- Univalence in locally cartesian
closed ∞-categories

With David Gepner. ArXiv:1208.1749**Abstract**: In the setting of presentable locally cartesian closed ∞-categories, we show that univalent families, in the sense of Voevodsky, form a poset isomorphic to the poset of bounded local classes, in the sense of Lurie. It follows that every ∞-topos has a hierarchy of "universal" univalent families, indexed by regular cardinals, and that n-topoi have univalent families classifying (n-2)-truncated maps. We show that univalent families are preserved (and detected) by right adjoints to locally cartesian localizations, and use this to exhibit certain canonical univalent families in ∞-quasitopoi (∞-categories of "separated presheaves"). We also exhibit some more exotic examples of univalent families, illustrating that a univalent family in an n-topos need not be (n-2)-truncated, as well as some univalent families in the Morel-Voevodsky ∞-category of motivic spaces, an instance of a locally cartesian closed ∞-category which is not an n-topos for any 0 ≤ n ≤ ∞. Lastly, we show that any presentable locally cartesian closed ∞-category is modeled by a type-theoretic model category, and conversely that the ∞-category underlying a type-theoretic model category is presentable and locally cartesian closed; moreover, univalent families in presentable locally cartesian closed ∞-categories correspond precisely to univalent fibrations in type-theoretic model categories. - Groupoids and Faà di Bruno formulae for Green functions in bialgebras of trees

With Imma Gálvez and Andy Tonks. ArXiv:1207.6404**Abstract**: We prove a Faà di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids. For suitable choices of P, the result implies also formulae for Green functions in bialgebras of graphs.

- Polynomial functors and polynomial monads

With Nicola Gambino.*Math. Proc. Cambridge Phil. Soc.***154**(2013), 153-192. ArXiv:0906.4931**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. - Categorification of Hopf algebras
of rooted trees
*Centr. Eur. J. Math.***11**(2013), 401-422. ArXiv:1109.5785**Abstract**: We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec N) whose semiring of functions is (a P-version of) the Connes—Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to Z and collapsing H_{0}). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring in the polynomial representation of the free monad on P. - Coherence for weak units

With André Joyal.*Documenta Math.***18**(2013), 71--110. ArXiv:0907.4553**Abstract**: We define weak units in a semi-monoidal 2-categoryas cancellable pseudo-idempotents: they are pairs (*C**I*,α) where*I*is an object such that tensoring with*I*from either side constitutes a biequivalence of, and α:*C**I*⊗*I*→*I*is an equivalence in. 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.*C* -
Local fibered right adjoints are polynomial

With Anders Kock.*Math. Struct. Comput. Sci.***23**(2013), 131-141. ArXiv:1005.4236**Abstract**: For any locally cartesian closed category E, we prove that a local fibered right adjoint between slices of E is given by a polynomial. The slices in question are taken in a well known fibered sense. - Double adjunctions and free monads

By Thomas Fiore, Nicola Gambino, and Joachim Kock.

Cahiers Topol. Géom. Différ. Catég.**53**(2012), 242-307. ArXiv:1105.6206**Abstract**: We characterize double adjunctions in terms of presheaves and universal squares, and then apply these characterizations to free monads and Eilenberg-Moore objects in double categories. We improve upon our earlier result in [Fiore-Gambino-Kock] to conclude: if a double category with cofolding admits the construction of free monads in its horizontal 2-category, then it also admits the construction of free monads as a double category. We also prove that a double category admits Eilenberg--Moore objects if and only if a certain parameterized presheaf is representable. Along the way, we develop parameterized presheaves on double categories and prove a double-categorical Yoneda Lemma. -
Feynman graphs, and nerve theorem for
compact symmetric multicategories (extended abstract)
With André Joyal.

Proceedings of the 6th International Workshop on Quantum Physics and Logic (Oxford 2009),*Electronic Notes in Theoretical Computer Science***270**(2) (2011), 105-113. ArXiv:0908.2675**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. -
Monads in double categories

With Tom Fiore and Nicola Gambino.

*J. Pure Appl. Alg.***215**(2011), 1174-1197. ArXiv:1006.0797**Abstract**: We extend the basic concepts of Street's formal theory of monads from the setting of 2-categories to that of double categories. In particular, we introduce the double category Mnd(C) of monads in a double category C and define what it means for a double category to admit the construction of free monads. Our main theorem shows that, under some mild conditions, a double category that is a framed bicategory admits the construction of free monads if its horizontal 2-category does. We apply this result to obtain double adjunctions which extend the adjunction between graphs and categories and the adjunction between polynomial endofunctors and polynomial monads. - Polynomial functors and trees

*Int. Math. Res. Not.***2011**(2011), 609-673. ArXiv:0807.2874**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. - Polynomial functors and opetopes

With André Joyal, Michael Batanin, and Jean-François Mascari.

*Adv. Math.***224**(2010), 2690-2737. ArXiv:0706.1033**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. - Elementary remarks on units in monoidal
categories

*Math. Proc. Cambridge Phil. Soc.***144**(2008), 53-76.

ArXiv:math.CT/0507349**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. - Note on commutativity in
double semigroups and two-fold monoidal categories

Mac Lane Memorial Volume,*J. Homotopy Rel. Struct.*,**2**(2007), 217-228. ArXiv:math.CT/0608452**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. - Weak units and homotopy 3-types
(with André Joyal)

*Street Festschriff: Categories in algebra, geometry and mathematical physics*, Contemp. Math**431**(2007), 257-276. ArXiv:math.CT/0602084

**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. - Weak
identity arrows in higher categories

*Internat. Math. Res. Papers*, vol. 2006, 1-54. ArXiv:math.CT/0507116.

**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.

- Simplicial
localisation of monoidal structures,
and a non-linear version of Deligne's conjecture
With Bertrand Toën.

*Compositio Math.*,**141**(2005), 253-261. ArXiv:math.AT/0304442**Abstract**: We show that if (*M*,⊗,*I*) is a monoidal model category then REnd_{M}(*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.

- Polynomial functors

Rough draft, 420pp. (The linked page has more info.)

- Notes on
Tamsamani categories, Segal categories, and applications.

(Notes from two talks of Bertrand Toën in Minneapolis, 2004.) - Spectra, supports, and Hochster
duality

(Letter to Giordano Favi and Paul Balmer, December 2007. 12 pages.)

Last updated: 2013-03-12 by Joachim Kock.