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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 ...

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Right whiskering, unknown naturality

Suppose $F,G:C\to D$ and $H:D\to E$ are functors and $\alpha:F\to G$ is a natural transformation.Let $H\circ \alpha:H\circ F\to H\circ G$ be the right whiskering $(H\circ \alpha)_A:H(FA)\to H(GA)$ ...
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Universal Property Of $\text{Bilin}(A,A',-):\text{Ab}\to\text{Set}$

In class, we had to state the universal property of $\text{Bilin}(A,A',-):\text{Ab}\to\text{Set}$, the functor taking an abelian group $B$ to the set of bilinear maps $f:U(A)\times U(A')\to U(B)$ ...
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Order of taking quotients?

Let $G$ be a group and $M,N$ be its normal subgroups. I am thinking about the relationship between $(G/M)/N$, $(G/N)/M$, $G/(M/N)$ and $(G/M\cup N)/(M\cap N)$ (they are not proper notions but I am ...
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coproduct of noncommutative algebra and commutative algebras

I have read the book "Rings with generalized identities" and I understand that the free product of asociative unital algebras are the coproduct of them, but I can't understand why this reduces to the ...
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Do adjunctions map to every object in a category?

I am trying to show that right adjoints preserve limits. So suppose we have $F: C \rightarrow D$ and $G: D \rightarrow C$ be 2 adjoint functors with $G$ the right adjoint. So suppose we have a limit $(...
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Looking for a terminology for “sameness” of functions

Consider the situation described in the following diagram, namely: $A$, $A'$, $B$, and $B'$ are sets. $\alpha:A\rightarrow A'$ and $\beta:B\rightarrow B'$ are bijections. $f:A\rightarrow B$ and $\ f':...
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Does the set Hom(X,Y), in any category, always form an Abelian group?

Given any category C and any two C-objects X and Y, is it always the case that the set Hom(X, Y) of morphisms from X to Y form an Abelian group? If so, and to be clear, is this always the case ...
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Presentation of $hS$ as in page 29 of HTT

At the page claims 29 of HTT, Lurie claims that the category $hS$ (we ignore all the enrichement) admits the following presentation by generators and relation : The objects of $hS$ are the vertices ...
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Effective epimorphisms in $G$-set

$\newcommand{\gset}{G\text{-}\mathsf{set}}$ Let $G$ be a group and $\gset$ the category of $G$-sets, whose morphisms are $G$-maps (i.e. set maps where the map commutes with $G$-action). What are the ...
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Fibred products and epimorphisms in $G$-set

$\newcommand{\gset}{G\text{-}\mathsf{set}}$ Let $G$ be any group and $\gset$ be the category of $G$-sets, with morphisms being $G$-maps. That is an object of $\gset$ is a pair $(X,\rho)$ where $\rho:G\...
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Multiset-Span monad

I have been thinking about multisets for a while. These are sets where elements can repeat, so $S =\{ a,a,b,c,b\}$ is a multiset on the set $A = \{a,b,c\}$. It is well known that there is a Monad on ...
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Category of Multisets and Spans

I have been thinking about multisets for a while. These are sets where elements can repeat, so $S =\{ a,a,b,c,b\}$ is a multiset on the set $A = \{a,b,c\}$. I have also been looking into morphisms ...
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Category theory using 2-sorted logic

Can anyone give some reference text that axiomatizes category theory using first order logic using 2 sorts (if that makes sense)? I have found reference about 1 sorted axiomatization but I also found ...
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Nearly locally presentable categories

Here1, in the remark $2.3 (1)$ how from the fact that ${\cal K}(A,-)$ does not preserve coproducts it follows that ${\cal K}(A,-)$ sends special $\lambda$-directed colimits to $\lambda$-directed ...
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Confused about Kleisli category target object (codomain)

According to Wikipedia Kleisli category is defined as: \begin{aligned}{\mathrm {Obj}}({{\mathcal {C}}_{T}})&={\mathrm {Obj}}({{\mathcal {C}}}),\\{\mathrm {Hom}}_{{{\mathcal {C}}_{T}}}(X,Y)&...
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Central Concept of Category Theory?

Is it reductionist to say "Universal properties are the central concept of Category Theory". And if so, is it a useful fiction to keep in ones head as they are learning the subject?
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Is every category isomorphic to a subcategory of the category of relations?

Is it the case that every (small) category is isomorphic to a subcategory of Rel, the category of relations?
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VAR, Algebra and local presentability

Here1, on the page 282, I would like to understand why precisely Examples of $k$-ary operations are all $k-\mathrm{lim}$, BUT $k-\textrm{colim}$ must be filtered. Where have we used that $k$ is ...
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Is Set, the category of sets, a self-dual category?

Because Rel, the category of relations, is self-dual, and because Rel has Set, the category of sets, as a subcategory, can we conclude that Set itself is self-dual? And, just as Rel is a dagger ...
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Categories with zero Grothendieck group

I am interested in examples of triangulated categories that have zero Grothendieck group but are somehow still interesting, say have non-zero Hochschild homology. More example, for what rings $R$ is ...
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Set-theoretic worries about functor transformation from $\text{Id}$ to $V\mapsto V^*$

Upon reading the wikipedia article for natural transformations of functors, I stumbled across the section on the dual of a vector space. This is not a question about why the transformation from the ...
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Is the category of $2$-topological quantum field theories locally small?

At first, looking at 2TQFT I see no reason to expect it to be locally small. But we know that 2TQFT is equivalent to the category of commutative Frobenius algebras cFA$_{\mathbb K}$, which is locally ...
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What is the $\infty$-category associated to a model category?

It is often said that model categories are but a shadow of an $\infty$-category. It is also often said that model categories should give rise to an $\infty$-category via their homotopies. In fact, ...
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What exactly is called “cone” in the category theory and how does it relate to a category of cones?

As far as I understand, cone is a pair of some object $X \in Obj(\mathcal{C})$ (which can be viewed as a $\Delta_X$ - constant functor from some other category to the $\mathcal{C}$) and a set of ...
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The Powerset Monad

I am struggling to prove commutativity of the diagrams for the powerset monad in Category $\mathbb{Set}$. To show that $\mu:\mathcal{P}^{2}\longrightarrow \mathcal{P}$ given by $\mu_{X}:\mathcal{P}^{...
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Notation for Geometric realization of simplicial sets

I am confused about some notation in the quick way of constructing the geometric realization of a simplicial set. Consider the simplex category $\Delta \downarrow X$ of a simplicial set $X$. The ...
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1answer
28 views

Are there such things as cyclically presented (arrows-only) categories?

Motivation: Let $w$ be an element of the free group $F_n$ on the generators $\{x_i\}_{i=0}^{n-1}$. Define a function $\theta: F_n\to F_n$ by $x_i\mapsto x_{i+1}$ (and extend this to all elements of $...
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Is there a categorification of “(virtually) solvable”?

If this question doesn't make sense or is otherwise poor quality, then I'm sorry. Motivation: As part of my research, I study virtually solvable (1) groups. These are goups that have a solvable ...
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1answer
138 views

Adjunctions and isomorphism.

I'm stuck with the next exercise. I don't know how can I solve it. Let $\mathcal{F}:\mathcal{C}\to\mathcal{D}$ and $\mathcal{E},\mathcal{G}:\mathcal{D}\to\mathcal{C}$ three adjoint functors $\...
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A Nerve functor into any $\infty$-comos $\mathcal{N}: Cat \to \mathcal{K}$

I believe there is a notion of a nerve functor into any $\infty$-cosmos $\mathcal{K}$. My inclination is that it would be defined as the colimit of the constant functor that sends all objects to the ...
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The category of special unitary groups

Is there such a thing as the category of special unitary groups and continous group homomorphisms? There is a category of abelian groups and This category factorizes the multi set monad. I am ...
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What is a formal full proof that “$ \mathbf{Set}$ is a Category” in the context of Category Theory?

I am learning category theory and couldn't find a full formal proof of the simplest example I could think of (that $\mathbf{Set}$ is a Category). What's seems the "hello world" of category theory? ...
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Every equalizer is monic: a specific detail

I have a very basic question which is driving me nuts. I was trying to show that an equalizer is always monic. I ended up doing something like in In Every equalizer is monic, namely: Suppose $i : ...
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Slice and coslice categories: duality

I've heard that slice and coslice categories are dual notions. But in what way exactly? My first idea was to see that $\mathsf{C^{op}}/c = c/\mathsf{C}$ for any object $c$ of a category $\mathsf{C}$, ...
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Motivation for topological categories

Define a topological category as a small category with topologies on the set of objects $C_0$ and on the set of arrows $C_1$, such that the domain map and the codomain map are continuous. One can ...
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Descent data and trivialization of bundles via coherent isomorphisms of fibers

In this MO question I tried to understand how a trivialization of a bundle $\begin{smallmatrix}A\\ \downarrow\\ B \end{smallmatrix}$ is related to a coherent family of isomorphisms between its fibers. ...
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Relationship between limits from 1 to functor category and limits of functors

Let $F:A\to C$ be a functor and let $U:1\to C^{A}$ be a functor from 1-object category given by $U(0)=F.$ What is the exact relationship between $\lim{U}$ and $\lim F$ ?
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A different definition of coproduct in a category

I am following Pavel et al book "Tensor Categories". They write that an additive category is a category $\mathcal{C}$ such that: My problem is with (A3). i know that what they are trying to say is ...
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Set theoretic issues in the definition of a site in Stacks Project

I've been learning about sites from the Stacks Project, which is generally very precise in its terminology, but I've found some of their conventions very confusing in this part. Their definition of a ...
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Elements of the Monoid in the category of endofunctors

Quoting from Categories for the Working Mathematician by Saunders Mac Lane: All told, a monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of ...
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Monoid in the category of endofunctors and Monoid as a category with one object

Quoting from Categories for the Working Mathematician by Saunders Mac Lane: All told, a monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of ...
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1answer
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Is $\text{Aut}_{\text{br}}(C)$ braided?

Let $C$ be a braided monoidal category. The category $\text{Aut}_{\text{br}}(C)$ of braided monoidal autoequivalences of $C$ is monoidal with tensor product functor given by the composition $\circ$. ...
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Sheave of sets, what does $\{f_i \} \mapsto \{f_i \mid_{U_i \cap U_j}\}$ mean?

... for an open covering $U = \bigcup U_i$, an $I$-indexed family of functions $f_i : U_i \to \Bbb{R}, \ i \in I$, is an element of the product set $\prod_i CU_i$, while the assignments $\{f_i\} \...
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The function from the $\{\}$ to an any other set?

In the $\mathcal{SET}$ - category of sets and maps between them - there is an initial object - the $\{\}$. It means that there is unique map from the $\{\}$ to an any other set (object of $\mathcal{...
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Morphism=monomorphism•epimorphism?

Is it true that any morphism in any category can be written as a combination of monomorphism and epimorphism? In SET and categories where monomorphism is an injective function and epimorphism is a ...
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If the objects of a category form a proper class, do the arrows necessarily form a proper class too?

In some categories, like $\text{Set}$ or $\text{Group}$, the objects are "constructed" out of sets (or are sets, possibly with additional structure). In order to avoid paradoxes, the collection of ...
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Grothendieck Universe

I have a question concerning the Grothendieck's universe: Let fix a GUniverse $U$. Is a $U$-set the same as a $U$-category or is there a subtle difference?
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Comma category construction?

I am getting closer to the essence of a comma category. Given 1) category $\mathcal{A}$ with objects $A, B$ and signle not-identity morphism $f: A \mapsto B$, and 2) single-object category with ...
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Does sheafification of bundles have a right adjoint?

Given a topological space $X$, let $\mathbf{Bundle}(X)=\mathbf{Top}/X$ be the category of bundles over $X$, and let $\mathbf{Sh}(X)$ be the category of sheaves over $X$. Then there's a sheafification ...
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Wedge sum of spheres is the quotient $X^n/X^{n-1}$

As in the title, I want to prove that $\bigvee_jS_j^n=X^n/X^{n-1};\ X$ is a $CW$ complex and $X^n$ and $X^{n-1}$ are the $n-$ and $n-1$-skeleta. Below, I present a sketch of an attempt using pushouts, ...