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For a category $mathbb{C}$, a small category $mathbb{I}$, and a pre-cover relation $sqsubset$ on $mathbb C$ we prove, under certain completeness assumptions on $mathbb C$, that a morphism $g: Bto C$ in the functor category $mathbb {C}^{mathbb I}$ admits an image with respect to the pre-cover relation on $mathbb C^{mathbb I}$ induced by $sqsubset$ as soon as each component of $g$ admits an image with respect to $sqsubset$. We then apply this to show that if a pointed category $mathbb{C}$ is: (i) algebraically cartesian closed; (ii) exact protomodular and action accessible; or (iii) admits normalizers, then the same is true of each functor category $mathbb{C}^{mathbb I}$ with $mathbb{I}$ finite. In addition, our results give explicit constructions of images in functor categories using limits and images in the underlying category. In particular, they can be used to give explicit constructions of both centralizers and normalizers in functor categories using limits and centralizers or normalizers (respectively) in the underlying category.
We make some beginning observations about the category $mathbb{E}mathrm{q}$ of equivalence relations on the set of natural numbers, where a morphism between two equivalence relations $R,S$ is a mapping from the set of $R$-equivalence classes to that
We show that doubly degenerate Penon tricategories give symmetric rather than braided monoidal categories. We prove that Penon tricategories cannot give all tricategories, but we show that a slightly modified version of the definition rectifies the s
Frobenius monoidal functors preserve duals. We show that conversely, (co)monoidal functors between autonomous categories which preserve duals are Frobenius monoidal. We apply this result to linearly distributive functors between autonomous categories.
The category of Hilbert spaces and contractions has filtered colimits, and tensoring preserves them. We also discuss (problems with) bounded maps.
We give a new syntax independent definition of the notion of a generalized algebraic theory as an initial object in a category of categories with families (cwfs) with extra structure. To this end we define inductively how to build a valid signature $