A homotopy-theoretic universal property of Leinster's operad for weak omega-categories

Operations in Leinster's Weak $ω$-Category Operad

Batanin defines a weak $ω$-category as an algebra for a certain operad. Leinster refines this idea and defines the weak $ω$-category operad as the initial object of a category of "operads with contraction". We demonstrate how a higher category structure arises from this definition by explicitly constructing various composites, associativity and coherence laws, and an Eckmann-Hilton braiding.

Monad interleaving: a construction of the operad for Leinster's weak $ω$-categories

We show how to "interleave" the monad for operads and the monad for contractions on the category \coll of collections, to construct the monad for the operads-with-contraction of Leinster. We first decompose the adjunction for operads and the adjunction for contractions into a chain of adjunctions each of which acts on only one dimension of the underlying globular sets at a time. We then exhibit mutual stability conditions that enable us to alternate the dimension-by-dimension free functors. Hence we give an explicit construction of a left adjoint for the forgetful functor $\owc \lra \coll$, fr…

代数理论的$ω$-范畴化

The $ω$-categorification of Algebraic Theories

本文通过将Leinster的球状operad扩展为球状PRO并引入收缩结构，构造了一个monad，其代数为任意等式代数理论的弱$ω$-范畴化，其中初始弱化给出完全弱化的$ω$-范畴化。

Graded Leinster monoids and generalized Deligne conjecture for 1-monoidal abelian categories

In our recent paper [Sh1] a version of the "generalized Deligne conjecture" for abelian $n$-fold monoidal categories is proven. For $n=1$ this result says that, given an abelian monoidal $k$-linear category $\mathscr{A}$ with unit $e$, $k$ a field of characteristic 0, the dg vector space $\mathrm{RHom}_{\mathscr{A}}(e,e)$ is the first component of a Leinster 1-monoid in $\mathscr{A}lg(k)$ (provided a rather mild condition on the monoidal and the abelian structures in $\mathscr{A}$, called homotopy compatibility, is fulfilled). In the present paper, we introduce a new concept of a ${\it graded}…

Weak omega-categories from intensional type theory

We show that for any type in Martin-Löf Intensional Type Theory, the terms of that type and its higher identity types form a weak omega-category in the sense of Leinster. Precisely, we construct a contractible globular operad of definable composition laws, and give an action of this operad on the terms of any type and its identity types.

Comparing operadic theories of $n$-category

We give a framework for comparing on the one hand theories of n-categories that are weakly enriched operadically, and on the other hand n-categories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (Cheng-Gurski) and examples of the latter are the definition by Batanin and variants (Leinster). We will show how to take a theory of n-categories of the former kind and produce a globular operad whose algebras are the n-categories we started with. We first provide a generalisation of Trimble's original theory that allows for the…