3:12 AM
i'm trying to understand the most basic example of equivariant colimits (a la Bourbon Book), namely a "$C_2$-pushout" where $C_2$ fixes the source term and permutes the two target terms. more precisely, this is indexed by a $G$-category $\underline{I} \to O_{C_2}^{op} \simeq (BC_2)^\triangleleft$ whose fiber over the initial object is $pt$ (the source term) and whose fiber over each object of $BC_2$ is the span category.
i believe that the story should be the same for genuine $C_2$-spaces or genuine $C_2$-spectra, so i'll just write $\underline{S}_{C_2} \to O_{C_2}^{op}$ for either. then an $\underline{I}$-diagram is essentially the data of a genuine $C_2$-object $X \in S_{C_2}$ and a morphism $X^e \to Y$ in $S$. i'm thinking of this as presenting a span $Y \leftarrow X^e \to Y$.
i'm guessing that taking the $C_2$-colimit should commute with passing to underlying objects, i.e. i'm guessing that the correct answer is the pushout of $pt \leftarrow X \to Y \times (C_2/e)$ [or maybe i should write $Y \wedge (C_2/e)_+$ for the spectra case?] in $S_{C_2}$.
question 1: i'm having trouble seeing this rigorously, i.e. i'd like to understand why this construction is indeed the parametrized left adjoint $Fun_G(\underline{I},\underline{S}_{C_2}) \to Fun_G(O_{C_2}^{op},\underline{S}_{C_2})$ (or maybe i should write $\underline{Fun}$) . can somebody help me out with this?
question 2: what happens if i happen to have genuine $C_2$-structures on all the terms in the pushout? i'd expect this is coming from a special case of $\underline{I}$-diagrams in $\underline{S}_{C_2}$, namely $\underline{I}$-diagrams in $S_{C_2} \times O_{C_2}^{op}$. do i get any additional structure in this case?

4 hours later…
7:16 AM
For question 1: in general the pushout can be described as $X\amalg_{ind\,X} ind\,Y$. the trick is to consider the following diagram
$\require{AMScd} \begin{CD} X @<<< X @>>> X\\ @AAA @AAA @AAA\\ X @<<< \varnothing @>>> X\\ @VVV @VVV @VVV\\ Y @<<< \varnothing @>>> Y\end{CD}$
with the $C_2$-action reflecting along the vertical axis (so the indexing $C_2$-category is just $\underline{S}\times I$ where $I$ is the usual span category), and then take the colimit in the two possible orders
For question 2: you get no additional structure, for exactly the same reason that $-×C_2/e$ (which is equivalent to $ind\,res$) forgets all the structure except the underlying object
Here $ind$ is the left adjoint to the functor $res:C_{C_2}→C_e$, and (this is a theorem) it represents the indexed coproduct
@AaronMazel-Gee This formula seems to assume we are in an additive category or somesuch, doesn't it? Otherwise I don't even see how it is true for the usual pushouts

7:55 AM
I should probably also mention that $ind\,Y$ is not always best thought of as something like $Y×C_2$, since if the C_2-action on $C_e$ is non-trivial, its underlying object is more like $Y\amalg \sigma Y$. For example, for the $C_2$-category of complex vector spaces with the semilinear Galois action, if $V$ is a complex vector space $ind\,V$ is equivalent to the real vector space $(V\oplus \bar V)^{C_2}$

15 hours later…
10:45 PM
@DenisNardin awesome, thanks for the responses. unfortunately my chatjax hasn't been working in some time, i'll have to figure out how to get that back up and running. but so anyways, a few questions...
is that grid-diagram taking place in $C_2$-spaces? if so, how did you get it (and is there a systematic approach for doing so)? is there an "equivariant finality" result that proves that this gives the same equivariant colimit?
@DenisNardin can you explain more here please?
@DenisNardin just to be sure: here, are you writing $C_G$ for an arbitrary category $C$ (likely spaces or spectra in practice)?
@DenisNardin yes, i don't stand by my formula at all -- i wouldn't be surprised if it is wrong, or wrong in general
@DenisNardin i'm trying to understand this comment. i guess you're thinking in a more general context than "$G$-objects in $S$=Spaces or Spectra", and rather $C$ is any $G$-category at all? in the former case, i believe the restriction over the full subcategory of $O_G^{op}$ on $(G/e)$ is simply $BG \times S$, so the action is trivial. and you are warning me that in general i should be more careful?