BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Claudio Pisani DTSTART:20261106T181500Z DTEND:20261106T190000Z DTSTAMP:20261030T095227Z UID:ThirdVWDC/8 DESCRIPTION:Title: Unbiased Algebra\nby Claudio Pisani as part of Third Virtual Worksho p on Double Categories\n\n\nAbstract\nThe main trust of the unbiased appro ach to algebra is that operations on a set X form a category whose objects are (finite) families I → X and are indexed by functions I → J. Furth ermore\, operations can be reindexed along pullbacks in finite sets.\n\nTh is inevitably leads to the notion of unbiased symmetric multicategory [1]: a double category M with parallel sums\, along with a sum-preserving doub le functor M → Pb (to the double category of pullback squares in finite sets) which is a discrete fibration in the tight direction. The loose arro ws of M should be thought of as the abstract operations of the algebraic t heory\, while tight arrows and cells give reindexing.\n\nOperads are the s ingle-sorted theories\, namely those M with just an object over a terminal set. If S is a species of structure in the sense of Joyal\, an S-function is a function endowed with an S-structure on each of its fibers. For many basic species\, S-functions can be naturally composed\, so that they form the operations of an operad\; in fact all operads arise in this way.\n\nA n algebra for M is itself a symmetric multicategory A with a double functo r A → M which is a discrete opfibration in the loose direction. The loos e arrows of A are the (concrete) operations of the algebra\, indexed by th e abstract operations in M. For instance\, an algebra for the terminal ope rad Pb → Pb is (the unbiased form of) a commutative monoid X: a concret e operation evaluates a family I → X along a function I → J (the under laying abstract operation) to give a family J →X. The algebras for the o perad Tot\, associated to the species of total orders\, are plain monoids: operations evaluate families along functions with totally ordered fibers. \n\nIt is well-known that operads (or more generally symmetric multicatego ries) do not capture all kinds of ordinary algebraic theories since\, roug hly\, one cannot duplicate or delete variables. To this end one introduces cartesian multicategories\, which have a very neat definition in the pres ent context. Namely\, they are the algebras for a natural monad cart on sy mmetric multicategories: the loose arrows of cart M are spans (actually\, "enhanced" spans [1]) formed by a tight arrow and a loose arrow in M. Thes e are composed in the usual way for spans\, except that we use M-cells in place of pullbacks. Thus\, a cartesian structure cart M → M provides a w ay to "sum" an operation along a function to give a new operation. \n\nMul ticategories which are both cartesian and cocartesian (a.k.a. sequential [ 2]) amount to categories enriched in commutative monoids. In particular\, cartesian and cocartesian operads amount to rigs. In the operad associated to a rig R\, an operation α over a function I → J\, is a family I → R\, and composition is given by multiplication in the underlying monoid (a s in the discrete distribution monad). The sum of α along f : I → K is obtained by adding the elements in the fibers of f.\n\nAlgebras for a cart esian operad M are required to satisfy a further compatibility condition\, corresponding to the fact that they can be seen as morphisms of cart-alge bras from M to the endomorphism operad End(X). The algebras for the cartes ian operad associated to a rig R are the R-modules.\nFor an operad M\, the free M-algebra monad is given again by spans in M but\, in a sense\, dual ly to cart. If M is cartesian\, free M-algebras are obtained by quotientin g with respect to a natural equivalence on spans. Similarly\, given N → M\, the relatively free M-algebras are obtained by quotienting spans in M. \nWe also illustrate the connections between cartesian operads and Lawvere theories.\n\nThe overall purpose of the talk is to show how the unbiased approach\, by avoiding the awkwardness which inevitably arises from an unn atural skeletal indexing\, turns out to be rather effective and opens up n ew perspectives.\n\n[1] C. Pisani\, Unbiased multicategory theory\, Theory and Applications of Categories 44 (2025)\, 826–868.\n[2] C. Pisani\, Se quential multicategories\, Theory and Applications of Categories 29 (2014) \, 496–541.\n LOCATION:https://researchseminars.org/talk/ThirdVWDC/8/ END:VEVENT END:VCALENDAR