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Discovery Projects - Grant ID: DP130101172

Discovery Projects - Grant ID: DP130101172
ARC | Discovery Projects
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  • Internal algebra classifiers as codescent objects of crossed internal categories

    Inspired by recent work of Batanin and Berger on the homotopy theory of operads, a general monad-theoretic context for speaking about structures within structures is presented, and the problem of constructing the universal ambient structure containing the prescribed internal structure is studied. Following the work of Lack, these universal objects must be constructed from simplicial objects arising from our monad-theoretic framework, as certain 2-categorical colimits called codescent objects....

    Algebraic Kan extensions along morphisms of internal algebra classifiers

    An algebraic left Kan extension is a left Kan extension which interacts well with the algebraic structure present in the given situation, and these appear in various subjects such as the homotopy theory of operads and in the study of conformal field theories. In the most interesting examples, the functor along which we left Kan extend goes between categories that enjoy universal properties which express the meaning of the calculation we are trying to understand. These universal properties say...

    Operads as polynomial 2-monads

    In this article we give a construction of a polynomial 2-monad from an operad and describe the algebras of the 2-monads which then arise. This construction is different from the standard construction of a monad from an operad in that the algebras of our associated 2-monad are the categorified algebras of the original operad. Moreover it enables us to characterise operads as categorical polynomial monads in a canonical way. This point of view reveals categorical polynomial monads as a unifying...

    Fa\`a di Bruno for operads and internal algebras

    Kock, Joachim; Weber, Mark (2016)
    Projects: ARC | Discovery Projects - Grant ID: DP130101172 (DP130101172)
    For any coloured operad R, we prove a Fa\`a di Bruno formula for the `connected Green function' in the incidence bialgebra of R. This generalises on one hand the classical Fa\`a di Bruno formula (dual to composition of power series), corresponding to the case where R is the terminal reduced operad, and on the other hand the Faa di Bruno formula for P-trees of G\'alvez--Kock--Tonks (P a finitary polynomial endofunctor), which corresponds to the case where R is the free operad on P. Following G...

    Regular patterns, substitudes, Feynman categories and operads

    We show that the regular patterns of Getzler (2009) form a 2-category biequivalent to the 2-category of substitudes of Day and Street (2003), and that the Feynman categories of Kaufmann and Ward (2013) form a 2-category biequivalent to the 2-category of coloured operads (with invertible 2-cells). These biequivalences induce equivalences between the corresponding categories of algebras. There are three main ingredients in establishing these biequivalences. The first is a strictification theore...

    Homotopy theory for algebras over polynomial monads

    We study the existence and left properness of transferred model structures for "monoid-like" objects in monoidal model categories. These include genuine monoids, but also all kinds of operads as for instance symmetric, cyclic, modular, higher operads, properads and PROP's. All these structures can be realised as algebras over polynomial monads. We give a general condition for a polynomial monad which ensures the existence and (relative) left properness of a transferred model structure for its...
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