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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5answers
80 views

Morphism epimorphism if and only if surjective

In the category of sets, I want to prove that a morphism is an epimorphism if and only if it is surjective. In both directions, I'm having a hard time approaching this problem. This is how far I ...
1
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1answer
49 views

Identity morphism requirement in categories

In order to verify a category, you need to show that the class of morphisms respect associativity and contains an identity morphism. I'm looking for a class of morphisms that doesn't contain an ...
6
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0answers
105 views

Strongly unbiased symmetric monoidal category

Let $\mathcal{C}$ be a category. Define a strongly unbiased symmetric monoidal structure on $\mathcal{C}$ to be a rule which associates to every finite set $I$ a functor $\mathcal{C}^I \to ...
2
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1answer
100 views

$\mathbf{Rng}$ is not a subcategory of $\mathbf{Ab}$?

I read in the book Categories and functors (Pareigis) that the category of rings Rng is not a subcategory of the category of abelian groups Ab. I didn't understand the argument the author used: ...
3
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1answer
67 views

Do we have accepted terms for semigroups and semigroupoids without identities?

I would like to call a semigroup without an identity a "pure semigroup" and a semigroupoid without any identity a "pure semigroupoid". I'm not sure whether such things have been defined in literature ...
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2answers
94 views

In definition of a category , what is the meaning of 'consists of'

A category $\mathsf C$ consists of the following three mathematical entities: A class $\operatorname{ob}(\mathsf{C})$, whose elements are called objects; A class $\hom(\mathsf{C})$, whose ...
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0answers
45 views

When modular tensor categories are equivalent?

I would like to know when we say that two modular tensor categories are equivalent. Is it true that two modular tensor categories are equivalent if they are equivalent as monoidal categories? Or do ...
5
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2answers
95 views

Why is $Set$ not equivalent to $Set^{op}$?

I understand this question already has an answer on here, but I do not understand how the answer justifies that $Set$ not equivalent to $Set^{op}$. The explanation is that if they were equivalent and ...
6
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1answer
59 views

Can we simultaneously freely adjoin both limits and colimits to a category?

I'm aware that given a category $C$, it's possible to take the free (co)completion of $C$ in order to freely adjoin (co)limits to $C$, in the sense that we can construct a left adjoint to the ...
3
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0answers
60 views

Category which is not a subcategory of a complete category

By Yoneda lemma every small category $\mathcal{C}$ can be embedded in the cocomplete (and complete) category $[C^{op} ,\textbf{Set}]$. Most examples I know of large categories which are not complete, ...
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1answer
71 views

Showing $\mathcal{A}^*$ is Well-Quasi-Ordered

I was told that the problem below is supposed to prove that $\mathcal{A}^*$ (defined below in "Attempted Solution") is well-quasi-ordered. Two problems come to my mind: How is this problem proving ...
5
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1answer
48 views

The category of Lie algebra representations

A representation of a Lie algebra $\mathfrak{g}$ on a vector space $V$ is a homomorphism of Lie algebras $\mathfrak{g} \to \mathfrak{gl}(V)$. We define morphisms between representations as ...
2
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1answer
36 views

Understanding this formula constructing the limit of any small diagram in $\mathbf{Set}$

I'm reading Limits In Category Theory by Scott Messick, and trying to understand the formula on page 8, for Theorem 5.2, which states: Let $F : \mathscr{J} \to \mathbf{Set}$ be any small diagram in ...
2
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1answer
29 views

Left adjoint to forgetful from modules to abelian groups

What is the left adjoint to the forgetful functor $U : R-\mathsf{Mod} \to \mathsf{Ab}$? Note here that $R$ is a general ring, not necessarily commutative. I've seen that they define it as $F A = R ...
5
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0answers
88 views

On Neeman's new axiom (GTR3) for triangulated categories

In the paper Some new axioms for triangulated categories, Neeman introduces a list of axioms on an additive category $\mathbf T$ with a given self-equivalence $\Sigma\colon \mathbf T\to \mathbf T$. ...
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0answers
44 views

In the category $\mathsf{Set}$ regular epis are stable under pullback

I read that in the category of sets regular epis are stable under pullback. I know that epis in $\mathsf{Set}$ are surjective maps, but I don't see why the pullback of a regular epi should be ...
1
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2answers
88 views

Why not just define equivalence relations on objects and morphisms for equivalent categories?

My question is regarding this blog post https://unapologetic.wordpress.com/2007/05/30/equivalence-of-categories/ which I will paraphrase below: The author gives the example of a category ...
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0answers
46 views

Equality on classes

On every set $A$ there is an equality $=_A$ defined. This is necessary to speak about injectivity, surjectivity, the group axioms and so ... In category theory, one uses classes to be collections of ...
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0answers
32 views

Size issues in 2-categories

I was playing a bit the 2-category Cat trying to have a better understanding of the notion of a 2-category (strict I guess). The usual definition of a category that I use assumes that $Hom(A,B)$ is a ...
1
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1answer
45 views

definition of the directed colimit of a functor

Let $(\Lambda,\le )$ be a directed set, which we can understand as a small category: The set of all objects is $\Lambda$ and for $\lambda,\lambda '\in \Lambda$ there exists an unique morphism ...
7
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1answer
76 views

Why would the category of topological spaces be a balanced category (i.e. monic epimorphisms are isomorphisms)?

I've just read on this page that For example, $\mathsf {Set}$ (the cateogry of sets), $\mathsf {Grp}$ (the category of groups), and $\mathsf {Top}$ (the category of topological spaces) are all ...
3
votes
2answers
109 views

What is the most general category in which exist short exact sequences?

Let $A,B,C$ be objects, $0$ the final object, and $f:A\to B$ and $g:B\to C$ morphisms in some category. Consider the sequence: $$ 0 \to A \to B \to C \to 0\;. $$ I would like to say something ...
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0answers
38 views

How to turn a product in a tensor product [duplicate]

I want to prove that: $$k[V_1 \times V_2] \simeq k[V_1] \otimes k[V_2]$$ where $\otimes$ is the tensor product of $K-$algebras, $V_i$ are varieties and $k[V_i]$ are the coordinate rings. I was ...
1
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1answer
24 views

Converse to: Equivalent conditions for a preabelian category to be abelian

In the following question: Equivalent conditions for a preabelian category to be abelian. How is the converse easily shown? I see why every monomorphism, f, is the kernel of the cokernel of f, but why ...
2
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0answers
68 views

Synthetic differential geometry and algebraic geometry

I am reading here and there about basic synthetic differential geometry. One of the central ideas seems to be that it should be developed in a suitable topos, hence, in particular, a cartesian closed ...
2
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1answer
59 views

A category whose classifying space has nontrivial higher homotopy groups

The classifying space of a category $\scr{C}$ is obtained by taking its nerve $N\scr{C}$, which is the simplicial set defined by $$ N\mathscr{C}_n:= \mathrm{Fun}([n],\mathscr{C}) $$ and the ...
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1answer
55 views

Topos $M_{2}$ from Goldblatt's book

Why in Topoi Goldblatts' book, on page 123, is said that the topos $M_{2}$ has only two truth-values and in the following, page 140 (example 4), we have truth tables for truth-morphisms in $M_{2}$ ...
4
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0answers
50 views

Examples of calculating perverse sheaves on algebraic varieties with easy stratification.

This question is also asked in mathoverflow http://mathoverflow.net/questions/232589/examples-of-calculating-perverse-sheaves-on-algebraic-varieties-with-easy-strati I have been learning intersection ...
6
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0answers
108 views

Most general form of Cayley's theorem?

For many classes of algebraic structures, there exists a family of structures such that any member of the class can be embedded in some member of the family (groups and symmetric groups, unital rings ...
2
votes
1answer
35 views

Tensor left adjoint as a bi-functor?

Working in the category of modules over a fixed ring $A$ $\operatorname{Mod}_A$, we have the adjunction: $$F(X) = X\otimes_A M \dashv G(X) = \operatorname{Hom}_A(M,X)$$ for a fixed module $M$. Is ...
1
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1answer
48 views

Why are the isomorphisms and bijective morphisms not identical in the category of Pos?

Let $\text{Pos}$ be the category of partially ordered sets and monotonic functions. A morphism $f$ is called an isomorphism if there is a morphism $g$ such that $f\circ g$ and $g\circ f$ are identity ...
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1answer
47 views

why is the inverse image defined by the right kan extension instead of the left?

Why don't we define the inverse image of a sheaf to be the left kan extension and then take the sheafification?
3
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1answer
46 views

Category of finitely presented $R$-algebras cartesian closed?

On page 26 of these notes, in the paragraph between formulas $(63)$ and $(64)$, the author says the category of finitely presented $R$-algebras is cartesian closed. I thought this category was ...
7
votes
1answer
78 views

The category of locally $P$ spaces

Let $P$ be a class of topological spaces (for example, compact spaces). The class of locally $P$ spaces consists of those spaces in which every point has a neighborhood basis consisting of $P$ ...
3
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0answers
40 views

Taking the topological dual in terms of category theory

Consider the categories $\mathbf{Vect}$ of vector spaces $X$ with linear maps and $\mathbf{TopVect}$ of topological vector spaces $(X, \tau)$ with continuous linear maps both over $\mathbb{R}$. ...
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1answer
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Simplicial Sets, Simplicial Monoids, and Simplicial Free Monoids [closed]

What is the difference between simplicial sets and simplical monoids? Does there exist an adjunction between these two structures? Does there exist a simplicial free monoid over some set which in turn ...
5
votes
2answers
107 views

Monad as not trivial adjunctions

It is well known that a monad $(T, \mu, \eta)$ can be factorized in multiple ways as adjunctions, and that in some sense, Kleisli is the initial factorization while Eilenberg-Moore is the final ...
1
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2answers
59 views

Arrow of Arrow Set is a Functor?

My question is whether arrows between arrows (or arrows in $\bf{Set}^{\rightarrow}$) are, by definition, functors who, in the general sense, map from $\bf{Set}$ to itself. The definition of a functor ...
2
votes
2answers
42 views

Why ${0,1}$ is subobject classifier in $Sets$

I don't want to use the fact that pullbacks in $Sets$ are just subset of $X \times Y$ Here $p_1$ and $q_1$ are unique (because $1$ is terminal object), $f(1) = 1$. We want to have unique $g$ by ...
0
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0answers
26 views

Image of presheaf morphism

I know the definition of the image of a morphism in a category, but I'm wondering if in my specific case there is a nice simplification. Let ...
0
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0answers
25 views

Forgetful functor preserves coequalizers.

Let $\mathcal{T}$ be an algebraic theory, i.e a category with a denumerable set of objects $\left\{T^0,T^1,\ldots,T^n,\ldots\right\}$ where each $T^n$ is the product of $T^1$. Let ...
6
votes
1answer
112 views

Does $f\otimes \operatorname{Id} = \operatorname{Id}$ imply $f= \operatorname{Id}$?

Let $R$ be a commutative ring, and $X$ an $R$-module. If an $R$-endomorphism of $X$ satisfies $f\otimes \operatorname{Id}_X = \operatorname{Id}_{X\otimes X}$, is it true that ...
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0answers
22 views

Three different definitions of Modular Tensor Categories

I have found three different definitions of Modular Tensor Categories. I want to know if anybody can give a sketch of proof for their equivalences (some parts are easy of course) A Modular Tensor ...
3
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0answers
32 views

An alternative of projective dimension in triangulated categories

Does anybody know anything about an alternative of the notion of projective dimension (defined in abelian categories with enough projective objects) in triangulated categories?
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0answers
51 views

Definitions of Weil Algebras

I am confused by several definitions of Weil Algebras and their connection to each other. Kock's book on synthetic differential geometry defines a Weil algebra over a ring $R$ as an $R$-algebra of ...
2
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0answers
39 views

monoid action on categorical limits

If $\mathscr D$ is a category and $\mathscr C$ is a complete category, consider a functor $F\colon \mathscr D \to \mathscr C$. Let $\Sigma_F$ be the collection of all functors $\sigma \colon \mathscr ...
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1answer
51 views

can a category,between any two of whose objects there are maps,not necessarily unique, be regarded as a preordered set?

In my book on category theory it has been stated that a category in which for each pair of its objects there is "at most" one map between them can be regarded as a preordered set. I do not know the ...
2
votes
2answers
56 views

Leinster question on isomorphic functor categories

Hi so the question is basically to prove that for categories A and B $$ [A,B]^{op} \simeq [A^{op},B^{op}] $$ where these are functor categories. So my first port of call was to write out what the ...
1
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2answers
39 views

Are operations upon index-classes (rather than index-sets) allowed in mathematics?

Why we always see extended operations, like arbitrary unions, products, etc. in different parts of mathematics in the form of extensions of finite ones upon arbitrary sets (called index set)? Why ...
3
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0answers
51 views

Leinster's Category theory model answers

Is there a source of model answers for the examples in Leinster's Basic Category theory? The book has an excellent set of questions but sadly no apparent answers are provided, despite the book being ...