arXiv subjects
Publications and source records attributed to Dominik Trnka.
We show that abelian categories can be characterized as structures consisting of a colax algebra and a lax algebra connected by a lax mixed rewriting rule. To this end we develop a theory of lax rewriting rules for pairs of lax-lax and colax-lax algebras over 2-monads.
We prove that a pointed category has kernels if and only if it is a lax algebra for the arrow 2-monad, and that this holds if and only if it is the d\'ecalage of a supercoherent structure. We will then interpret categories with kernels as the sought-after weak version of unary operadic categories.
We describe a Grothendieck construction for non-symmetric operads with values in categories, and hence in groupoids and posets. The construction produces a 2-category which is operadically fibered over the category D of finite non-empty ordinals and surjections. We describe an inverse for the construction, yielding an equivalence of constant-free non-symmetric categorical operads and operadic 2-categories (split-)fibered over D, which resembles the correspondence of categorical presheaves and fibered categories. The result provides a new characterization of non-symmetric categorical operads and tools to study them.
We introduce unary operadic 2-categories as a framework for operadic Grothendieck construction of a categorical O-operad, O being a unary operadic category. The construction is a fully faithful functor $\int_O$ which takes categorical O-operads to operadic functors over O, and we characterize its essential image by certain lifting properties. Such operadic functors are called operadic fibrations. Our theory is an extension of the discrete (unary) operadic case and, in some sense, of the classical Grothendieck construction of a categorical presheaf. For the terminal unary operadic category $\odot$, a categorical $\odot$-operad is a strict monoidal category V and its Grothendieck construction $\int_\odot V$ is connected to the `para' construction appearing in machine learning. The 2-categorical setting provides a characterization of O-operads valued in V as operadic functors $O \to \int_\odot V$. Last, we describe a left adjoint to $\int_\odot$.
We present a Markl-style definition of operads colored by a small category. In the presence of a unit these are equivalent to substitudes of Day and Street. We show that operads colored by a category are internal algebras of a certain categorical operad of functors. We describe a groupoid-colored quadratic binary operad, whose algebras are non-unital Markl operads in the context of operadic categories. As a by-product we describe the free internal operad construction.