Let $X$ be a nice space, and view it as an $\infty$-groupoid via its singular simplicial set. Consider the constant functor $\mathbb{S}$ valued functor to Spectra, mapping all simplices to the sphere spectrum.
I would like to understand how to compute the (co)limits of this functor. I'm not very familiar with this language, especially on a workable level, so this question may have serious conceptual errors. If I understand the theory correctly however, the limit should output $X$ (or maybe $X_+$?) as a suspension spectrum, but I'm not sure about the colimit.
To compute the (co)limit, my understanding is that we need to resolve our indexing diagram, then the can take an ordinary (co)limit. The examples of homotopy (co)limits I am comfortable with however are all quite simple indexing categories, so I'm not sure how one ought to actually do this.
As a precise question, let $X$ be the circle $S^1$, how would one compute these (co)limits in this instance? Can this be seen more easily if we use a triangulation/CW decomposition of the circle? Where does the extra basepoint come from, from a formal perspective?
First, we need to understand colimits of diagrams of shape $\mathcal{X}$ with value $\Delta^0$ in the (∞, 1)-category $\mathbf{S}$ of spaces. Fortunately the answer is well known: the colimit is just the ∞-groupoid completion $\left| \mathcal{X} \right|$ of $\mathcal{X}$, i.e. the localisation $\mathcal{X} [\mathcal{X}^{-1}]$. (If you are working with simplicial sets and $\mathcal{X}$ is a quasicategory, then this is just $\mathcal{X}$ again but now regarded as an object in the Kan–Quillen model structure rather than the Joyal model structure.)
Now the general case follows by abstract nonsense. Given an (∞, 1)-category $\mathcal{C}$ and an object $A$ in $\mathcal{C}$, we have $$\begin{aligned} \textstyle \mathcal{C} (C, \varprojlim_\mathcal{X} A) & \cong \textstyle \varprojlim_\mathcal{X} \mathcal{C} (C, A) \\ & \cong \textstyle \varprojlim_\mathcal{X} \mathbf{S} (\Delta^0, \mathcal{C} (C, A)) \\ & \cong \textstyle \mathbf{S} (\varinjlim_{\mathcal{X}^\textrm{op}} \Delta^0, \mathcal{C} (C, A)) \\ & \cong \mathcal{C} (C, A^{\left| \mathcal{X}^\textrm{op} \right|}) \\ \end{aligned}$$ in the sense that $\varprojlim_\mathcal{X} A$ exists if and only if the power (= cotensor) $A^{\left| \mathcal{X}^\textrm{op} \right|}$ exists, and they are equivalent. Dually, $\varinjlim_\mathcal{X} A$ exists if and only if the copower (= tensor) $\left| \mathcal{X} \right| \cdot A$ exists.
For the specific case where $\mathcal{C}$ is the (∞, 1)-category $\textbf{Sp}$ of spectra and $A$ is the sphere spectrum $\mathbb{S}$, $\varinjlim_\mathcal{X} \mathbb{S}$ is the stabilisation $\Sigma^\infty \left| \mathcal{X} \right|_+$ and $\varprojlim_\mathcal{X} \mathbb{S}$ is the mapping spectrum $F (\Sigma^\infty \left| \mathcal{X}^\textrm{op} \right|_+, \mathbb{S})$. (If it is not obvious why we have to add a basepoint, consider the case where $\mathcal{X} = \emptyset$, and then the case where $\mathcal{X}$ is contractible.)