# Questions tagged [category-theory]

Categories are structures containing objects and arrows between them. Most mathematical structures can serve as objects of a category, with structure morphisms as arrows. Many constructions are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category theory, too.

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### Monomorphisms and epimorphisms in the category of morphisms

Let $\mathcal{A}$ be (an abelian, I don't think it should matter?) category. Then let $\operatorname{Mor}(\mathcal{A})$ be the category of morphisms, that is the category with objects given by ...
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### How to prove that two fusion systems are equal?

How to prove that two fusion systems $F$ and $F'$ are equal? the collection of objects of $F$ = the collection of objects of $F'$ and for any morphism $f$ in $F$, it is a morphism in $F'$, and ...
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### Categorical general topology — reference request

I am looking for literature describing general topology in terms of category theory. I would prefer literature which does not assume too much familiarity with category theory, but would appreciate any ...
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### structure-preserving: a history?

I am familiar with the occurrence of structure-preserving morphisms in Category Theory, but I would love to know more about the history of the concept, where it started and so on? I suspect it might ...
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### About the complement of a subobject in a topos

Let $\mathcal{E}$ be a topos and let $X$ be an object of $\mathcal{E}$. Let $S \to X$ be a subobject of $X$. We only know that the category of the subobject of $X$ is a Heyting Algebra, so we do not ...
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### What are the domains of the multiplication and unit morphisms of a monoid object?

I'm trying to understand what a Monoid is from category theory perspective, but I'm a bit confused with notation used to describe it. Here is Wikipedia: In category theory, a monoid (or monoid ...
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### Left adjoint for forgetful functor

I am aware that forgetful functors usually have a left adjoint, but I'm more interested in developing my technique. Let $\mbox{Cat}$ be the category of small categories and $\mbox{Set}$ the category ...
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### Inversion on an internal groupoid

Let $\mathcal{C}$ be a category with pullbacks, with $\mathscr{G}=({\bf Ob}_\mathscr{G},{\bf Hom}_\mathscr{G},cod,dom,{\bf 1}, \circ_{\mathscr{G}},-^{-1})$ an internal groupoid in $\mathcal{C}$. I'm ...
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### Image in category of sets, existence of pushouts

First question: According to definition from wikipedia: definition What is $v$ in category $Set$? Second question: Let $C$ be a category. Show that binary coproducts and coequalizers in $C$ exist of ...
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### Two questions about the Hom functors

Given a locally small category $\mathcal{C}$, Wikipedia defines the Hom functors as At my lectures (for the more specific case of $\mathcal{C}=\operatorname{R-Mod}$ -- the category of R-modules for ...
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### Suggestion for seminar about rings of continuous functions [on hold]

I have to do a seminar about the rings of continuous functions, it will be a part of a course in topology. The main topic of my seminar will be the functor from the topological space and the ...
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### How to define pivot columns?

When you use Gaussian elimination to solve a homogeneous system of linear equations, you end up with "pivot variables" and "non-pivot variables". The non-pivot variables have the property that they ...
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### Idempotence of Lawvere-Tierney topology induced by Grothendieck topology

I'm hoping someone can elucidate a step in the proof of V.1.2 in Mac Lane & Moerdijk's Sheaves in Geometry and Logic. Let $\mathcal{C}$ be a small category and $J$ a Grothendieck topology. $J$ ...
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### how are characterized categories that are determined by they graph up to isomorphism?

I know that thin categories have this property, but i also know that they are not the only ones. What are other kinds of categories that have this property? Is there any characterization of that fact?(...
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### What is name we need to use for endofunction if in its definition endomorphism were not in the category of sets?

An endofunction (or self-mapping or other variation) is an endomorphism in the category of sets, that is a function from a set X to X itself. Ok, what new names we have now for endofunction if ...
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