FUNCTORIAL SEMANTICS
OF
ALGEBRAIC THEORIES
AND
SOME ALGEBRAIC PROBLEMS IN THE
CONTEXT OF FUNCTORIAL SEMANTICS OF
ALGEBRAIC THEORIES
F. WILLIAM LAWVERE
c F. William Lawvere, 1963, 1968. Permission to copy for private use granted.
Contents
A
Authors comment
Theory and Applications of Categories, Vol. 13, No. 10, 2004, pp. 164168.
FUNCTORIAL CONCEPTS OF COMPLEXITY
FOR FINITE AUTOMATA
For Aurelio, an exacting colleague and a treasured friend since 1972,
when he was one of the unfailingly enthusiastic four who
Foreword
The study that was initiated by Birkhoff in 1935 was named general algebra
by Kurosh in his classic text; the subject is also called universal algebra, as in
the text by Cohn. The purpose of general algebra is to make explicit common
features of
Theory and Applications of Categories, Vol. 20, No. 14, 2008, pp. 497503.
CORE VARIETIES, EXTENSIVITY, AND RIG GEOMETRY
F. WILLIAM LAWVERE
Abstract. The role of the Frobenius operations in analyzing finite spaces, as well
as the extended algebraic geometr
OPEN PROBLEMS IN TOPOS THEORY
Bill Lawvere
4 April 2009 * updated July 2016
For Martin Hyland and Peter Johnstone in honor of their sixtieth birthdays.
Here are seven problems that I have not yet been able to solve. Clarification
on them would further adv
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arXiv.org > math > arXiv:math/0209005
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6.080/6.089 GITCS
Mar 11, 2008
Lecture 10
Lecturer: Scott Aaronson
1
Scribe: Yinmeng Zhang
Administrivia
Work hard on the homework, but dont freak out. Come to office hours. There will be a short (1
week) pset before spring break, and an exam on the Thurs
Zero-One Permanent is #P -Complete, A Simpler Proof
Amir Ben-Dor
Shai Haleviy
Dept. of Computer Science
Technion
Haifa, Israel 32000
February 22, 1995
y
This research was supported by United States-Israel Binational Science Foundation grant 88-00282
This
6.080/6.089 GITCS
March 6, 2008
Lecture 9
Lecturer: Scott Aaronson
1
Scribe: Ben Howell
Administrivia
1.1
Problem Set 1
Overall, people did really well. If youre not happy with how you did, or theres something you
dont understand, come to office hours on
Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 4149.
AXIOMATIC COHESION
F. WILLIAM LAWVERE
Abstract. The nature of the spatial background for classical analysis and for modern
theories of continuum physics requires more than the partial
Topology Atlas Document # topd-65
John Isbell's Adequate Subcategories
F. W. Lawvere
From TopCom, Volume 11, #1
For mathematicians of my age, the theory of rings of continuous functions was one of the first exciting
research topics we encountered. Many re
Reprints in Theory and Applications of Categories, No. 11, 2005, pp. 135.
AN ELEMENTARY THEORY
OF THE CATEGORY OF SETS (LONG VERSION)
WITH COMMENTARY
F. WILLIAM LAWVERE
Received by the editors 2005-04-01.
Transmitted by M. Hyland, A. Kock, R. Rosebrugh. R
Theory and Applications of Categories, Vol. 8, No. 9, 2001, pp. 253283.
HOW ALGEBRAIC IS ALGEBRA ?
J. ADAMEK
, F. W. LAWVERE AND J. ROSICKY
ABSTRACT. The 2-category VAR of nitary varieties is not varietal over CAT . We
introduce the concept of an algebrai
Theory and Applications of Categories, Vol. 11, No. 11, 2003, pp. 252282.
CONTINUOUS CATEGORIES REVISITED
)
)
J. ADAMEK
, F. W. LAWVERE, J. ROSICKY
ABSTRACT. Generalizing the fact that Scotts continuous lattices form the equational
hull of the class of a
An Interview with F. William Lawvere
You have written a paper, published for the first time in 1986, entitled Taking
categories seriously1. Why should we take categories seriously ?
In all those areas where category theory is actively used the categorical
Algebra univers. 49 (2003) 3549
0002-5240/03/010035 15
c Birkh
auser Verlag, Basel, 2003
Algebra Universalis
On the duality between varieties and algebraic theories
mek, F. W. Lawvere, and J. Rosicky
J. Ada
Abstract. Every variety V of nitary algebras is
The Bulletin of Symbolic Logic
Volume 9, Number 2, June 2003
FOUNDATIONS AND APPLICATIONS:
AXIOMATIZATION AND EDUCATION
F. WILLIAM LAWVERE
Abstract. Foundations and Applications depend ultimately for their existence on each
other. The main links between t
ARTICLE IN PRESS
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S0315-0860(04)00059-X/BRV AID:2460 Vol.()
YHMAT:m2 v 1.24 Prn:24/09/2004; 10:35
[DTD5] P.1 (1-7)
by:Vita p. 1
Historia Mathematica ()
www.elsevier.com/locate/hm
Review
Hermann Gnter Grassmann, A new branch of mathematics, The A
Studia Scieutiaruni Mathernaticarum Huugarica 1 (1966) 2 1 5-235 .
ON A PROBLEM OF GRAPH THEORY
by
P . ERDS , A . RNYII and V . T . SS 2
0 . Introduction
Let G be a non-directed graph having n vertices, without parallel edges and
slings . Let the verti
Math 565: Combinatorics and Graph Theory
Professor: David E Speyer
Fall 2013
Course meets: Tuesdays and Thursdays, 11:30-1:00, 3088 East Hall
Text: Introduction to Graph Theory, Doug West, ISBN 9780130144003
I expect to jump around a lot in the text, and
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H
Solution Set 8
Problem 1 Consider a (2n + 1) (2n + 1) checker board, with the corners colored black. Suppose
that we remove any black square from the board, living 4n2 + 4n squares behind. Show that the
remaining squares can be tiled with dominos. (A dire
Problem Set 7 due November 5th
Please see the course website for homework policy.
Problem 1 Fix a positive integer d; in this problem we will study d-regular graphs on n vertices.
Let A(G) be the adjacency matrix and i its eigenvalues. On the previous pro
SOLUTION SET 9
1. Take the shortest path which visits every point. Let H be the set of edges contained in this
path. Since H is a connected graph, it contains a spanning tree. The length of this tree is s, so
t s.
On the other hand, let T be a minimal spa
SOLUTION SET 1
1. Basic enumeration and binomial coefficients
1. (A Course in Combinatorics, Problem 13A) We want to place the integers 1, 2, . . . , r into
a circular array with n positions so that they occur in order, clockwise, and such that consecutiv
SOLUTION SET 2
1. The generating function for f (n) is
f (n)xn =
1
.
1 xai
Notice that all the roots of the denominator lie on the unit circle. So every term in the partial
fraction expansion looks like a/(1 rx)k for some r on the unit circle. The corresp
SOLUTION SET 3
1. Let F (x) = W (x)/x. Set Q(x) = x/W (x) = 1/F (x), and let P (x) be Q1 (x). By Lagrange
inversion, we have
[coecient of xn ]P (x) =
1
1
[coecient of x1 ]F (x)n = .
n
n
So
xn
= log(1 x).
n
Then Q is the inverse function. Writing down log(
SOLUTION SET 4
1. As suggested in the hint, suppose for the sake of contradiction that, for
every k, there are two sequences wk and vjk which differ only in the kth
coordinate. For every k, choose one such pair (ik,jk). Create a graph G
with n vertices,
SOLUTION SET 5
1.a and b
1
2
SOLUTION SET 5
1.c We choose to rst do the loop at 000 and then leave it. Here is the resulting tour; the key point
is to always follow the edge in the tree second.
000, 000, 001, 011, 110, 101, 011, 111, 111, 110, 100, 001, 0