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# Questions tagged [category-theory]

Categories are structures containing objects and arrows between them. Many mathematical structures can be viewed as objects of a category, with structure preserving morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seemingly different areas of mathematics.

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### Showing that $\varinjlim (L_i \otimes M_i) \cong \varinjlim L_i \otimes \varinjlim M_i$

I am currently self-studying category theory and I was trying to solve the following problem Given a filtered set $I$ and a ring $A$, if $(L_i, f_{ji})$ and $(M_i, g_{ij})$ are inductive systems of ...
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### Should homomorphisms between algebraic structures be considered functions?

Consider the map $\varphi:\{0,1,2,3\}\to\{0,1,2,3\}$ given by $\varphi(x)=x$. If $\{0,1,2,3\}$ is endowed with the structure of $\mathbb Z/4\mathbb Z$, then $\varphi$ may be considered to be a group ...
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### What are isomorphisms in different categories of vector spaces and algebras?

I start this post to collect different categories of vector spaces and operator algebras and the isomorphisms in them. There are a variety of vector spaces (normed, Banach, Hilbert etc.) and a variety ...
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### Presheaf problem

I need to show that the global element of a presheaf $P\in \widehat{\mathcal{O}(X)}$ is equivalent to global section of bundle of germs $\Lambda_P\xrightarrow{p}X$. In particular, I need help to show ...
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### In a category $C$, is Hom$(X, Y)$ the same as (or in bijection with) Hom$(Y, X)$?

I want to understand the notion of an opposite category in a very specific case, say the category $Grp$ of groups with homomorphisms as morphisms. In the opposite category of groups $Grp^\circ,$ for ...
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### Quillen pair with derived equivalence $\implies$ Quillen equivalence

Let $L\dashv R$ a Quillen pairs an adjunction between model categories $M,M'$ with $L$ preserving cofibrations and $R$ fibrations. From then we can construct an adjunction $\mathbb LL\dashv \mathbb RR$...
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### Hilbert spaces and directed colimit

How to prove that a Hilbert space is the directed colimit of its finite-dimensional subspaces? Does this imply that the category of Hilbert spaces (and bounded linear maps) is the Ind-completion (see ...
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### Seifert-van-Kampen theorem for groupoid and group

We know two versions of Seifert-van-Kampen theorem, one for fundamental groupoids and the other for groups. How do these two relate to each other? I know that the case for groups can be derived from ...
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### Pullback of the identity is 'basically' the identity?

I encountered the titular statement in a discussion on subobject classifiers (in particular: in the proof that the domain of such a subobject classifier is terminal). However, I think his 'basically' ...
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### In preadditive categories, initial objects are terminal

I want to prove: in a preadditive category $\mathscr{C}$, if an object of $\mathscr{C}$ is initial, thent it is also terminal (hence a zero object). So, let's start a proof. I am reading Borceux's ...
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### Amnestic Functors in Concrete Categories

I am reading the book: Joy of Cats. I am trying to understand the notion of an amnestic functor in a concrete category. In the book it is defined in terms of fibres. What’s the basic notion of an ...
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### What does $D(D(f))=D(f)=C(D(f))$ it mean in the category of generalized monoid.

The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes In chapter 7 of Arbib and Manes about Functors. The authors introduce the category ...
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### Introduction to topology in modern (categorical) language?

Have studied traditional point-set topology, but find there's a fairly large gap between the preparation typical point-set courses give you, and the level assumed in algebraic topology texts. Looking ...
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### Representable presheaves on the slice category

$\def\sfC{\mathsf{C}} \def\op{\mathrm{op}} \def\set{\mathsf{Set}} \def\psh{\operatorname{PSh}} \def\ob{\operatorname{Ob}} \def\hom{\operatorname{Hom}}$Let $\sfC$ be a category. A presheaf over $\sfC$ ...
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### A pair of adjoint functors and unicity

I was working on the following problem, Given $F_1,F_2:\mathscr{C}\rightarrow\mathscr{D}$ and $G_1,G_2:\mathscr{D}\rightarrow\mathscr{C}$ such that $(F_1,G_1)$ and $(F_2,G_2)$ form adjoint pairs, ...
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### Show that $\phi$ is surjective in commutative staircase diagram

I'm having some trouble with the following problem involving a sort of staircase diagram and I would really appreciate some help. Assume the following commutative diagram of modules and module ...
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### Question about notation used for a proof about Vector space Product in category theory

The following are taken from $\textit{Arrows, Structures and Functors the categorical imperative}$ by Arbib and Manes $\text{Definition 1:}$ A $\textbf{product}$ in the category $\textbf{K}$ of a a ...
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### Can higher dimensional categories be represented as hypergraph and if not why?

As the diagrams of categories can be represented as graphs of objects and morphisms, I was wondering if (the diagrams of) higher-dimensional categories could be represented as hypergraphs, and if not ...
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### How category of diffeological spaces is related to the classifying category of HoTT? Maybe they are the same?

I am trying to read "Homotopy Type Theory: The Logic of Space" https://www.cambridge.org/core/books/abs/new-spaces-in-mathematics/homotopy-type-theory-the-logic-of-space/...
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### are comma-objects semi-coflexible

I need someone to read through my proof, because I feel very uncertain about 2-categorical limits. A strict indexed category $C:\mathscr S^{op}\to Cat$ is semi coflexible when every pseudo-...
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### Is there a cojoin, the dual construction of the join of topological spaces?

The join $X\star Y$ of topological spaces $X$ and $Y$ can alternatively be written as a homotopy pushout of the canonical diagram $X\leftarrow X\times Y\rightarrow Y$: X\star Y =X\...
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### When will a countable group be the inductive limit of its finite subgroups

It is proved in this post that every group is the inductive limit of the family of its finitely generated subgroups (where the partial order is the inclusion). Indeed, given $G_i, G_j$ two finitely ...
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### Slick way to check that a diagram is a pullback square (Sheaves in Geometry and Logic, Chapter 2, Proposition 1)

I'm reading Chapter 2 of Mac Lane and Moerdijk's Sheaves in Geometry and Logic. At some point our guys claim that given a topological space $X$ and a sheaf $F$ on $X$, a subsheaf $S$ of $F$ ...
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### Stackification of finite categories

Assume I have a base category $\mathscr S$ with finite limits and a geometric morphism $\gamma :\mathscr S\to Fin$ into the category of finite sets (for example because $\mathscr S$ is positive ...
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### If the presheaf Hom$_\mathcal{C}(- \times A, B) : \mathcal{C}^\text{op} \to \textbf{Set}$ is representable, then $\mathcal{C}$ is ccc

We consider a small category $\mathcal{C}$ with binary products, and we consider, for any objects $A, B$ of $\mathcal{C}$, the assignments \begin{eqnarray*} F :\,\, &\mathcal{C}^\text{op} &\...
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