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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On the terminology concerning images in category theory
Reading the notion of image in category theory, I've found on nLab the following one:
Let $\mathfrak{C}$ be a category and let $M \subseteq Mono(\mathfrak{C})$ be a subclass of the monomorphisms in $\...
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Intuition behind large diagrams in category theory
I am attempting to read "Tensor Categories" by Pavel Etingof et. al.
The following pentagon axiom is a part of the definition of a monoidal category:
The following diagram is part of the ...
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Proving the injectivity of a certain map of sets
Let $I$ be the subset $[0,1]\subset \mathbb R$. For any set $S$, let $i_S:S\to I\times S$ be the map defined by $i_S(s)=(0,s)$ for all $s\in S$.$ \require{AMScd}$
Let $i:A\to X$ be a map of sets, and ...
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Understanding the solution of Exercise 4.1.32 in Tom Leinster "Basic Category Theory".
Here is the exercise and its solution:
1-Is there a typo and $\varphi$ should be $\psi$?
2- I do not understand how by exercise 2.1.14 we will get the first equation in the solution of 4.1.32. And ...
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Does the application functor have adjoints?
For $K$ and $C$ categories, with $C$ complete and cocomplete, we define the application functor on a fixed $k:K$ as $\ @\ k : [K,C]\rightarrow C$
$$D\ @\ k = D\ k$$
$$\alpha\ @\ k = \alpha_k$$
for $D:[...
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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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Is the trivial category cocomplete?
The trivial category $1$ has only one object $*$, and the only arrow is the identity arrow. Is $1$ cocomplete ? In other words, does any functor $\mathbb{C}\overset{F}\to 1$ from a small category $\...
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Show that $C^*(\mathbb Z)$ is the universal $C^*$-algebra generated by a unitary.
Show that $C^*(\mathbb Z)$ is the universal $C^*$-algebra generated by a unitary. I.e.,show that for any unital $C^*$-algebra $A$ containing a unitary $u$, there is a unique unital *-homomorphism $\...
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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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Show that $C^*\colon G\to C^*(G)$ is a functor from $Gr$ to Cstar
Let $Gr$ denote the category of discrete groups where the morphisms are group homomorphisms, and let Cstar denote the category of $C^∗$-algebras where the morphisms are $∗$-homomorphisms. Show that $C^...
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Sheaf Hom, what is the codomain of this "sheaf" on $X$
The main refference: Prove that sheaf hom is a sheaf.
Without even knowing about the codomain of
$$\mathcal Hom(\mathscr F,\mathscr G):Open(X)^{op}\to (???)$$
Defining $\mathcal Hom(\mathscr F,\...
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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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Show the canonical morphism from the pullback to the product is monic
I'm working through "The Joy of Abstraction" and this is problem 19.2.2
I've solved 19.2.1, and 19.2.3:
See if you can show that there is always a canonical morphism from the pullback $a \...
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A left adjoint for the regular representation functor
Consider the following category:
objects are triples $(G,A,\alpha)$ where $G$ is a group, $\alpha : G\times A\to A$ is a left action of $G$ on a set $A$;
morphisms $(G,A,\alpha)\to (H,B,\beta)$ are ...
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AEC as categories
I have 3 questions on Abstract Elementary Classes.
what is the necessary and sufficient condition that a category $\cal K$ is equivalent to and AEC.
why is it necessary that some faithful functor $U:...
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Is every topos a De Morgan topos?
I think I'm getting something wrong, but I don't see what that is. Does anyboy see where is the error in the following argument.
Let $\mathcal{E}$ be a nondegenerate topos. The topos $\mathcal{E}_{\...
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Corepresentable functor definition.
I have been searching for the definition of a Corepresentable functor here https://ncatlab.org/nlab/show/small+object but still I did not get what exactly is its definition. I also love the definition ...
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Equivalence relation functor representable?
Let $E : \mathbf{Set} \to \mathbf{Set}$ be the contravariant functor taking a set $X$ to the set $E(X)$ of distinct equivalence relations on $X$. It takes (I assume) a function $f:X \to Y$ to a ...
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Sheaf topos is localisation at covering sieve inclusion
I know that the sheafification functor $a:Pr(C) \to Sh(C,J)$ is up to equivalence the localisation $Pr(C) \to Pr(C)[W^{-1}]$ at the class of those morphisms which $a$ inverts. But is it also the ...
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Existence of biproducts in pretriangulated dg-categories
I'm studying dg-categories, and mostly following Bernhard Keller (https://arxiv.org/abs/math/0601185). I'm trying to understand how for a pretriangulated dg-category $\mathcal{A}$, the category $H^0(\...
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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 $X\leftarrow X\times Y\rightarrow Y$:
\begin{equation}
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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Extending an adjunction using colimits
I'd like a confirmation of a fact about colimits and adjunction; the motivation is that I think that this fact is used implicitly sometimes in Kerodon, but I've never seen it written out. $\require{...
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Is there a projection functor $\mathbf{Set}/X^2 \to \mathbf{Set}/X$
I have an object in the slice / arrow category $\mathbf{Set}/X^2$ and I want to transport it to $\mathbf{Set}/X$ by forgetting the second element of the index. Reading up on slice categories I've only ...
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Partial arrow classifier as an (adjunction-related) universal arrow
Goldblatt gives a brief overview of adjunctions in his "Topoi", and one of the exercises asks to characterise the partial arrow classifier in terms of some universal arrow.
Well, I gave it ...
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pushout in the category of topological space is closed under product with compact set
Assume we have the following pushout, where $X_0, X_1, X_2, X$ are topological spaces:
$$\require{AMScd} \begin{CD} X_0 @>{}>> X_1\\ @V{}VV @VV{}V\\ X_2 @>>{}> X\end{CD}$$
Now let $K$...
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Is a locally free sheaf of finite rank a projective object? [duplicate]
Let $X$ be a locally ringed space, in the category of $\mathscr O_X$- modules, is a locally free sheaf of finite rank a projective object?
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Possibly easy diagram chase of the $3 \times 3$ lemma
Given the following diagram
which is exact on all horizontal rows and the left and middle columns. I'm trying to show that it is exact at $A''$ which amount to showing that $\ker(\varphi_3) = 0$ that ...
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A particular presheaf on a small category. What if it's representable?
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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References on history of topology and categorical aspects of topology.
I am currently writing the conclusions of my bachelor's thesis on convergence spaces and there are a couple of points I would like to make, but lack the proper references to cite in order to do so.
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