Symmetric Monoidal Categories for Operads - Index of

Chapter 1
Symmetric Monoidal Categories
for Operads
Introduction
The notion of a symmetric monoidal category is used to give a general
background for the theory of operads. Standard examples of symmetric
monoidal categories include categories of modules over a commutative ring,
categories of differential graded modules, various categories of coalgebras,
the category of sets together with the cartesian product, the category of simplicial
sets, and the category of topological spaces. Other possible examples
include the modern categories of spectra used to model stable homotopy, but
in applications operads come often from the category of topological spaces
or simplicial sets and categories of spectra are not used as the base category
in our sense (see next).
The first purpose of this chapter is to survey definitions of symmetric
monoidal categories.
To set our constructions, we fix a base symmetric monoidal category, usually
denoted by C, in which all small colimits, all small limits exist, and we
assume that the tensor product of C preserves all colimits (see §1.1.1). In the
sequel, operads are usually defined within this base category C. But, in our
constructions, we use naturally algebras over operads in extensions of the base
category to which the operad belongs. For this reason, we review definitions
of symmetric monoidal categories to have an appropriate axiomatic background
for our applications. The axiomatic structure, needed to generalize
the definition of an algebra over an operad, consists of a symmetric monoidal
category over the base category C, an object under C in the 2-category of
symmetric monoidal categories (see §1.1.2).
Definitions of symmetric monoidal categories are reviewed in §1.1 with this
aim in mind. The structure of a symmetric monoidal category over the base
category is made explicit in that section.
In §1.2, we recall the definition and properties of particular colimits
(namely, reflexive coequalizers and filtered colimits). We use repeatedly
that these colimits are preserved by tensor powers in symmetric monoidal
categories.
B. Fresse, Modules over Operads and Functors, Lecture Notes in Mathematics 1967, 21
DOI: 10.1007/978-3-540-89056-0 1, c○ Springer-Verlag Berlin Heidelberg 2009