Let Sp be the category of symmetric spectra, regarded as having the stable model structure. We prove that the category of small categories enriched over Sp admits a model category structure. The method of proof applies to other categories than symmetric spectra as well, as long as these categories are linked to the category of simplicial sets via a certain strong monoidal Quillen pair.

We formulate an elementary condition on an involutive quantaloidQ under which there is a distributive law from the Cauchy completion monad over the symmetrisation comonad on the category of Q-enriched categories. For such quantaloids, which we call Cauchy-bilateral quantaloids, it follows that the Cauchy completion of any symmetric Q-enriched category is again symmetric. Examples include Lawver...

This paper continues the development of a simplicial theory of weak ω-categories, by studying categories which are enriched in weak complicial sets. These complicial Gray-categories generalise both the Kan complex enriched categories of homotopy theory and the 3-categorical Gray-categories of weak 3-category theory. We derive a simplicial nerve construction, which is closely related to Cordier ...

One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads [1] to this task. We present a general construction of a tensor product on the category of n-globular sets from any normalised (n + 1)-operad A, in such a way that the algebras for A ma...

Four requirements are suggested for an axiomatic system S to provide the foundations of category theory: (R1) S should allow us to construct the category of all structures of a given kind (without restriction), such as the category of all groups and the category of all categories; (R2) It should also allow us to construct the category of all functors between any two given categories including t...

Event sequence visualization aids analysts in many domains to better understand and infer new insights from event data. Analysing behaviour before or after a certain of interest is common task scenarios. In this paper, we introduce, formally define, position double trees as domain-agnostic tree approach for task. The shows the sequences that led on left, those followed right. Moreover, our enab...

The most familiar example of higher, or vertically iterated enrichment is that in the definition of strict n-category. We begin with strict n-categories based on a general symmetric monoidal category V. Motivation is offered through a comparison of the classical and extended versions of topological quantum field theory. A sequence of categorical types that filter the category of monoidal catego...

Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice.

Applying (enriched) categorical structures we define the notion of ordered sheaf on a quantaloid Q, which we call ‘Q-order’. This requires a theory of semicategories enriched in the quantaloid Q, that admit a suitable Cauchy completion. There is a quantaloid Idl(Q) of Q-orders and ideal relations, and a locally ordered category Ord(Q) of Q-orders and monotone maps; actually, Ord(Q) = Map(Idl(Q)...

Lyubashenko has described enriched 2–categories as categories enriched over V–Cat, the 2–category of categories enriched over a symmetric monoidal V. This construction is the strict analogue for V–functors in V–Cat of Brian Day’s probicategories for V–modules in V–Mod. Here I generalize the strict version to enriched n–categories for k–fold monoidal V. The latter is defined as by Balteanu, Fied...