Related papers: The Category TOF
We study categories whose objects are the braid representations, i.e. strict monoidal functors $F\colon B\rightarrow Mat$ from the braid category $B$ to the category of matrices $Mat$. Braid representations are equivalent to solutions to…
We construct a monoidal version of Lurie's un/straightening equivalence. In more detail, for any symmetric monoidal $\infty$-category $\mathbf C$, we endow the $\infty$-category of coCartesian fibrations over $\mathbf C$ with a (naturally…
We consider the problem of the variational quantum circuit synthesis into a gate set consisting of the CNOT gate and arbitrary single-qubit (1q) gates with the primary target being the minimization of the CNOT count. First we note that…
We prove, over any base ring, that the infinity-category of strictly unital A-infinity-categories (and strictly unital functors) is equivalent to the infinity-category of unital A-infinity-categories (and unital functors). We also identify…
We detail a construction of a symmetric monoidal structure, called the reduced tensor product on the 2-category of braided tensor categories $\mathbf{BTC}(\mathcal{A})$ containing a fixed symmetric fusion subcategory $\mathcal{A}$. The…
In fairly elementary terms this paper presents, and expands upon, a recent result by Garner by which the notion of topologicity of a concrete functor is subsumed under the concept of total cocompleteness of enriched category theory.…
We consider commutative Frobenius pseudomonoids in the bicategory of spans, and we show that they are in correspondence with 2-Segal cosymmetric sets. Such a structure can be interpreted as a coherent 2-dimensional topological quantum field…
For a (possibly large) realized limit sketch $\mathcal{S}$ such that every $\mathcal{S}$-model is small in a suitable sense we show that the category of cocontinuous functors $\mathsf{Mod}(\mathcal{S}) \to \mathcal{C}$ into a cocomplete…
We analyse topological orbifold conformal field theories on the symmetric product of a complex surface M. By exploiting the mathematics literature we show that a canonical quotient of the operator ring has structure constants given by…
We develop a general theory of (extended) inner autoequivalences of objects of any 2-category, generalizing the theory of isotropy groups to the 2-categorical setting. We show how dense subcategories let one compute isotropy in the presence…
We define a class of non-compact Fano toric manifolds, called admissible toric manifolds, for which Floer theory and quantum cohomology are defined. The class includes Fano toric negative line bundles, and it allows blow-ups along fixed…
The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are…
The reversible implementation of classical functions accounts for the bulk of most known quantum algorithms. As a result, a number of reversible circuit constructions over the Clifford+$T$ gate set have been developed in recent years which…
This paper considers the possible underlying multicategories for a symmetric monoidal category, and shows that, up to canonical and coherent isomorphism, there really is only one. As a result, there is a well-defined forgetful functor from…
We prove a result of equivalence invariance of formal category theory for statements that can be expressed within an equipment. To do this, we exploit Henry and Bardomiano Mart\'inez's link between Makkai's FOLDS (first order logic with…
For a category B with finite products, we first characterize pseudofunctors from B to Cat whose corresponding opfibration is cartesian monoidal. Among those, we then characterize the ones which extend to pseudofunctors from internal groups…
The object of this paper is to prove that the standard categories in which homotopy theory is done, such as topological spaces, simplicial sets, chain complexes of abelian groups, and any of the various good models for spectra, are all…
We present two deterministic schemes for constructing a CNOT gate and a Toffoli gate on photon-atom and photon-atom-atom hybrid quantum systems assisted by bad cavities, respectively. They are achieved by cavity-assisted photon scattering…
A simple criterion for a functor to be finitary is presented: we call $F$ finitely bounded if for all objects $X$ every finitely generated subobject of $FX$ factorizes through the $F$-image of a finitely generated subobject of $X$. This is…
Categories of partial functions have become increasingly important principally because of their applications in theoretical computer science. In this note we prove that the category of partial bijections between sets as an…