The Normal Decomposition of Morphisms in Categories With Finite Limits and Colimits
This paper presents a generalized construction of the normal decomposition of morphisms in categories with finite limits and colimits, extending the concept beyond pointed categories to encompass a wider range of mathematical structures.
κ-完備化與函子範疇：探討 Indκ(CI) ≃ Indκ(C)I 成立的條件
本文探討了在何種條件下，函子範疇 CI 的 κ-完備化等價於從 I 到 C 的 κ-完備化的函子範疇。
When is the Ind-Completion of a Functor Category Equivalent to the Functor Category of the Ind-Completion?
This paper investigates when the $\kappa$-ind completion of a functor category, denoted $\operatorname{Ind}\kappa(C^I)$, is equivalent to the category of functors from $I$ to the $\kappa$-ind completion of $C$, denoted $\operatorname{Ind}\kappa(C)^I$. While a previous theorem incorrectly claimed this holds for any Cauchy complete category $C$ and $\kappa$-small category $I$, this paper disproves that claim and provides two corrected theorems. The first corrected theorem states the equivalence holds if $C$ has $\kappa$-small colimits and $I$ is $\kappa$-small. The second corrected theorem states the equivalence holds if $C$ is an arbitrary category and $I$ is well-founded and $\kappa$-small. The paper provides counterexamples and discusses the optimality of these conditions.
擬度量空間範疇上的有界理想單子及其代數
擬度量空間上的有界理想構成擬度量空間範疇上的一個單子，其代數是 Mislove 局部dcpo 的度量版本。
The Bounded Ideal Monad on the Category of Quasi-Metric Spaces and its Algebras: A Characterization via Formal Balls
This research paper explores the concept of bounded ideals in quasi-metric spaces, demonstrating that a quasi-metric space is an algebra of the bounded ideal monad if and only if it is standard and its set of formal balls forms a local dcpo.
A Concise Proof of the Full Faithfulness of Rezk's Nerve Functor
The Rezk nerve functor from the (∞,1)-category of (∞,1)-categories to the (∞,1)-category of complete Segal spaces is fully faithful.
Directed Equality in Category Theory: A Semantic Analysis Using Dinaturality
Dinaturality allows for a semantic validation of introduction and elimination rules for directed equality, which can be characterized as a relative adjoint to a functor that contracts two natural variables into a single dinatural one. Ends and coends can be interpreted as "directed quantifiers" in this setting.
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