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The non-commutative sequoid operator $\oslash$ on games was introduced tocapture algebraically the presence of state in history-sensitive strategies ingame semantics, by imposing a causality relation on the tensor product ofgames. Coalgebras for the functor $A \oslash \_$ - i.e. morphisms from $S$ to$A \oslash S$ - may be viewed as state transformers: if $A \oslash \_$ has afinal coalgebra, $!A$, then the anamorphism of such a state transformerencapsulates its explicit state, so that it is shared only between successiveinvocations.  We study the conditions under which a final coalgebra $!A$ for $A \oslash \_$is the carrier of a cofree commutative comonoid on $A$. That is, it is a modelof the exponential of linear logic in which we can construct imperative objectssuch as reference cells coalgebraically, in a game semantics setting. We showthat if the tensor decomposes into the sequoid, the final coalgebra $!A$ may beendowed with the structure of the cofree commutative comonoid if there is anatural isomorphism from $!(A \times B)$ to $!A \otimes !B$. This condition isalways satisfied if $!A$ is the bifree algebra for $A \oslash \_$, but ingeneral it is necessary to impose it, as we establish by giving an example of asequoidally decomposable category of games in which plays will be allowed tohave transfinite length. In this category, the final coalgebra for the functor$A \oslash \_$ is not the cofree commutative comonoid over A: we illustratethis by explicitly contrasting the final sequence for the functor $A \oslash\_$ with the chain of symmetric tensor powers used in the construction of thecofree commutative comonoid as a limit by Melli\'es, Tabareau and Tasson.

Sequoidal Categories and Transfinite Games: A Coalgebraic Approach to Stateful Objects in Game Semantics