of opposites in the title is essentially that between logic and geometry,
and there are compelling reasons for maintaining that geometry is the leadi
Reprints in Theory and Applications of Categories, No. 15, 2006, pp. 113.
DIAGONAL ARGUMENTS
AND
CARTESIAN CLOSED CATEGORIES
F. WILLIAM LAWVERE
Author Commentary
In May 1967 I had suggested in my Chicago lectures certain applications of category
theory to
Reprints in Theory and Applications of Categories, No. 9, 2005, pp. 17.
CATEGORIES OF SPACES MAY NOT BE GENERALIZED SPACES
AS EXEMPLIFIED BY DIRECTED GRAPHS
F. WILLIAM LAWVERE
Author commentary: When this paper was distributed at the 1986 international ca
Math 429, Quiz 6.
Your name:
1. Prove that if X is connected and has the same homotopy type as Y then Y
is also cconnected.
2. For a path f let f be the path given by
f (t) = f (1 t).
Prove that f
g (relcfw_0, 1) if and only if f
g (relcfw_0, 1).
Math 429, Quiz 4.
Your name:
1. Dene on R by x y if and only if x y is rational. Show that is an
equivalence relation and that R/ with the quotient topology is not Hausdor.
2. Let X be a Hausdor space and A, B X be two compact subsets. Show
that A B is co
Math 429, Quiz 3.
Your name:
1. Suppose that X is a Gspace. Prove that the function g given by g (x) = g x
is a homeomorphism from X to itself for all g G.
2. Consider R with equivalence relation x x. Show that R/ is homeomorphic to [0, ).
Math 429, Quiz 2.
Your name:
1. Show that subset (a, b) in R with induved topology is homeomorphic to R.
2. Give an example of topological spaces X , Y with a continuus bijection
f : X Y such that f 1 is not continuous.
Math 429, Midterm 3.
Your name:
1. Prove that any continuous map f : D2 D2 has a xed point; i.e., a point
such that f (x) = x.
2. Let p : X X be a covering and let f , f : Y X be two liftings of
f : Y X.
Assume Y is connected and f (y0 ) = f (y0 ) for som
Math 429, Midterm 2.
Your name:
1. (8.7) Prove that compact subset of Hausdor space is closed.
2. (9.3) Show that interval [0, 1] is connected.
3. Let be a relation on the points of a space X dened by saying that x y
if and only if there is a path joining
Total internal reflection ellipsometry: principles
and applications
Hans Arwin, Michal Poksinski, and Knut Johansen
A concept for a measurement technique based on ellipsometry in conditions of total internal reflection is
presented. When combined with sur
Categorical Dynamics
F. William Lawvere
Abstract.
In the (Chicago 1967) setting of a cartesian closed category E of spaces, with a
given pointed infinitesimal space T , we call secondorder infinitesimal the symmetric
square W = T 2 /2! whose two axes coa
Intrinsic CoHeyting Boundaries and the Leibniz Rule
in Certain Toposes
F. William Lawvere
Department of Mathematics, S.U.N.Y. at Buffalo
Buffalo, NY 14214
Certain lattices, Such as that of all closed subsets of a topological
space or that of all subtopos
C AHIERS DE
TOPOLOGIE ET GOMTRIE DIFFRENTIELLE
CATGORIQUES
F. W ILLIAM L AWVERE
Toward the description in a smooth topos of the
dynamically possible motions and deformations
of a continuous body
Cahiers de topologie et gomtrie diffrentielle catgoriques,
t
Ordinal Sums and Equational Doctrines
F. William Lawvere
Our purpose is to describe some examples and to suggest some directions for the study
of categories with equational structure. To equip a category A with such a structure means
roughly to give certa
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Perugia Notes
Luned` Nov. 13, 1972
17:15  19:00
Programma Provvisorio del Corso di
Teoria delle Categorie Sopra un Topos di Base
Prof. F. W. Lawvere
Il Corso si compone di due parti. La prima, di carattere pi`
u elementare,
sar`a svolta nei primi due o t