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We prove that the 2-category of closed categories of Eilenberg and Kelly is equivalent to a suitable full 2-subcategory of the 2-category of closed multicategories.

Forward Automatic Differentiation (AD) is a technique for augmenting programs to both perform their original calculation and also compute its directional derivative. The essence of Forward AD is to attach a derivative value to each number, and propagate these through the computation. When derivatives are nested, the distinct derivative calculations, and… (More)

- CLOSED MULTICATEGORIES, OLEKSANDR MANZYUK
- 2012

We prove that the 2-category of closed categories of Eilenberg and Kelly is equivalent to a suitable full 2-subcategory of the 2-category of closed multicategories.

We prove that three definitions of unitality for A ∞-categories suggested by the first author, by Kontsevich and Soibelman, and by Fukaya are equivalent.

- UNITAL A∞-CATEGORIES, VOLODYMYR LYUBASHENKO, OLEKSANDR MANZYUK
- 2008

Assuming that B is a full A ∞-subcategory of a unital A ∞-category C we construct the quotient unital A ∞-category D ='C/B'. It represents the A u ∞-2-functor A → A u ∞ (C, A) mod B , which associates with a given unital A ∞-category A the A ∞-category of unital A ∞-functors C → A, whose restriction to B is contractible. Namely, there is a unital A… (More)

We prove existence of equalizers in certain categories of cocomplete cocate-gories. This allows us to complete the proof of the fact that A ∞-functor categories arise as internal Hom-objects in the category of differential graded cocomplete augmented cocategories.

- FREE A∞-CATEGORIES, VOLODYMYR LYUBASHENKO, OLEKSANDR MANZYUK
- 2006

For a differential graded k-quiver Q we define the free A ∞-category FQ generated by Q. The main result is that the restriction A ∞-functor A ∞ (FQ, A) → A 1 (Q, A) is an equivalence, where objects of the last A ∞-category are morphisms of differential graded k-quivers Q → A. A ∞-categories defined by Fukaya [Fuk93] and Kontsevich [Kon95] are… (More)

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