A Note on the Composition Product of Symmetric Sequences

We consider the composition product of symmetric sequences in the case where the underlying symmetric monoidal structure does not commute with coproducts. Even though this composition product is not a monoidal structure on symmetric sequences, it has enough properties to be able to define monoids (which are then operads on the underlying category) and make a bar construction. The main benefit of this work is in the dual setting, where it allows us to define a cosimplicial cobar construction for cooperads. Introduction If (C,∧, S) is a symmetric monoidal category in which ∧ commutes with finite coproducts (which happens, for example, when C is closed symmetric monoidal), then the composition product is a monoidal structure on the category of symmetric sequences in C. In this case, an operad is precisely a monoid with respect to this monoidal structure. If ∧ does not commute with finite coproducts, the composition product need not be associative (or even unital). Here we consider this case and show that, nonetheless, enough structure exists to be able to make sense of the notion of a ‘monoid’. We use the term ‘pseudomonoidal’ to refer to the relevant structure and we show that an operad in the underlying category C is precisely such a ‘monoid’ for the pseudomonoidal structure on the category of symmetric sequences in C. Our main motivation is in fact cooperads. A cooperad in C is equivalent to an operad in C (with the canonical symmetric monoidal structure on the opposite category). Although many interesting symmetric monoidal categories are closed symmetric monoidal (for example, compactly-generated topological spaces and the S-modules of EKMM), their opposite categories are rarely so. This is reflected in the fact that the monoidal product does not generally commute with finite products in these topological examples. In an algebraic setting, such as the category of modules over a commutative ring, finite products and coproducts are isomorphic and so the issue addressed in this note does not arise. The reason we are interested in viewing operads as monoids is that we can then use the standard machinery of the simplicial bar construction (in its reduced, one-sided and twosided forms). Here we show that there is an appropriate analogue of the bar construction for monoids in these pseudomonoidal categories. In particular, this means that there is in general a cosimplicial cobar construction for a cooperad in any symmetric monoidal category. Here is an outline of the note. In §1 we define pseudomonoidal structures. In §2 we show that the composition product forms part of a pseudomonoidal structure on the category of symmetric sequences. In §3 we say what a monoid for a pseudomonoidal structure is and show that an operad is precisely such an object for the composition product. Finally, in §4 we describe the simplicial bar construction in this setting. One thing to note: there is a close relationship between the ideas of this paper and the ‘functor operads’ of McClure-Smith [5, §4]. In fact, what we call a ‘pseudomonoidal structure’ would be more or less a ‘non-symmetric functor cooperad’ in the McClure-Smith