Various structures are studied in category theory using properties of objects and morphisms between them. 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 ...

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relationships of topological and C* concepts in noncommutative topology

According to wikipedia, noncommutative topology is " a term used for the relationship between topological and C*-algebraic concepts". Can somebody expand on this, give examples/theorems/results and ...
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Separable algebra in tensor category

I am trying to understand the definition of a separable algebra in a tensor category. In the book $Tensor\ Categories$ by Etingof, Gelaki, Nikshych and Ostrik they provide the definition: "Definition ...
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equivalence of categories between modules and vector space

Let $Map_k$ the category where the objects are triples $(V,W,f)$, where $V$ and $W$ are finite dimensional $K$-vector spaces and $f:V\rightarrow W$ is a $K$-linear map. A morphism from $(V,W,f)$ to $(...
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2answers
61 views

What does “is natural in $A$” mean in this context?

While reading Bredon's Topology and Geometry, I've come across the following claim: Naturality in $A$ of the sequence defining $\text{Ext}(A,G)$ shows that $\text{Ext}(A,G)$ is a contravariant ...
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1answer
31 views

Let $\mathcal{A}$ be an Abelian category (as defined in Stacks), then all monomorphisms are kernels.

I'm struggling to prove this statement, using the definitions below (I'm assuming the proof for the statement about epimorphisms is analogous). I know that a morphism $f:x\to y$ is monic if and only ...
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Do the effective descent morphisms w.r.t the codomain fibration hint at the “right topology”?

An intuitive approach to basic descent theory for me started with open covers $ \left\{ U_i \right\}$, replaced them with a singleton covering $\coprod _iU_i\rightarrow X$, and then generalized to ...
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27 views

Morphisms as zigzags, composition as concatenation

I am trying to find any references to the following construction that relates to zigzag categories (https://ncatlab.org/nlab/show/zigzag+category). Let C be a category. Let Z(C) be the category with ...
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1answer
51 views

Grothendieck ring of varieties

In the context of Grothendieck ring of varieties there is often used notion of variety over variety (for example here -2.2.1). I always used only varietes over field. My question is : how is it define ...
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33 views

Day convolution on the category of copresheaves on the opposite of a monoidal closed category

When $\mathcal{C}$ is symmetric monoidal closed, the Day convolution gives a symmetric monoidal closed structure to the category $[\mathcal{C},\mathbf{Set}]$. Suppose that instead, it is $\mathcal{C}...
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3answers
42 views

Why not use the Cartesian product for the monoidal category of modules? Why use the tensor product?

At least for vector spaces, the Cartesian product is a direct sum (categorical product and coproduct) operation. Looking at the definition of monoidal category on Wikipedia, the Cartesian product/...
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1answer
29 views

Simple question regarding the monad whose algebras are the category of sets

The category $\mathbf{Set}$ of sets can be viewed as the category of models for a Lawvere theory, and hence it is equivalent to the Eilenberg-Moore category of algebras of an (associated to the ...
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84 views

For which category (if any) are Lie algebras the algebras of a monad?

I was reading about monads recently, and it came to me that the purpose of the category of algebras of a monad seems to be to switch to a "representation" which is easier for computations. Soon after ...
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1answer
40 views

Initial object is limit of identity functor: converse

A known theorem in category theory is Suppose $\mathscr{C}$ has an initial object $c$. Then $c$, along with its unique maps, forms the limit of the identity functor $\mathscr{C} \to \mathscr{C}$...
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34 views

Question about generators and Hom functor

In a given category $\mathcal{C}$ I want to prove the following statement: If $U$ es a generator in the category $\mathcal{C}$ if and only if the left-exact functor $Hom_{\mathcal{C}}(U,-): \mathcal{...
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1answer
53 views

Are functors (from small categories) functions?

I am looking for either (1) confirmation that the following is true, (2) the mistake making it false pointed out to me: Let $F:\mathcal{C} \to \mathcal{D}$ be a functor from a small category $\...
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1answer
101 views

Is there a general way to tell whether two topological spaces are homeomorphic?

We know that if two topological spaces $X$ and $Y$ are homeomorphic, then they have the same fundamental groups, and the same homology. In other words, we have functors $$\pi_1 : \mathsf{Top} \to \...
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25 views

Diagonal Functor an Isofibration?

Let $C$ be a category and let $\Delta:C\rightarrow C\times C, \Delta=(id_c,id_c)$ be the diagonal functor. Recal that an isofibration is a functor p: E→B such that for any object $e\in E $ and any ...
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35 views

Algebra of Endomorphisms of a functor $\mathcal{F}$

Recently, I was reading a paper and stumbled upon something for which I don't find the formal definition unfortunately in any textbook that have in my hands. It's the notion of the endomorphism ...
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3answers
100 views

“Alternatives” to Natural Transformations

I would like someone to either (1) point out the mistake in what follows or (2) confirm what is said is correct. This would be accomplished by addressing the part in yellow only. The rest of the ...
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20 views

Are exponential objects examples of (co)universal objects which are not (co)limits?

This is in some sense a follow-up to previous questions I have had asking about the relationship between products and exponential objects. Products can be written as, and in often are defined to be, ...
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30 views

Is the tensor product of BAOs a kind of extended BAO?

I've been reading "Boolean algebras with operators. Part I." (Jonsson, Tarski) where, given a subalgebra of a Boolean Algebra, they define its perfect extension. As far as I understand it can be ...
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3answers
45 views

A question about the product functor on finite sets

I am a beginner in Category Theory so please excuse me if this is a trivial question. Let $\mathbf{FSet}$ denote the category of finite sets. The product functor $X\times -:\mathbf{FSet}\to \mathbf{...
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3answers
60 views

Is there a mathematical difference between currying and partial application?

I know the following example doesn't make what I am saying rigorous, but hopefully it clarifies to some extent what I mean. For various computer implementations, dividing by 2 and multiplying by 0....
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1answer
53 views

Evaluating a colimit of functors

This is my first question here. I am working on some categories and this question regarding functors seemed very natural: Let $\{G_\alpha \rightarrow F\}_{\alpha \in \Gamma}$ with $F$ and $\{G_\alpha\...
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66 views

Categorical Interpretation of Strongest/Weakest Topology

One way to define the product topology is as the weakest topology for which all projection maps are continuous. Strongest/weakest topologies satisfying a given property are ubiquitous in topology and ...
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54 views

Is the assignment of a root system to a semisimple Lie algebra functorial?

As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such ...
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1answer
51 views

Algebras with a self-dual congruence lattice

The well known (Mal'tsev) conditions that characterize certain properties of the congruence lattice of an algebra. The existence of a 3-ary term $p$ together with familiar identities $p(x,y,y) \sim x \...
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53 views

Relationships between categories of elements giving relationships between the functors

Suppose $\mathscr{C} \underset{G}{\overset{F}{\rightrightarrows}} \mathsf{Set}$ are such that $\operatorname{El}F$ and $\operatorname{El}G$ have initial objects which lie in fibers over the same ...
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How to show these two definitions of Moss' language are equivalent? (Coalgebraic logic)

Definition 1: Suppose $T:\mathbb Set \to \mathbb Set$ is set-enofunctor that preserves weak pullbacks. The $\kappa$-Moss' language, written by $\mathcal L_T^\kappa$ is the carrier set of $(\mathcal ...
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Why care about relation liftings under (covariant) power-set functor? (coalgebraic logic)

Moss' motivation to use the notion of relation lifting was in their importance when applied to membership relation, which leads to define the semantics of $\nabla$ operator. Now in the whole process, ...
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1answer
32 views

How to show the $\kappa$-small functor is $\kappa$-accessible? (coalgebraic logic)

A $\mathtt {Set}$-functor $T:\mathtt {Set} \to \mathtt {Set}$ is defined to be $\kappa$-accessible for a regular cardinal $\kappa$ iff for all sets $X$ and all $x\in TX$ there exists a subset $Y\...
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35 views

Categorical definition of closure operation

Consider a finitely complete category $\mathscr{B}$. A universal closure operation on $\mathscr{B}$ consists in giving, for every subobject $S\rightarrowtail B$ in $\mathscr{B}$, another subobject $\...
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1answer
41 views

Circular definition in slice category?

I am reading Aluffi (Algebra Chapter 0) there he introduces the slice category in a kind of excercise: When thinking about it I got confused about the "nature" of the $Z$ (and $A$). Since they are ...
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1answer
55 views

What is an injection in a topos?

I am reading the paper of Andrej Bauer that proves there is a realizability topos in which there is an injection from the internal Baire Space $\mathbb{N}^\mathbb{N}$ to the natural numbers $\mathbb{N}...
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2answers
45 views

Principal bundle as homotopy fiber universally self-trivializes

In this MO answer, I was told the definition of principal bundle as a homotopy fiber of its classifying map precisely says that it's the universal bundle which trivializes itself. However, I'm having ...
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1answer
53 views

Almost an Adjunction…

The contravariant power-set functor $P:Set^{op} \longrightarrow Set$, together with its dual $P^{op}:Set \longrightarrow Set^{op}$, almost constitute an adjunction: there is natural monomorphism $\...
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31 views

Does limit functor preserve isomorphism in inverse systems?

Let $I$ and $I'$ be an inverse systems for which limit exists (For example R-modules) spanned by some indexing categories with order-relation $\lambda$ and $\lambda'$ respectively. Let $\phi:\lambda' \...
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169 views

Why would I learn modern category theory if my interest mainly is structured sets, what would I have to gain? [closed]

A long time ago I studied mathematics at the University of Stockholm. I had a romantic view of modern algebra and manage to make the first two algebra courses by self studies in order to immediately ...
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1answer
54 views

Translation of the Axiom schema of Separation into purely category-theoretic terms.

It is well-known the the category $\mathcal Set$ is a topos. In this topos, how would one translate the Axiom schema of Separation into purely category-theoretic terms? I ask this question because ...
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2answers
64 views

Difference between Product and Exponential Object?

Do the exponential object and the product coincide for $Set$? If they do, then why are they different for general categories? If they do not coincide for $Set$, can the reason why be extended to ...
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30 views

Invertible product of noninvertible morphisms

This question may be too broad. Under what conditions is a product of noninvertible morphisms invertible? Suppose that we model a finite number of different acts of observation (i.e., a thermometer ...
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33 views

image of direct sum

Let $\mathcal A$ be an abelian category. Show that $$\mathrm{Im}(\bigoplus_{1\leqslant i\leqslant n}\ \varphi_i)\simeq \bigoplus_{1\leqslant i\leqslant n}\mathrm{Im}\varphi_i$$ There are two ...
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1answer
80 views

Does this category have a name? (Relations as objects and relation between relations as morphisms)

Given two relations $R\subseteq A\times B$ and $R'\subseteq A'\times B'$. Is it known/used that every relation $r\subseteq R\times R'$ can be characterized by two relations $\alpha\subseteq A\times A'$...
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1answer
33 views

Group homomorphism in category theory [closed]

Too often i come across the statement a functor between two groups as categories is the homomorphism between the corresponding groups. This may be trivial, but has anybody proved is necessary and ...
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2answers
29 views

What are the subcategories of ordered sets / groups?

This is an exercise from Tom Leinster's Basic Category Theory. It asks: 1)What are the subcategories of an ordered set? Which are full? 2)What are the subcategories of a group?Which are full? I'm ...
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122 views

On the separation axiom in a Lawvere or “generalized” metric space

According to the nLab, Lawvere metric spaces occur rather naturally (that is as certain enriched categories). A Lawvere metric space is a set $X$ equipped with a function $d : X\times X \to [0,\infty]$...
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Alternative Definition of Contravariant Functor

Given two categories, $C$ and $D$, a covariant functor is usually defined as a regular functor $C \to D$, whereas a contravariant functor is usually defined as a regular functor $C^{op} \to D$. ...
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The completeness of a category of additive functors between additive categories

In what follows $\textbf{preadditive}$: a category $\mathscr{C}$ is preadditive when $\forall\ A,B,\ \mathscr{C}(A,B)$ is an abelian group and the morphisms composition is a group homomorphism on ...
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A cartesian diagram?

let $k$ be a field, and $X$ and $Y$ varieties over $k$. Let $L$ be an extension of $k$, and $X_L=X\times_k L$. Is the diagram $$\require{AMScd} \begin{CD} X_L\times Y_L\times X_L @>>> X\...
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2answers
42 views

Morphisms between biproducts in additive categories

I can't understand the following description of a morphism between biproducts in an additive category, which I found in Borceux, Vol.2 If $A_1,A_2,B_1,B_2$ are four objects in an additive category $\...