I like category theory. I do category theory. I (usually) fail at category theory, but sometimes I write something. Give me something, I put functors in it. Address: London, Ontario, Canada
We exploit the equivalence between t-structures and normal torsion theories on stable infinity-ca... more We exploit the equivalence between t-structures and normal torsion theories on stable infinity-categories to unify two apparently separated constructions in the theory of triangulated categories: the characterization of bounded t-structures in terms of their hearts and semiorthogonal decompositions on triangulated categories. In the stable infinity-categorical context both notions stem from a single construction, the Postnikov tower of a morphism induced by a $\mathbb{Z}$-equivariant multiple (bireflective and normal) factorization system $\{\mathbb{F}_i\}_{i\in J}$. For J = $\mathbb{Z}$ with its obvious self-action, we recover the notion of Postnikov towers in a triangulated category endowed with a t-structure, and give a proof of the the abelianity of the heart in the infinity-stable setting. For J is a finite totally ordered set, we recover the theory of semiorthogonal decompositions.
The present note is a recollection of the most striking and useful applications of co/end calculu... more The present note is a recollection of the most striking and useful applications of co/end calculus. We put a considerable effort in making arguments and constructions rather explicit: after having given a series of preliminary definitions, we characterize co/ends as particular co/limits; then we derive a number of results directly from this characterization. The last sections discuss the most interesting examples where co/end calculus serves as a powerful abstract way to do explicit computations in diverse fields like Algebra, Algebraic Topology and Category Theory. The appendices serve to sketch a number of results in theories heavily relying on co/end calculus; the reader who dares to arrive at this point, being completely introduced to the mysteries of co/end fu, can regard basically every statement as a guided exercise.
]: Una categoria triangolata consta del dato di una coppia (T, S), dove T è una categoria additiv... more ]: Una categoria triangolata consta del dato di una coppia (T, S), dove T è una categoria additiva, ed S è un funtore di sospensione (spesso chiamato funtore di shift o traslazione), dove sia possibile isolare una famiglia (eventualmente una classe) di triangoli, detti triangoli distinti, caratterizzati dagli assiomi seguenti:
We exploit the equivalence between t-structures and normal torsion theories on stable ∞-categorie... more We exploit the equivalence between t-structures and normal torsion theories on stable ∞-categories to unify two apparently separated constructions in the theory of triangulated categories: the characterization of bounded t-structures in terms of their hearts and semiorthogonal decompositions on triangulated categories. In the stable ∞-categorical context both notions stem from a single construction, the Postnikov tower of a morphism induced by a Z-equivariant multiple (bireflective and normal) factorization system {F i } i∈J . For J = Z with its obvious self-action, we recover the notion of Postnikov towers in a triangulated category endowed with a t-structure, and give a proof of the the abelianity of the heart in the ∞-stable setting. For J is a finite totally ordered set, we recover the theory of semiorthogonal decompositions. References 24
We characterize t-structures in stable ∞-categories as suitable quasicategorical factorization sy... more We characterize t-structures in stable ∞-categories as suitable quasicategorical factorization systems. More precisely we show that a t-structure t on a stable ∞-category C is equivalent to a normal torsion theory F on C, i.e. to a factorization system F = (E, M) where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts.
We develop the theory of recollements in a stable ∞-categorical setting. In the axiomatization of... more We develop the theory of recollements in a stable ∞-categorical setting. In the axiomatization of Beȋlinson, Bernstein and Deligne, recollement situations provide a generalization of Grothendieck's "six functors" between derived categories. The adjointness relations between functors in a recollement D 0
We exploit the equivalence between t-structures and normal torsion theories on stable infinity-ca... more We exploit the equivalence between t-structures and normal torsion theories on stable infinity-categories to unify two apparently separated constructions in the theory of triangulated categories: the characterization of bounded t-structures in terms of their hearts and semiorthogonal decompositions on triangulated categories. In the stable infinity-categorical context both notions stem from a single construction, the Postnikov tower of a morphism induced by a $\mathbb{Z}$-equivariant multiple (bireflective and normal) factorization system $\{\mathbb{F}_i\}_{i\in J}$. For J = $\mathbb{Z}$ with its obvious self-action, we recover the notion of Postnikov towers in a triangulated category endowed with a t-structure, and give a proof of the the abelianity of the heart in the infinity-stable setting. For J is a finite totally ordered set, we recover the theory of semiorthogonal decompositions.
The present note is a recollection of the most striking and useful applications of co/end calculu... more The present note is a recollection of the most striking and useful applications of co/end calculus. We put a considerable effort in making arguments and constructions rather explicit: after having given a series of preliminary definitions, we characterize co/ends as particular co/limits; then we derive a number of results directly from this characterization. The last sections discuss the most interesting examples where co/end calculus serves as a powerful abstract way to do explicit computations in diverse fields like Algebra, Algebraic Topology and Category Theory. The appendices serve to sketch a number of results in theories heavily relying on co/end calculus; the reader who dares to arrive at this point, being completely introduced to the mysteries of co/end fu, can regard basically every statement as a guided exercise.
]: Una categoria triangolata consta del dato di una coppia (T, S), dove T è una categoria additiv... more ]: Una categoria triangolata consta del dato di una coppia (T, S), dove T è una categoria additiva, ed S è un funtore di sospensione (spesso chiamato funtore di shift o traslazione), dove sia possibile isolare una famiglia (eventualmente una classe) di triangoli, detti triangoli distinti, caratterizzati dagli assiomi seguenti:
We exploit the equivalence between t-structures and normal torsion theories on stable ∞-categorie... more We exploit the equivalence between t-structures and normal torsion theories on stable ∞-categories to unify two apparently separated constructions in the theory of triangulated categories: the characterization of bounded t-structures in terms of their hearts and semiorthogonal decompositions on triangulated categories. In the stable ∞-categorical context both notions stem from a single construction, the Postnikov tower of a morphism induced by a Z-equivariant multiple (bireflective and normal) factorization system {F i } i∈J . For J = Z with its obvious self-action, we recover the notion of Postnikov towers in a triangulated category endowed with a t-structure, and give a proof of the the abelianity of the heart in the ∞-stable setting. For J is a finite totally ordered set, we recover the theory of semiorthogonal decompositions. References 24
We characterize t-structures in stable ∞-categories as suitable quasicategorical factorization sy... more We characterize t-structures in stable ∞-categories as suitable quasicategorical factorization systems. More precisely we show that a t-structure t on a stable ∞-category C is equivalent to a normal torsion theory F on C, i.e. to a factorization system F = (E, M) where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts.
We develop the theory of recollements in a stable ∞-categorical setting. In the axiomatization of... more We develop the theory of recollements in a stable ∞-categorical setting. In the axiomatization of Beȋlinson, Bernstein and Deligne, recollement situations provide a generalization of Grothendieck's "six functors" between derived categories. The adjointness relations between functors in a recollement D 0
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