Abstract: We explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellative idempotent (in a 1-categorical sense). This notion is more economical than the usual notion in terms of left-right constraints, and is motivated by higher category theory. To start, we describe the semi-monoidal category of all possible unit structures on a given semi-monoidal category and observe that it is contractible (if nonempty). Then we prove that the two notions of units are equivalent in a strong functorial sense. Next, it is shown that the unit compatibility condition for a (strong) monoidal functor is precisely the condition for the functor to lift to the categories of units, and it is explained how the notion of Saavedra unit naturally leads to the equivalent non-algebraic notion of fair monoidal category, where the contractible multitude of units is considered as a whole instead of choosing one unit. To finish, the lax version of the unit comparison is considered. The paper is self-contained. All arguments are elementary, some of them of a certain beauty.
Abstract: A concrete computation — twelve slidings with sixteen tiles — reveals that certain commutativity phenomena occur in every double semigroup. This can be seen as a sort of Eckmann-Hilton argument, but it does not use units. The result implies in particular that all cancellative double semigroups and all inverse double semigroups are commutative. Stepping up one dimension, the result is used to prove that all strictly associative two-fold monoidal categories (with weak units) are degenerate symmetric. In particular, strictly associative one-object, one-arrow 3-groupoids (with weak units) cannot realise all simply-connected homotopy 3-types.
Abstract: We show that every braided monoidal category arises as End(I) for a weak unit I in an otherwise completely strict monoidal 2-category. This implies a version of Simpson's weak-unit conjecture in dimension 3, namely that one-object 3-groupoids that are strict in all respects, except that the object has only weak identity arrows, can model all connected, simply connected homotopy 3-types. The proof has a clear intuitive content and relies on a geometrical argument with string diagrams and configuration spaces.
Abstract: There are a dozen definitions of weak higher categories, all of which loosen the notion of composition of arrows. A new approach is presented here, where instead the notion of identity arrow is weakened — these are tentatively called fair categories. The approach is simplicial in spirit, but the usual simplicial category Δ is replaced by a certain `fat' delta of `coloured ordinals', where the degeneracy maps are only up to homotopy. The first part of this exposition is aimed at a broad mathematical readership and contains also a brief introduction to simplicial viewpoints on higher categories in general. It is explained how the definition of fair n-category is almost forced upon us by three standard ideas.
The second part states some basic
results about fair categories, and give examples.
The category of
fair 2-categories is shown to be equivalent to the category of
bicatgeories with strict
composition law. Fair 3-categories correspond to tricategories
with strict composision laws.
The main motivation for the theory is Simpson's
weak-unit conjecture according to which n-groupoids with strict
composition laws and weak units should model all homotopy n-types.
A proof of this conjecture in dimension 3 is announced,
obtained in joint work with A. Joyal.
Technical details and a fuller treatment of the applications will
appear elsewhere.
Abstract: We show that if (M,⊗,I) is a monoidal model category then REndM(I) is a (weak) 2-monoid in sSet. This applies in particular when M is the category of A-bimodules over a simplicial monoid A: the derived endomorphisms of A then form its Hochschild cohomology, which therefore becomes a simplicial 2-monoid.
Abstract: We describe a category of Feynman graphs and show how it relates to compact symmetric multicategories (coloured modular operads) just as linear orders relate to categories and rooted trees relate to multicategories. More specifically we obtain the following nerve theorem: compact symmetric multicategories can be characterised as presheaves on the category of Feynman graphs subject to a Segal condition. This text is a write-up of the second-named author's QPL6 talk; a more detailed account of this material will appear elsewhere.
Abstract: We define weak units in a semi-monoidal 2-category C as cancellable pseudo-idempotents: they are pairs (I,α) where I is an object such that tensoring with I from either side constitutes a biequivalence of C, and α: I ⊗ I → I is an equivalence in C. We show that this notion of weak unit has coherence built in: Theorem A: α has a canonical associator 2-cell, which automatically satisfies the pentagon equation. Theorem B: every morphism of weak units is automatically compatible with those associators. Theorem C: the 2-category of weak units is contractible if non-empty. Finally we show (Theorem E) that the notion of weak unit is equivalent to the notion obtained from the definition of tricategory: α alone induces the whole family of left and right maps (indexed by the objects), as well as the whole family of Kelly 2-cells (one for each pair of objects), satisfying the relevant coherence axioms.
Abstract: We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship with operads and other related notions is explored.
Abstract: We explore the relationship between polynomial functors and trees. In the first part we characterise trees as certain polynomial functors and obtain a completely formal but at the same time conceptual and explicit construction of two categories of rooted trees, whose main properties we describe in terms of some factorisation systems. The second category is the category Ω of Moerdijk and Weiss. Although the constructions are motivated and explained in terms of polynomial functors, they all amount to elementary manipulations with finite sets. Included in part 1 is also an explicit construction of the free monad on a polynomial endofunctor, given in terms of trees. In the second part we describe polynomial endofunctors and monads as structures built from trees, characterising the images of several nerve functors from polynomial endofunctors and monads into presheaves on categories of trees. Polynomial endofunctors and monads over a base are characterised by a sheaf condition on categories of decorated trees. In the absolute case, one further condition is needed, a projectivity condition, which serves also to characterise polynomial endofunctors and monads among (coloured) collections and operads.
Abstract: We give an elementary and direct combinatorial definition of opetopes in terms of trees, well-suited for graphical manipulation (e.g. drawings of opetopes of any dimension and basic operations like sources, target, and composition); a substantial part of the paper is constituted by drawings and example computations. To relate our definition to the classical definition, we recast the Baez-Dolan slice construction for operads in terms of polynomial monads: our opetopes appear naturally as types for polynomial monads obtained by iterating the Baez-Dolan construction, starting with the trivial monad. Finally we observe a suspension operation for opetopes, and define a notion of stable opetopes. Stable opetopes form a least fixpoint for the Baez-Dolan construction. The calculus of opetopes is also well-suited for machine implementation: in an appendix we show how to represent opetopes in XML, and manipulate them with simple Tcl scripts.
This is a study of the "fat delta", the basic shape category for
the theory of fair categories. This little category is very interesting
in itself, and it has many different descriptions, among which: as the
category of coloured (semi)-ordinals; as the
category of
epimorphisms in Δ; as the category of planar trees
of height 2; dual to the category of subdivided finite strict
intervals; as a Grothendieck construction or moduli space for all
coherent colourings of finite ordinals...