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Notions of `operad' and `multicategory' abound. This work provides a single framework in which many of these various notions can be expressed. Explicitly: given a monad * on a category S, we define the term `(S,*)-multicategory', subject to…
We show that a $\mathbb{P}$-object and simple configurations of $\mathbb{P}$-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient…
We study $\omega$-weak equivalences between weak $\omega$-categories in the sense of Batanin-Leinster. Our $\omega$-weak equivalences are strict $\omega$-functors satisfying essential surjectivity in every dimension, and when restricted to…
It is shown how double categories provide a direct abstract approach to coloured operads; namely, product-preserving normal lax functors from (Pb C)^op (the opposite of the double category of pullback squares in C) to Cat (the double…
We prove certain conjecture holds true for a finite category which has M\"obius inversion. The conjecture states a relationship between the zeta function of a finite category and the Euler characteristic of a finite category.
Many Properties of a category X, as for instance the existence of an adjoint or a factorization system, are a consequence of the cowellpoweredness of X. In the absence of cowellpoweredness, for general results, fairly strong assumption on…
We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in…
Since the time when the first optical instruments have been invented, an idea that the visible image of an object under observation depends on tools of observation became commonly assumed in physics. A way to formalize it in mathematics is…
In this paper, we prove that there is a canonical homotopy $(n+1)$-algebra structure on the shifted operadic deformation complex $Def(e_n\to\mathcal{P})[-n]$ for any operad $\mathcal{P}$ and a map of operads $f\colon e_n\to\mathcal{P}$.…
The operad of moulds is realized in terms of an operational calculus of formal integrals (continuous formal power series). This leads to many simplifications and to the discovery of various suboperads. In particular, we prove a conjecture…
A folklore result in category theory is that a (weakly) Cartesian closed category with finite co-products is distributive. Usually, the proof of this small result is carried on using the fact that the exponential functor is right adjoint to…
In this paper we present a new extension of the theory of well-bounded operators to cover operators with complex spectrum. In previous work a new concept of the class of absolutely continuous functions on a nonempty compact subset $\sigma$…
We use type-theoretic techniques to present an algebraic theory of $\infty$-categories with strict units. Starting with a known type-theoretic presentation of fully weak $\infty$-categories, in which terms denote valid operations, we extend…
We present a general construction of the derived category of an algebra over an operad and establish its invariance properties. A central role is played by the enveloping operad of an algebra over an operad.
An implicit operation of a class of similar algebras $\mathsf{K}$ is a collection of first order definable partial functions on the members of $\mathsf{K}$ that is globally preserved by homomorphisms. For instance, "taking inverses" can be…
Delta lenses are functors equipped with a suitable choice of lifts, generalising the notion of split opfibration. In recent work, delta lenses were characterised as the right class of an algebraic weak factorisation system. In this paper,…
It is a classical theorem that the free product of ordered groups is orderable. In this note we show that, using a method of G. Bergman, an ordering of the free product can be constructed in a functorial manner, in the category of ordered…
We define and study binary operations for homotopy groups with coefficients. We give conditions to prove that certain binary operations are the homomorphic image of the generalized Whitehead product. This allows carrying over properties of…
We prove that the category of algebras over a cofibrant operad admits a closed model category structure. This leads to the notion of "virtual operad algebra" - the algebra over a cofibrant resolution of the given operad. In particular,…
We show how an effect algebra $\mathcal{X}$ can be regarded as a category, where the morphisms $x \rightarrow y$ are the elements $f$ such that $x \leq f \leq y$. This gives an embedding $\mathbf{EA} \rightarrow \mathbf{Cat}$. The interval…