# ct.category theory – Invert a suspension object in a stable monoidal category

Suppose we are given a stable symmetrical full closed monoidal $$( infty, 1)$$-Category $$mathcal {C}$$ with suspension $$Sigma$$and let $$X in mathcal {C}$$ to be dualisable. I would like to create a new stable monoidal symmetric cocomplete category $$mathcal {C}[Sigma X^{-1}]$$ at the same time as a symmetrical and symmetric monoidal functor $$mathcal {C} to mathcal {C}[Sigma X^{-1}]$$ with the following property: For any symmetric monoidal stable category $$mathcal {D}$$, the induced functor $$Fun_ {ex, cont} ^ { otimes} ( mathcal {C}[Sigma X^{-1}], mathcal {D}) to Fun_ {ex, cont} ^ { otimes} ( mathcal {C}, mathcal {D})$$ induces equivalence of $$Fun_ {ex, cont} ^ { otimes} ( mathcal {C}[Sigma X^{-1}], mathcal {D})$$ with the full subcategory of $$Fun_ {ex, cont} ^ { otimes} ( mathcal {C}, mathcal {D})$$ consisting of functors $$mathcal {C} to mathcal {D}$$ who send $$Sigma X$$ to an invertible object in $$mathcal {D}$$. Here the notation $$Fun_ {ex, cont} ^ { otimes} ( mathcal {C}, mathcal {D})$$ refers to the category of monoidal functors that are both accurate and continuous.

My idea was to locate the card game $$( Sigma ( iota)) otimes id_Y$$, or $$iota$$ is the map $$X otimes X ^ * to mathbb {1}$$ derived from duality data for $$X$$. Right here $$Y$$ extends on all objects in $$mathcal {C}$$. However, I do not know if the resulting category would meet all the requirements listed above.

EDIT: Invert $$Sigma X$$ is clearly equivalent to reverse $$X$$.