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RicciflowsRicciflows
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5 months ago
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Article

Is any essentially surjective morphism of categories an epimorphism?

My question: Let $\cal{C},\cal{D}$ be categories (resp. stacks). Let $F:\cal{C}\rightarrow\cal{D}$ be an essentially surjective functor, i.e. surjective on isomorphism classes of objects. Then is $F$ an epimorphism in the category of small categories (resp. stacks)?Answer: No. For example, any functor between categories with one object is essentially surjective, but e.g. if $M_1, M_2$ are two nonzero monoids then the inclusion $M_1 \to M_1 \oplus M_2$ of a direct summand, thought of as a functor ...

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NekomusumeNekomusume
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6 months ago
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Article

Must the morphisms of the category be structure-preserving？I found something different in a textbook.

My question: It is well known that morphisms between the objects of the category are structure-preserving, but I found that in a textbook it said that morphisms are often structure-preserving. Does this mean that there can be a morphism that is not structure-preserving?Answer 1: A category doesn't have to consist of sets with some additional structure and maps between those preserving the structure.Examples for categories that are not of this kind are:Given any group $G$, we can form a category ...

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RicciflowsRicciflows
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5 months ago
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Article

Is fully faithful functor between groupoids injective on objects?

My question: Let $\cal{C}$ and $\cal{D}$ be two groupoids, i.e. the category whose morphisms are isomorphisms. Let $F:\cal{C}\rightarrow\cal{D}$ be a fully faithful functor from $\cal{C}$ to $\cal{D}$. Then is $F$ injective on objects? In other words, is the object function $F:{\rm{Ob}}(\cal{C})\rightarrow{\rm{Ob}}(\cal{D})$ injective?Answer: No. Given any set $X$ we can construct a groupoid called the indiscrete groupoid on $X$, which has a unique (iso)morphism $x \to y$ for $x, y \in X$. Every ...

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NekomusumeNekomusume
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a year ago
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Article

A short introduction to category theory

This short note originated from a short talk of basic category theory in 2022. Category theory was first introduced by Samuel Eilenberg and Saunders Mac Lane in the 20th century. It rapidly became a strong tool in almost all domains of mathematics. Category theory studies the abstract structures of different mathematical objects, and it also has applications on computer science, including AI.1. CategoriesIn this section, we will first lay out the general definition of categories, then we give so ...

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