Several [sources](https://www.google.it/search?client=opera&q="left+t-structure"&sourceid=opera&ie=UTF-8&oe=UTF-8) I see speak of a "left t-structure", but lack a precise definition. Where can I find a reference for this?



Left and right $t$-structures
When you try to put a t-structure, say, on the homotopy category of complexes $H(\mathcal{A})$, where $\mathcal{A}$ is "almost an abelian category" (more than additive, but the coimage must not be isomorphic to the image; I do not remember all the assumptions, but they are easy to find), there are two ways to do that. The first way to consider the subcategory $H^{\leq0}(\mathcal{A})$ of complexes split in positive degrees (make sure to define correctly what split is) and proceed with the usual construction. This is how you get the left one. Now, you can repeat this construction for the opposite category $\mathcal{A}^{op}$ and look at the left t-structure on $H(\mathcal{A}^{op})$. Under the equivalence $H(\mathcal{A})^{op}\simeq H(\mathcal{A}^{op})$ you get a second t-structure on $H(\mathcal{A})$, the right one. If you start with an abelian category, these will be the same. However, if you category is "almost abelian", these might be different (one can describe the hearts of both explicitly).


As for references, the key words are *Schneiders* and *quasi-abelian categories*.


