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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7
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1answer
96 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 \...
0
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3answers
33 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/...
0
votes
1answer
25 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 ...
0
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2answers
68 views

Silly question about descent

Most sources say descent is defining an object over $S$ using objects over $U_i$ for some cover $\left\{ U_i \right\}$ of $S$. If I replace the covering family with a single arrow $\coprod _i U_i\...
2
votes
1answer
37 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 $\mathscr{C} \to \mathscr{C}$...
1
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1answer
74 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 ...
2
votes
1answer
52 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\...
3
votes
3answers
98 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 ...
0
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0answers
38 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 $\...
0
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0answers
32 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{...
4
votes
0answers
106 views

On comparing two different notions of compactly generated space

I have encountered, in different circumstances, the following two slightly different categories: The full category of $\mathsf{Top}$ consisting of all objects that are: a) topological spaces ...
2
votes
0answers
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 ...
0
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0answers
24 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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2answers
82 views

Converting topological problems to algebraic problems [closed]

We can convert topological problems to algebraic problems by functors. Can you give any example about this situation?
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0answers
34 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 ...
2
votes
0answers
52 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 ...
0
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0answers
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, ...
1
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3answers
40 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{...
13
votes
5answers
1k views

Mnemonic for the fact that a right(left) adjoint functor preserves limits(colimits)

A right adjoint functor preserves limits. Dually a left adjoint functor preserves colimits. I often forget which is which. Of course, you can look up a book on category theory or use internet. But it'...
1
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3answers
59 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....
1
vote
2answers
63 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 ...
1
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0answers
37 views

Is the lattice of normal subobjects of algebraic theories always modular?

The lattice of (sided) ideals of a ring is always modular, basically by the distributivity of the powerset lattice. I know this argument also works for norma subgroups. I was wondering whether this ...
1
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0answers
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 ...
3
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0answers
66 views

Unit and co-unit of Exponential Product and Sets

Let $\cal C$ be a category with binary products and let $Y$ and $Z$ be objects of $\cal C$. The exponential object $Z^Y$ can be defined as a universal morphism from the functor $–×Y$ to $Z$. ...
1
vote
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\...
1
vote
1answer
52 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 ...
2
votes
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 ...
1
vote
0answers
45 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 \...
4
votes
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}...
0
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0answers
11 views

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, ...
0
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0answers
10 views

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 ...
1
vote
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 ...
1
vote
1answer
33 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 $\...
2
votes
2answers
91 views

Universal property of tensor products / Categorial expression of bilinearity

Let $V$ and $W$ be linear spaces. According to Wikipedia, the universal definition of the tensor product $V \otimes W$ satisfies the following property: There is a bilinear map (i.e., linear in each ...
2
votes
0answers
63 views

On direct sum and direct product of groups

I understood the definitions of direct sums and direct products of groups (rings, modules, etc). The definitions of them can be given in terms of universal properties - certain commutative diagram in ...
2
votes
1answer
52 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 $\...
2
votes
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 $\alpha\subseteq A\times A'$...
1
vote
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 ...
3
votes
1answer
224 views

Semisimple objects in abelian categories

Let $\mathcal A$ be any Grothendieck abelian category and $0 \neq M \in \cal A$ an object. It is true that $M$ admits a simple subquotient? It is certainly true for $\mathcal A=R-Mod$ since $M$ ...
2
votes
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 ...
1
vote
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 ...
0
votes
0answers
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' \...
1
vote
1answer
95 views

A category of relations - or two different?

Objects in the category Rel2 (my notation) are the relations $r\subseteq A\times B$, $r'\subseteq A'\times B'$ (the morphisms in Rel) and morphisms are pair of relations $\alpha\subseteq A\times A'$ ...
4
votes
3answers
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 $d : X\times X \to [0,\infty]$...
0
votes
2answers
22 views

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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0answers
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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votes
0answers
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 ...
0
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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 ...
6
votes
1answer
412 views

Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
2
votes
1answer
40 views

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 $C^{op} \to D$. ...