The intuition for multicategories There are many different species - PowerPoint PPT Presentation

The intuition for multicategories There are many different species of multicategory. A multicategory Unwirings and exponentiability for multicategories of a given species is like a category, but with the following tweaks (the exact details of which will depend on the species): ◮ The sources of the arrows are not just objects, they are Nathan Bowler combinations of objects (the manner of combination depends on the species of multicategory). The targets are still just objects. October 24, 2010 ◮ The classes of objects and arrows may have extra structure, which will also depend on the species. The definition of multicategories The associativity and identity laws Identity law: A species of multicategory is determined by a weak double a 1 a 1 a 1 a 0 a 0 a 0 a 0 Ta 0 Ta 0 Ta 0 Ta 0 category E and a monad T on that weak double category. An η a 0 η Ta 0 ⇓ ids ⇓ η a 1 ⇓ 1 ⇓ T ids T η a 0 ( E , T )-multicategory is given by 1 1 1 Ta 1 T 2 a 0 Ta 1 T 2 a 0 a 1 a 1 a 0 a 0 ◮ a 0-cell a 0 of E . Ta 0 Ta 0 = ⇓ 1 = 1 1 a 1 ◮ a horizontal 1-cell a 0 → Ta 0 of E . µ a 0 µ a 0 − 1 ⇓ comp 1 ⇓ comp ◮ 2-cells a 0 a 0 a 0 Ta 0 a 1 Ta 0 Ta 0 a 1 a 1 Ta 1 T 2 a 0 a 1 Associativity law: a 0 a 0 a 0 Ta 0 T 2 a 1 T 2 a 1 T 3 a 0 η a 0 µ a 0 Ta 1 T 2 a 0 ⇓ ids and ⇓ comp a 1 a 1 Ta 1 1 1 a 0 T 3 a 1 a 0 T 2 a 0 Ta 0 Ta 0 a 0 a 0 Ta 0 Ta 0 µ a 0 µ Ta 0 ⇓ µ a 1 T µ a 0 1 ⇓ comp 1 ⇓ 1 1 ⇓ T comp a 1 a 1 Ta 1 T 2 a 0 a 1 a 1 Ta 1 a 0 a 0 T 2 a 0 Ta 0 = Ta 0 satisfying certain conditions, called the associativity and identity µ a 0 µ a 0 ⇓ comp ⇓ comp 1 1 laws. a 0 a 0 Ta 0 Ta 0 a 1 a 1

A few species of multicategory Unwirable maps of algebras and unwirings Let T be a cartesian monad on a cartesian category C . A map f : ( A , α ) → ( B , β ) of T -algebras is unwirable iff Tf TB TA is a pullback. E T ( E , T )-multicategories β α Span ( Set ) identity categories B A Span ( Set ) list plain multicategories f Dropping the requirement that B have a T -algebra structure, an Span ( Gph ) path virtual double categories unwiring of ( B , f ) is a map ν : B × A TA → TB making the Span ( C / C 0 ) T / C ( Span ( C ) , T )-multicategories over C following diagrams commute in C : ultrafilter topological spaces Rel B × A η A B × A TA ν TA T ν B × A T 2 A T ( B × A TA ) T 2 B B ν η B µ B B × A µ A π ′ B × A TA TB TA TB ν Tf Unwirings which are isomorphisms correspond to unwirable maps of T -algebras. Unwirings as distributive laws Unwirings as exponentiability-lifters g For any object C → A of C / A , pulling back ν along − T π : T ( B × A C ) → TB gives a diagram l ( ν ) g T ( B × A C ) For any unwiring ν of B , l ( ν ) gives B × A − the structure of a T π ′ B × A TC TC cartesian colax map of monads from T / A to itself. Suppose that f B × A Tg Tg T π is exponentiable as a map in C . Let m ( ν ) be the mate of l ( ν ) with respect to the adjuction B × A − ⊣ ( − B ) A . Then (( − B ) A , m ( ν )) is B × A TA TB TA . ν Tf a lax map of monads from ( T / A ) to itself. Thus the functor ( − B ) A lifts to an endomorphism of the category T -Alg / ( A , α ) of Since the maps l ( ν ) g are formed in this way by pullback, they ( T / A )-algebras. In particular, any unwirable map of T -algebras is collectively form a cartesian natural transformation exponentiable. l ( ν ): B × A T − → T ( B × A − ). For any unwiring ν of B as above, l ( ν ) is a distributive law of the comonad B × A − over T / A . This gives a correspondence between unwirings ν of B and cartesian distributive laws of B × A − over T / A .

Unwirability for multicategories in the sense of Leinster Cartesian 2-cells in double categories Suppose that in some double category we have a 2-cell Now let T be a suitable monad (in the sense of Leinster) on a m B ′ cartesian category C . So we have a ‘free T -multicategory’ monad B T + on C -Gph . A map f of T -multicategories is unwirable iff the g ⇓ θ g ′ squares C ′ . C n comp B 1 ids B 0 B 1 B 1 ◦ B 1 We say θ is cartesian iff for any f : A → B and f ′ : A ′ → B ′ , any and f 0 f 1 f 1 ∗ f 1 f 1 other 2-cell A 0 A 1 A 1 ◦ A 1 comp A 1 l ids A ′ A l are both pullbacks. In fact, if the second of these squares is a A ′ ⇓ b A f f ′ φ factors through θ pullback then the first must also be. m uniquely as B ′ g · f ⇓ φ g ′ · f ′ B A T -multicategory is unwirable iff the unique map ! from it to the C ′ g terminal T -multicategory is unwirable. C ⇓ θ g ′ n C ′ . C n Translating from Leinster’s world to Cruttwell and Cartesian cells in Span ( Set ) Shulman’s For example, a 2-cell The unit map of the terminal T -multicategory is the component of η at 1, and so in this case the first pullback square in the definition r of an unwirable T -multicategory factors as p q h 1 ! u b ′ a 0 a 0 1 b s t η a 0 g η 1 g ′ ids a 1 c c ′ Ta 0 T 1 . s T ! in Span ( Set ) is cartesian iff r is the limit of the diagram a 0 u b ′ b The right hand square is a 1 1 ids s t pullback, so the whole thing is a g g ′ is cartesian. a 0 a 1 a 0 pullback iff the left hand square c c ′ is, that is iff t s η a 0 1 with p , h and q being legs of the limit cone a 0 Ta 0

Normalised and unwirable multicategories Normalised and unwirable topological spaces Definition A multicategory is normalised iff the identity 2-cell is cartesian. A topological space is normalised iff it is T 1 . Definition A normalised multicategory is unwirable iff the composition 2-cell A T 1 topological space X is unwirable iff for any point x in X and is also cartesian. any open neighbourhood U of x there is another open A correspondence between kinds of T -algebra in the horizontal neighbourhood V of x such that any open cover of U has a finite bicategory and these structures subset covering V . Such a topological space is called quasi locally lax multicategories compact . The quasi locally compact spaces are precisely the weak unwirable multicategories exponentiable objects of Top . colax unwirings References Conjecture Nathan Bowler. For sufficiently friendly species of multicategory, amongst the A unified approach to the construction of categories of games . normalised multicategories it is precisely the unwirable ones which PhD thesis, University of Cambridge, 2010. are exponentiable. G. S. H. Cruttwell and Michael A. Shulman. This is more useful than it appears because A unified framework for generalized multicategories, 2009. Theorem Tom Leinster. For every species of multicategory we can construct another Higher Operads, Higher Categories . species so that the multicategories of the first species are exactly Number 298 in London Mathematical Society Lecture Note the normalised multicategories of the second. Series. Cambridge University Press, 2004.