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Given a recollement $$ \mathbf{D}^0 \underset{\underset{i_R}\leftarrow}{\overset{\overset{i_L}\leftarrow}\to} \mathbf{D} \underset{\underset{q_R}\leftarrow}{\overset{\overset{q_L}\leftarrow}\to} \mathbf{D}^1 $$ the machinery exposed in BBD's Faisceaux Pervers (Astérisque 100), and in a more modern language in

Banagl, Markus. Topological invariants of stratified spaces. Springer Science & Business Media, 2007.

provides one with a $t$-structure on $\mathbf D$ obtained by "gluing par recollement" a $t$-structure $(\mathcal{D}_\ge^0, \mathcal{D}_<^0)$ on $\mathbf D^0$ and a $t$-structure $(\mathcal{D}_\ge^1, \mathcal{D}_<^1)$ on $\mathbf D^1$.

Edit: Apparently this construction deserves a more detailed explanation. When you have a t-structure $\mathcal{D}^i_{\ge}$ (identified with its left aisle) on $\mathbf{D}^i$ you can define the gluing $$ (\mathcal{D}^0_\ge\wr\mathcal{D}_\ge^1, \mathcal{D}^0_<\wr\mathcal{D}_<^1) $$ exploiting the functors $(i_L\dashv i\dashv i_R), (q_L\dashv q\dashv q_R)$: define $$\begin{gather} \{ X\in\mathbf{D}\mid qX\in \mathcal{D}_\ge^1, \; i_L X \in\mathcal{D}_\ge^0\}\\ \{ Y\in\mathbf{D}\mid qY \in\mathcal{D}_<^1,\; i_R Y \in\mathcal{D}_<^0 \} \end{gather}$$ It can be shown that these two classes form a new t-structure.

Now, [BBD] and Banagl book, starting from the classical geometric example of a stratification $\varnothing\subset U\subset X$ for a topological space, both insist on the fact that

By applying the gluing Theorem [i.e. the construction providing the glued $t$-structure] inductively, the notion of a p-perverse t-structure quickly generalizes to spaces with more than two strata

The possibility of doing this ultimately relies into the fact that gluing par recollement is an associative operation, namely (in some sense to be specified) if we denote the [left aisle of the] glued $t$-structure above as $\mathcal{D}^0_\ge\wr\mathcal{D}_\ge^1$, we have $$ (\mathcal{D}^0_\ge\wr\mathcal{D}_\ge^1)\wr \mathcal{D}^2_\ge = \mathcal{D}^0_\ge\wr(\mathcal{D}_\ge^1\wr \mathcal{D}^2_\ge). $$ In the geometric setting, I expect this result ultimately depend on the commutation of some diagrams of adjoint functors. Namely, the diagram

enter image description here

is commutative (in an obvious sense), where a squiggly arrow $f\colon \mathbf{D}\to \mathbf{D}'$ denotes a triple of adjoints $(f_L\dashv f\dashv f_R)$. In the geometric case, these maps are direct/inverse images, and the presence of a stratification $\varnothing U\subset V\subset X$ gives the desired compatibility.

Now,

What happens in the general setting of an "abstract" recollement in a generic triangulated (or better, stable) category? Is there any property of recollements ensuring that the glued $t$-structure $\mathcal{D}_0 \wr\dots \wr \mathcal{D}_n$ exists in a definite sense?

Note: this question is motivated by the same interest.

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  • $\begingroup$ Could you explain what gluing data do you have for your $D_i$? My guess it that $D_i$ should have "well-defined images" in the "glued category" $D$. $\endgroup$ Mar 16, 2015 at 19:50
  • $\begingroup$ The gluing arises as the classical construction described for example in Banagl's book. You have a recollement like the one above, $t$-structures $t_i$ on $\mathbf D_i$ and you define $\mathcal D_{\ge 0}^0\wr \mathcal{D}_{\ge 0}^1$ as $\{X\in\mathbf{D}\mid q(X)\in \mathcal{D}_{\ge 0}^1,\; i_L(X)\in \mathcal{D}_{\ge 0}^0\}$. Dually, $\mathcal D_{\le 0}^0\wr \mathcal{D}_{\le 0}^1 = (\mathcal D_{\ge 0}^0\wr \mathcal{D}_{\ge 0}^1)^\perp = \{Y\in\mathbf{D}\mid q(Y)\in \mathcal{D}_{\le 0}^1,\; i_R(X)\in \mathcal{D}_{\le 0}^0\}$. $\endgroup$
    – fosco
    Mar 16, 2015 at 20:04
  • $\begingroup$ [continued] Part of the problem is to establish, in the general case, what is the structure needed for the iterated gluing to make sense. Being the exposition (again in Banagl book) conducted in the full generality of triangulated categories, and since there is no mention of particular properties asked to the various $D_i$... I would expect (and I hope) the result is true in abstract situations, more general than the geometric case. What do you think? $\endgroup$
    – fosco
    Mar 16, 2015 at 20:16
  • $\begingroup$ Definitely no geometry is necessary here; yet could you describe the setting you consider in more detail? $\endgroup$ Mar 16, 2015 at 21:09
  • $\begingroup$ Sure. What is precisely the clarification you need? The geometric case suggests that for three $t$-structures $\mathcal{D}_\ge^i$, $i=0,1,2$ like above you need four distinct (but "compatible" in a sense) recollements. I can expand on this, if it's what you mean $\endgroup$
    – fosco
    Mar 16, 2015 at 21:14

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I didn't check that thoroughfully, but it seems that in the ("abstract") setting you are interested in there exist (exact) "projections" $j^{i*}:D\to D^i$ and $j^{i!}:D\to D^i$ ($D$ is the "big" triangulated category). Then the halves of the "glued $t$-structure" for $D$ are given by the following formulas: $D^{\le 0}=\{M\in Obj D:\ j^{i*}M\in D^{i, \le 0}\forall i\}$, $D^{\ge 0}=\{M\in Obj D:\ j^{!*}M\in D^{i, \ge 0}\forall i\}$. This certainly does not depend on the order on the gluing.

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