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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Morphisms of a category with one object, which is a group

I've read two articles(1,2), but still have a question about structure of group, say $G$, as an one-object category. Let's call this category $C$. I understand that morphisms of $G$, which is ...
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1answer
37 views

limits' definition

Please, can somebody help me? I was given the following definition of the LIMIT: Let $I$ be a small category and $F:I \to C$ a covariant functor ( where $C$ is a category), $K \in Ob(C)$, ...
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0answers
13 views

When does the first cohomology group commute with inverse limit?

Let $M_i,i\in\mathbb{N}$ be an inverse system of continous, discrete G-modules and let $M=\varprojlim M_i$. Under what conditions on $M$ and $M_i$ do we have $\varprojlim H^1(G, M_i) \cong H^1(G, M)$? ...
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3answers
639 views

Proving the existence of a proof without actually giving a proof

In some areas of mathematics it is everyday practice to prove the existence of things by entirely non-constructive arguments that say nothing about the object in question other than it exists, e.g. ...
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0answers
29 views

cokernel in the pointed set category $Set.$ [on hold]

Please, can someone give me an example of cokernel in the pointed set category? Thanks!
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1answer
25 views

Quasicoherent sheaf on the functor of points is the same as on the scheme itself

I've seen the definition of a quasicoherent sheaf $\mathcal{F}$ on an arbitrary functor $$ X : CRing \to Sets $$ as a specification of an $R$-module $\mathcal{F}(x)$ for each $R$-point $x \in X(R)$ ...
7
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1answer
45 views

Is I-adic completion a ring epimorphism?

Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For any $n \ge 1$, the ring homomorphism $R \rightarrow R/I^n$ is surjective, hence an epimorphism in the category of rings. What about ...
2
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2answers
32 views

Defintion of the $\mathrm{hom}$ functor in category theory

I am going through some notes 'Physics, Topology, Logic and Computation: A Rosetta Stone', on category theory. We first define the opposite category: Given a category $C$, we define the opposite ...
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0answers
33 views

Frobenius-Perron dimension on a fusion category

Let $C$ be a fusion category with simple objects $V_i\in I$. The fusion rule is $V_i\otimes V_j \cong N_{i,j}^k V_k$. The Frobenius-Perron dimension of a simple object $V_i$, $\mathrm{FPdim}(i)$, is ...
4
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1answer
41 views

Tensor of cocomplete categories

Let $C$, $D$ and $E$ be cocomplete categories. Is there a construction $C \otimes D$ such that there is a correspondence between functors $C \otimes D \to E$ preserving colimits and functors $C ...
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0answers
31 views

an example of direct product in Ab

Can someone help me with this? Are there direct products in the category $Ab$ (the category of the abelian groups)? If yes, please, can you give me an example? Thank you?
2
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1answer
48 views

“Coforgetful” functors?

Let $F : \mathbf{Sets} \to \mathbf{Cat}$ be the free functor that takes the elements of a set to the objects of a discrete category. Does it has a left adjoint? Awodey (2010 p.249 ex.8) says it does, ...
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1answer
940 views

Which is the most powerful language, set theory or category theory? [on hold]

As far as I know, mathematics is written based on a language which can be for example set theory or category theory. My concern is about the power of these languages. How can we realize which language ...
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2answers
76 views

Is there any comprehensive definition of the opposite category $\mathcal C^{op}$ of a category $\mathcal C$?

In the definition of the opposite category $\mathcal C^{op}$ of a category $\mathcal C$, it is said that : objects remain the same and arrows' directions are changed (that is, and arrow $f:A \to B$ in ...
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0answers
60 views

Order-Preserving Bijection $f:A\to A^*$?

Let $A$ be a well-quasi-ordered infnite set. Does there exist an order-preserving bijection $f:A\to A^*$, where $A^*$ is the free monoid over $A$ under the subword ordering? Would this subword ...
1
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1answer
38 views

Exactness of a right adjoint functor

Let $F: \mathcal{A} \longrightarrow \mathcal{B}$, $G: \mathcal{B} \longrightarrow \mathcal{A}$ be two additive functors between abelian categories, such that $(F, G)$ is an adjoint pair. I want to ...
10
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0answers
76 views

Category Theory Zoo

There are a few very useful websites when it comes to either finding a specific object with certain properties (and maybe lacking other properties) or finding out which properties a certain object ...
3
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0answers
79 views

Name for categories with a certain property on coproducts

Is there a name for categories with the following property: The category has zero morphisms, coproducts, and for each family $(X_i)_{i \in I}$ of objects the natural map $$\hom(Y,\bigoplus_{i \in I} ...
11
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3answers
718 views

Example of a category without product

A question was raised in our class about the non-existence of product in a category. The two examples that came up in the discussion was the category of smooth manifolds with boundary and the category ...
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1answer
27 views

Is the braid category biclosed and bicomplete?

Let $\mathcal{B}$ be the braid category, as in Categories for the Working Mathematician §XI.4 p.262 (objects are natural numbers and morphisms are the braids $n\to n$). Then this can be given a ...
7
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1answer
76 views

How does Hartshorne's definition of group schemes encode the law for the neutral element?

Hartshorne's Algebraic Geometry says A scheme $X$ with a morphism to another scheme $S$ is a group scheme over $S$ if there is a section $e\colon\;S\to X$ (the identity) and a morphism ...
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1answer
42 views

Confusion regarding definition of adjoint functor - Hilton and Stammbach

While defining Adjoint functors in their book A Course in Homological Algebra, Hilton and Stammbach said the follwing: Let $F:\mathfrak{S}\rightarrow \mathfrak{M}_{\Lambda}$ be the free functor ...
3
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1answer
56 views

Recovering $\mathsf{Ab}$ from $\mathbb{Z}$-mod in a closed symmetric monoidal category

Given the usual definition of a module over a ring it is trivial to show that a $\mathbb{Z}$-module is an abelian group (it is just by definition). But my question concerns recovering this idea in a ...
9
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1answer
61 views

Categorification of algebra structures

This might be a bit of a soft question. Take a $\mathbb{C}$-linear category. Form the complex vector space spanned by its objecs modulo exact sequences. This construction is, as far as I know, the ...
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0answers
29 views

Complete atomic boolean algebras as coalgebras of some endofunctor on Set

I was hoping to use the fact that CABAs are powersets with extra structure on the morphisms to find an endofunctor $F:\text{Set}\to\text{Set}$ with $\text{Set}^{op}\simeq\text{Coalg}F$. I started by ...
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1answer
56 views

Hartshorne's notation $s: U \to \coprod_{\mathfrak{p} \in U} A_{\mathfrak{p}}$

I understand that Hartshorne is defining the sections on an open set of $\operatorname{Spec} A$ as functions from the points $\mathfrak{p} \in U$ into their localizations $A_{\mathfrak{p}}$ that are ...
0
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1answer
61 views

How do you know two morphisms are equal (without using elements)

Given two morphisms in some category, which is to say that you are told that $f$ and $g$ are in the cat $C$ and nothing more, how can you know if they are equal? Normally we appeal to the elements of ...
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1answer
36 views

Is the associated bundle construction a bifunctor?

Let $\mathsf{Prin}_G$ be the category of (right) $G$-principal bundles, with a morphism from the bundle $p: P \to M$ to the bundle $p': P' \to M'$ being a pair of arrows $\chi: P \to P'$ and ...
1
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1answer
35 views

Is this definition of $(c\downarrow G)$ the slice category or is it something else?

I'm learning about the slice and coslice category constructs, and I think I understand the basics from Wikipedia. However, in this lecture script (in German), there's another definition given, which ...
2
votes
1answer
23 views

In the coslice category, why are the morphisms from the terminal category element inclusions?

Wikipedia's definition of the coslice category uses the symbols $i_B$ for the objects $(B,i_B)\in(A\downarrow\mathcal C)$. Similarly, the definition of the slice category uses the symbol $\pi_B$. ...
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0answers
47 views

Presheaves,simplicial sets, evaluation functor, Yoneda lemma,hom-functor

In the bottom of page 8 in this paper, how it follows from the Yoneda lemma, that the evaluation functors $E_C$ are precisely of the form $\mathrm {hom(hom(-,}C),-)$ ?
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1answer
63 views

How to simulate power sets in structural set theory (ETCS)?

How to simulate power sets in structural set theory (ETCS)? (nlab) It turns out that one of the primary attributes of a structural set theory is that the elements of a set have no “internal” ...
0
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4answers
57 views

Defining natural transformations based on generalized elements?

Let $F : \mathbf{C} \to \mathbf{D} : G$ be two functors between categories $\mathbf{C}$ and $\mathbf{D}$. A natural transformation $\eta$ from $F$ to $G$ is a collection of morphisms $\eta : FC \to ...
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1answer
37 views

Is there a connection between local soundness and completeness in proof theory, and free objects in category theory?

I was watching Frank Pfenning's lecture series on proof theory, where he described the notions of local soundness, and local completeness. He described local soundness of a logical connective as, ...
2
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1answer
37 views

Proving a set is open in a locally ringed space $(X,\mathscr O_X)$

Let $(X, \mathscr O_X)$ be a locally ringed space and let $A = \Gamma(X,\mathscr O_X)$ be the global sections. For $f\in A$, define the "distinguished open base" as: $$D(f) = \{x\in X : \pi_x(f) ...
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0answers
60 views

Factorization to Prove Properties Shared between Categories

Question: Let $[n]$ be the finite chain category $[0\to1\to\cdots\to n]$. Then define an infinite sequence $$\mathcal{F}: \text{Seq}\to C$$ in $C$ to be essentially finite if $\mathcal{F}$ factors ...
4
votes
1answer
63 views

Name of a category constructed from the action of a group on a category

Let $G$ be a group acting on a category $C$. That is we have a morphism of groups $G \to Aut(C)$. We can now form a new category as follows: Its objects are tuples consisting of an object $x$ of ...
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0answers
46 views

Data about a morphism and Data about a category [closed]

I have been trying to develop or find theorems about probabilities over categories. This would include probabilistic categories, where morphisms and equations over morphisms are assigned ...
1
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1answer
51 views

How are non-bijective morphisms “reversed”?

I have the following general definitions: 1) The dual graph $G^\mathrm{op}$ of a graph $G$ is obtained by simply reversing all the arrows (e.g. by interchanging the roles of the source and target ...
6
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0answers
69 views

cofree coalgebra: explicit description from its universal property

Let $k$ be a commutative ring. There is a forgetful functor $$U : \mathsf{Coalg}_k \to \mathsf{Mod}_k$$ from $k$-coalgebras to $k$-modules. This has a right adjoint $R : \mathsf{Mod}_k \to ...
0
votes
1answer
32 views

How do you define such map $(C^B \times B^A) \to C^A$?

Suppose that $\mathbf{C}$ be cartesian closed and $B$ is an object of it. We define two functors $\mathbf{C} \times \mathbf{C} \to \mathbf{C}$ by $$ C^B \times B^A \qquad\text{and}\qquad ...
3
votes
1answer
67 views

The cofree coalgebra using adjoint functor theorems

Let $k$ be a commutative ring. There is a forgetful functor $$U : \mathsf{Coalg}_k \to \mathsf{Mod}_k$$ from $k$-coalgebras to $k$-modules. This has a right adjoint, called the cofree coalgebra on a ...
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1answer
40 views

How to define morphisms involved with empty sets

I am beginning to learn something about categories and I have a little question. In the category theory, how to define Mor(A, B) if A or/and B is/are empty set(s)? And why they are defined those ...
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0answers
20 views

Subcategories of Spaces via Orthogonality

I remember seeing a post on math stack exchange describing how to define subcategories of topological spaces via lifting and orthogonal properties with respect to simple continuous maps. Now I cannot ...
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0answers
27 views

Cohomology induces a functor

Let $\mathcal{A}$ be an abelian category and $(X^{\cdot}, d_{x}^{\cdot})$ a cochain complex in $\mathcal{A}$. Let $Ch(\mathcal{A})$ be the category of cochain complexes in $\mathcal{A}$. I define the ...
3
votes
1answer
90 views

Adjoint to $\mathsf{Proj}$? - A quest to understand categories of graded objects.

I've been having a hard time with graded objects in algebraic geometry for some time. Lately I realized a lot of my difficulties come from not having any idea at all of where graded objects live. ...
4
votes
2answers
54 views

In what sense right dual and braiding structure respect the tensor product structure in a monoidal category?

Throughout let $(\mathscr{C}, \otimes, \mathbf{1})$ be a monoidal category (I suppressed unitors and associators for simplicity). The usual definition a rigid monoidal category is done in two steps: ...
4
votes
1answer
39 views

Automorphism of a category sends objects to isomorphic objects?

quick question for my better understanding: Assume you have an additive category $\mathcal{C}$ and an automorphism $\Sigma$ of this category. Does $\Sigma$ send objects to isomorphic objects? If it ...
1
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1answer
50 views

Exact sequence splitting naturally

So I encountered a term that I don't quite recognize from lecture. The professor stated that a certain short exact sequence splits naturally, but I don't understand what the naturally condition is in ...
3
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1answer
44 views

If $F : \mathbf{C} \to \mathbf{D}$ is an equivalence, does $\alpha_{GD} = G(\beta_D)$ hold in general?

Let $F : \mathbf{C} \rightleftarrows \mathbf{D} : G$ be an equivalence with natural isos $\alpha : 1_\mathbf{C} \to GF$ and $\beta : 1_\mathbf{D} \to FG$ witnessing the referred equivalence. I wonder ...