categories: Weak algebraic structures

[Tom Leinster <T.Leinster@dpmms.cam.ac.uk>]

The following two papers on homotopy-algebraic structures - or "weakened"
algebraic structures, if you prefer - are now available.

The first one, "Up-to-Homotopy Monoids", is 8 pages long and is essentially
a set of notes for the talk I gave at the Louvain-la-Neuve PSSL in
October. It can serve as an introduction to the second one, "Homotopy
Algebras for Operads" (100 pages, but don't let that scare you: it should be
easy for category theorists). Abstracts are below.

Tom Leinster

* * *

"Up-to-Homotopy Monoids"

Informally, a homotopy monoid is a monoid-like structure in which properties
such as associativity only hold `up to homotopy' in some consistent way. This
short paper comprises a rigorous definition of homotopy monoid and a brief
analysis of some examples. It is a much-abbreviated version of the paper
`Homotopy Algebras for Operads', and does not assume any knowledge of
operads.

Available on the mathematics archive:
http://xxx.lanl.gov/abs/math.QA/9912084

* * *

"Homotopy Algebras for Operads"

We present a definition of homotopy algebra for an operad, and explore its
consequences.

The paper should be accessible to topologists, category theorists, and anyone
acquainted with operads. After a review of operads and monoidal categories,
the definition of homotopy algebra is given.  Specifically, suppose that M is
a monoidal category in which it makes sense to talk about algebras for some
operad P. Then our definition says what a homotopy P-algebra in M is,
provided only that some of the morphisms in M have been marked out as
`homotopy equivalences'.

The bulk of the paper consists of examples of homotopy algebras. We show that
any loop space is a homotopy monoid, and, in fact, that any n-fold loop
space is an n-fold homotopy monoid in an appropriate sense. We try to
compare weakened algebraic structures such as A_infinity-spaces,
A_infinity-algebras and non-strict monoidal categories to our homotopy
algebras, with varying degrees of success. We also prove results on `change
of base', e.g. that the classifying space of a homotopy monoidal category is
a homotopy topological monoid. Finally, we reflect on the advantages and
disadvantages of our definition, and on how the definition really ought to be
replaced by a more subtle infinity-categorical version.

Available on the mathematics archive:
http://xxx.lanl.gov/abs/math.QA/0002180