symmetric monoidal (∞,1)-category of spectra
Let $M$ be a commutative monoid (in the category $\mathbf{Sets}$). A $M$-graded monoid $\Phi$ in a symmetric monoidal category $\mathcal{V}$ with unit object $I$ is the data of
such that the obvious associativity and unit axioms hold.
Thus, a $M$-graded monoid is in particular a $M$-graded object. In fact, a $M$-graded monoid is just a monoid in the monoidal category of $M$-graded objects of $\mathcal{V}$, hence a lax monoidal functor $M \to \mathcal{V}$ by this proposition (where $M$ is viewed as a discrete monoidal category).
We say that a $M$-graded monoid is connected if
If $\Phi$ is a $M$-graded monoid in $\mathcal{V}$, then $(\Phi_{0},\nabla_{0,0},\eta)$ is a monoid in $\mathcal{V}$.
If $\mathcal{V}$ is a symmetric monoidal category which is also a enriched over pointed sets, and if $(A,\nabla,\eta)$ is a monoid in $\mathcal{V}$, then we get a $\mathbb{N}$-graded monoid $\Phi$ by defining:
The two examples above in the category of groups are connected (by considering that $\Sigma_{0} = 1$).
Last revised on April 20, 2023 at 17:49:23. See the history of this page for a list of all contributions to it.