Related papers: On some classes of lazy cocycles and categorical s…
Cartesian reverse differential categories (CRDCs) are a recently defined structure which categorically model the reverse differentiation operations used in supervised learning. Here we define a related structure called a monoidal reverse…
We survey some basic mathematical structures, which arguably are more primitive than the structures taught at school. These structures are orders, with or without composition, and (symmetric) monoidal categories. We list several `real life'…
We prove that for certain monoidal (Quillen) model categories, the category of comonoids therein also admits a model structure.
In this paper, we classify finite categories with two objects such that one of the endomorphism monoids is a group. We prove that having a group on one side affects the structure of the other endomorphism monoid, and we prove that it is…
In this survey article, we review some conceptual approaches to the cyclic category $\Lambda$, as well as its description as a crossed simplicial group. We then give a new proof of the model structure on cyclic sets, work through the…
In this article we investigate which categorical structures of a category C are inherited by its arrow category. In particular, we show that a monoidal equivalence between two categories gives rise to a monoidal equivalence between their…
A compact closed bicategory is a symmetric monoidal bicategory where every object is equipped with a weak dual. The unit and counit satisfy the usual "zig-zag" identities of a compact closed category only up to natural isomorphism, and the…
In the paper "Triangulations, orientals, and skew monoidal categories", the free monoidal category Fsk on a single generating object was described. We sharpen this by giving a completely explicit description of Fsk, and so of the free skew…
Monoids and groupoids are examples of poloids. On the one hand, poloids can be regarded as one-sorted categories; on the other hand, poloids can be represented by partial magmas of partial transformations. In this article, poloids are…
We classify certain categories of partitions of finite sets subject to specific rules on the colorization of points and the sizes of blocks. More precisely, we consider pair partitions such that each block contains exactly one white and one…
We offer two proofs that categories weakly enriched over symmetric monoidal categories can be strictified to categories enriched in permutative categories. This is a "many 0-cells" version of the strictification of bimonoidal categories to…
Within a subclass of monoids (with zero) a structural characterization is given of those that are associated to topologically transitive subshifts with Property (A).
We identify a categorical structure of the set of all CFTs. In particular, we show that the set of all CFTs has a natural monoidal strict $2$-category structure with the $1$-morphisms being sequences of deformations and $2$-morphisms…
Given an additive equational category with a closed symmetric monoidal structure and a potential dualizing object, we find sufficient conditions that the category of topological objects over that category has a good notion of full…
We define the category of tidy symmetric multicategories. We construct for each tidy symmetric multicategory Q a cartesian monad (E_Q,T_Q) and extend this assignation to a functor. We exhibit a relationship between the slice construction on…
We classify primitive, rank 1, omega-categorical structures having polynomially many types over finite sets. For a fixed number of 4-types, we show that there are only finitely many such structures and that all are built out of finitely…
Recall that the definition of the $K$-theory of an object C (e.g., a ring or a space) has the following pattern. One first associates to the object C a category A_C that has a suitable structure (exact, Waldhausen, symmetric monoidal, ...).…
We decribe the correspondence between normalised $\omega$-operads and certain lax monoidal structures on the category of globular sets. As with ordinary monoidal categories, one has a notion of category enriched in a lax monoidal category.…
We define the Hochschild complex and cohomology of a ring object in a monoidal category enriched over abelian groups. We interpret the cohomology groups and prove that the cohomology ring is graded-commutative.
In this paper we study the categories of braided categorical associative algebras and braided crossed modules of associative algebras and we relate these structures with the categories of braided categorical Lie algebras and braided crossed…