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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeAug 23rd 2020
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexCreated page with idea and the only extant references I know of. <a href="https://ncatlab.org/nlab/revision/generalized+polycategory/1">v1</a>, <a href="https://ncatlab.org/nlab/show/generalized+polycategory">current</a>

   Created page with idea and the only extant references I know of.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorvarkor
   • CommentTimeOct 25th 2022
   • PermaLink
   Author: varkor
   Format: MarkdownItexAdd a reference to Burroni's D-categories. <a href="https://ncatlab.org/nlab/revision/diff/generalized+polycategory/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/generalized+polycategory/2">v2</a>, <a href="https://ncatlab.org/nlab/show/generalized+polycategory">current</a>

   Add a reference to Burroni’s D-categories.

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorvarkor
   • CommentTimeDec 2nd 2023
   • (edited Dec 2nd 2023)
   • PermaLink
   Author: varkor
   Format: MarkdownItexThere is a joint generalisation of the setting of Koslowski and that of [[structured cospans]] (or rather structured spans): instead of specifying two monads $S$ and $T$, one specifies endofunctors $S$, $T$, $U$, $V$ such that $UT \cong VS$, along with the distributive laws necessary to express the composite of two $(T, S)$-spans by applying $U$ to the left span and $V$ to the right span and taking a pullback. Considering monads in the resulting double category (or, if we don't require pullbacks, "co-virtual double category") of spans gives rise to a notion of generalised polycategory that appears to capture some interesting examples not captured by existing frameworks.

   There is a joint generalisation of the setting of Koslowski and that of structured cospans (or rather structured spans): instead of specifying two monads and , one specifies endofunctors , , , such that , along with the distributive laws necessary to express the composite of two -spans by applying to the left span and to the right span and taking a pullback. Considering monads in the resulting double category (or, if we don’t require pullbacks, “co-virtual double category”) of spans gives rise to a notion of generalised polycategory that appears to capture some interesting examples not captured by existing frameworks.