

Categoriļ¬cations

The word “categoriﬁcation” gained
popularity in the 21st century. It was introduced by
Crane
[
Cra95
,
CF94
]
in the mid 90s. The popularity seems to be partly due
to Khovanov’s discovery of a
categoriﬁcation of the
Jones polynomial
.
However, analogous ideas, without using the word
categoriﬁcation, already appeared long ago. For
example, the ﬁrst of three memos in
Propp’s
[
Pro
]
begins with the following sentence:
A combinatorialist’s fundamental model of a
nonnegative integer
n
is a set of
n
points.
In
combinatorics
, it is important to ﬁnd a bijective proof for
an equality of values of two functions. For example,
the binomial theorem can be proved by counting the
number of subsets of a given ﬁnite set in two
different ways. Inequalities can be proved by
injections or surjections. Zeilberger
[
Zei
]
calls such proofs as combinatorializations. These are
typical examples of categoriﬁcations.
I recommend the article
[
BD98
]
by Baez and Dolan as a general introduction to
categoriﬁcations. See also
[
CY98
]
,
[
Ion
]
, and
[
BD01
]
.
[BD98]
John C. Baez and James Dolan.
Categoriﬁcation. In
Higher
category theory (Evanston, IL, 1997)
, volume 230 of
Contemp.
Math.
, pages 1–36. Amer. Math. Soc., Providence,
RI, 1998,
arXiv:math/9802029
.
[BD01]
John C. Baez and James Dolan. From
ﬁnite sets to Feynman diagrams. In
Mathematics unlimited—2001 and beyond
, pages 29–50. Springer, Berlin, 2001,
arXiv:math.QA/0004133
.
[CF94]
Louis Crane and Igor B. Frenkel.
Fourdimensional topological quantum ﬁeld
theory, Hopf categories, and the canonical bases.
J. Math.
Phys.
, 35(10):5136–5154, 1994.
[Cra95]
Louis Crane. Clock and category: is quantum gravity
algebraic?
J.
Math. Phys.
, 36(11):6180–6193, 1995.
[CY98]
Louis Crane and David N. Yetter. Examples of
categoriﬁcation.
Cahiers Topologie G
éom. Diff
érentielle Cat
ég.
, 39(1):3–25, 1998.
[Ion]
Lucian M. Ionescu. On Categoriﬁcation,
arXiv:math.CT/9906038
.
[Pro]
James Propp. Euler measure as generalized
cardinality,
arXiv:math.CO/0203289
.
[Zei]
Doron Zeilberger.
(
5
2
)
Proofs that
(
n
k
)
≤
(
n
k
+1
)
if
k < n∕
2,
arXiv:1003.1273
.


