Exercise 11. Typeset David Benson's\Higher-Dimensional Graph Path:" a d. # "x1 y1. // !z1 ... Exercise 12. Typeset the following categorical de nition of adjoints:.
EXERCISES IN XY-PIC
Kristoffer H. Rose
1. Introduction
This is a collection of `real-life' diagram typesetting problems and their solutions using the XY-pic macro package. The exercises are meant to complement the XY-pic Manual [r92:typesetting] version 2.6 (x-references in the text below refer to sections and subsections of this manual). The exercises are not much fun unless you have a running TEX system with the XY-pic macros available.1 We present the exercises in two groups: The rst exercises are fetched from `diagramathematics,' i.e., the art of presenting lemmas, proofs, etc. of mathematics (and theoretical computer science) using diagrams. Most of these have easy solutions using XY-pic since the package was designed for just this kind of diagrams. The second group contains complex diagrams from a variety of sources. I would like to thank David Benson, Eva Rose, Marc Andries, and Jesper J rgensen for making their diagrams available for this collection. 2. Mathematical Diagrams Exercise 3.
Typeset the axioms of category theory: f
A
B ~
~ f iB ~~
inverse:
g
C
/
B
/
C
h /
C
Typeset the following two diagrams depicting the relation between a function and its x O
Exercise 5.
and
g
~
f
@@ @@ g @g@;h f ;g @@ @
# "! B
Exercise 4.
A@
/
Typeset a `ligtning:'
f
f ?1
f(x) and x
f
o
f ?1 /
f(x)
A
u u
B
1 The package (including the manual) is available from any good TEX archive near you as well as by anonymous ftp from ftp.diku.dk [129.142.96.1] in the directory /pub/TeX/misc/xypic.tar.Z. You can use XY-pic with any TEX format, e.g., AMS-TEX, LATEX, or plain TEX. Installation requires opqrstuq except for a generic 300 dpi laserprinter.
Typeset by AMS-TEX
CB? ? ? ? ? ?? ?? ? ?? ?? ?? ? ? ? ? ?#!" @A
2 Exercise 6.
KRISTOFFER H. ROSE
Typeset the following characterization of the identity: ]]]]] aaaaa
p
Exercise 7.
Typeset the Church-Rosser theorem as formulated in [b84:lambda, p.282]: || || M 0B B
M @@
@
}
}
Exercise 8.
tid
A
n
N0
~
~
M; M 0; N; N 0 2
N
Typeset the strip lemma as formulated in [b84:lambda, p.282]: M HH
HH HH H
M0 I
$
$
I
M; M 0 ; N; N 0 2
N
I
I $
$
N0
9. Complex Diagrams
Typeset this diagram used to prove the strip lemma in [b84:lambda, p.282]. Hint : This is dicult. It is easier if the slight lifting of the M and N entries is ignored.
Exercise 10.
M SS ~ BBBSSSSSS ~ BB SSS ~ SSS Bj j ~~ SSS SSS ~ bb ~ bbbbbbbbbb M S S S b 0 S S N M SSS S S A S S S S A jj S S S SA S S S S b b b bN 0 b b b `
~
)
*
q
`
`
# "# " ! ! )
)
)
)
Exercise 11.
N
q
q
Typeset David Benson's\Higher-Dimensional Graph Path:" x1
