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Questions tagged [category-theory]

Categories are structures containing objects and arrows between them. Many mathematical structures can be viewed as objects of a category, with structure preserving morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seemingly different areas of mathematics.

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Closed model categories in the sense of Quillen [1969] vs the modern sense

The modern definition of (closed) model category differs in two ways from Quillen's 1969 definition: Model categories are now required to be complete and cocomplete, whereas Quillen only asked for ...
Zhen Lin's user avatar
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33 votes
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Semirings induced by symmetric monoidal categories with finite coproducts

A symmetric monoidal category with finite coproducts is by definition a symmetric monoidal category $(\mathcal{C},\otimes,1,\dotsc)$ such that the underlying category $\mathcal{C}$ has finite ...
Martin Brandenburg's user avatar
30 votes
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1k views

Sheaves of categories

I recently read an answer on this MO post explaining that one reason people are interested in higher category theory is to make reasonable sense of something like a "sheaf of categories" on a ...
Alex Saad's user avatar
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27 votes
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How does the internal language of a topos come to be?

There are several books and articles on topos theory which mention the internal language, but I can't manage to see the big picture from any of them. I would like a soft explanation of how the ...
Arrow's user avatar
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25 votes
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Does the category of algebraically closed fields of characteristic $p$ change when $p$ changes?

EDIT I've now posted this question on mathoverflow. It probably makes sense to post answers over there, unless someone prefers posting here. Let $\mathrm{ACF}_p$ denote the category of algebraically ...
tcamps's user avatar
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24 votes
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611 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 has....
Tim's user avatar
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23 votes
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499 views

Is there an endofunctor of the category of sets that maps $\kappa$ to $\kappa^+$?

For already some time I am slightly bothered by the following question about endofunctors of the category of sets: Is there an endofunctor of set which maps each infinite cardinal $\kappa$ to a set of ...
Jakub Opršal's user avatar
22 votes
0 answers
2k views

Limits and colimits in the category of fields

It is said that the category of fields $\mathsf{Fld}$ is ill-behaved, for example it is not an algebraic theory, does not have initial or terminal objects. In particular it is not presentable. On the ...
Martin Brandenburg's user avatar
20 votes
0 answers
524 views

What is the structure preserved by strong equivalence of metrics?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are (topologically) equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
Keshav Srinivasan's user avatar
19 votes
0 answers
1k views

Learning roadmap to Topological Quantum Field Theories from a mathematics perspective

I want to learn TQFT's and am looking for review articles or books. My mathematics knowledge is limited to one year of graduate course in Algebra (Groups,Rings,Fields,Categories, Modules and ...
user90041's user avatar
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19 votes
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Free medial magmas

A medial magma is a set $M$ with a binary operation $*$ satisfying $$(a*b)*(c*d) = (a*c)*(b*d)$$ for all $a,b,c,d \in M$. Medial magmas constitute a finitary algebraic category $\mathsf{Med}$, ...
Martin Brandenburg's user avatar
15 votes
0 answers
242 views

In which algebraic theories do 'free' and 'projective' coincide?

Free models of algebraic theories are always projective objects in the category of models, but the converse is not always true. For instance, some (actually, all) projective modules are direct ...
Arrow's user avatar
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14 votes
0 answers
324 views

Is there a Measure-Theoretic Proof of this new Result from Categorical Probability?

Recently, I stumbled across a new paper in categorical probability. Interestingly, they prove a result which may be formulated in purely measure-theoretic terms about which they note that "As ...
Small Deviation's user avatar
14 votes
0 answers
300 views

Transfinite horizontal composition

Suppose that you have two sequences $\{F_n\}$ and $\{G_n\}$ of endofunctors of $\bf A$ arising as strings of adjoints $\cdots\dashv F_{n-1}\dashv F_n\dashv F_{n+1}\dashv \cdots$, $\cdots\dashv G_{n-1}\...
fosco's user avatar
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13 votes
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534 views

Natural transformation = parametric polymorphic function in “structure categories”?

By a “structure category” I mean a concrete category that contains as objects all spaces of a particular type of structure, and as morphisms, functions that preserve that type of structure. I.e. the ...
user56834's user avatar
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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 $\varprojlim H^1(G, M_i) \cong H^1(G, M)$? ...
Sameer Kulkarni's user avatar
13 votes
1 answer
323 views

Is there such a thing as 'overtification' (dual to compactification)?

The dual to the notion of compactness for spaces is overtness. This duality is not manifest in the category of spaces but rather in the quantifiers used to define these notions. Is there a process ...
David Myers's user avatar
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13 votes
1 answer
440 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 \mathsf{...
Martin Brandenburg's user avatar
12 votes
0 answers
333 views

Intuition for AB5 and Grothendieck categories

I'm trying to get some intuition for AB5 categories and Grothendieck categories by asking primitive questions. First of all, why ask for exact filtered colimits? Are they there simply to have some ...
Arrow's user avatar
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12 votes
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205 views

Practical approaches to working with nonplanar commutative diagrams?

The 4-associahedron is the 4-dimensional version of Mac Lane's pentagon diagram. If you look at Trimble's notes on tetracategories, you can see the obvious difficulty in working with such a diagram (...
user avatar
11 votes
0 answers
486 views

Good reference for abelian category in Weibel's homological algebra

I want to learn some homological algebra from Weibel's book. After reading section 1.1 and 1.2, I find I lack knowledge of abelian category. In section 1.2, Weibel says the constructions on modules in ...
Z. He's user avatar
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11 votes
0 answers
213 views

Example where the associated sheaf does not exist

What is a simple example of a topological space $X$ and a complete category $\mathcal{C}$ such that the inclusion $\mathbf{Sh}(X,\mathcal{C}) \hookrightarrow \mathbf{PSh}(X,\mathcal{C})$ has no left ...
Martin Brandenburg's user avatar
11 votes
0 answers
106 views

Topology and Categories for the Visually Impaired

Are there any books (and/or other reference material) for teaching topology and/or category theory for the visually impaired? For example, a teacher may have experience with tactile learning tools ...
ZxJx's user avatar
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0 answers
189 views

Algebraic Geometric Analogue of Brown's Representability

Brown's representability theorem is very usefull to show that the functor $$X \rightarrow H^i(X,A)$$ is representable. I would be interested to see if there exists an analogue of this statement in ...
curious math guy's user avatar
11 votes
0 answers
574 views

Applications of Jordan-Holder theorem in an abelian category

The Jordan-Holder theorem says that any chain of subobjects of a finite lenght object can be refined to a composition series, and that any composition series has the same lenght. This theorem holds ...
less's user avatar
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11 votes
0 answers
143 views

Relationship between Haefliger structures and principal $\Gamma^q$-bundles

A Haefliger structure on a smooth manifold is a cocycle with coefficient in the Haefliger groupoid $\Gamma^q$. This generalises the notion of foliation of codimension $q$. We know that the classifying ...
W. Rether's user avatar
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11 votes
0 answers
581 views

Uses of category-theory

I am a graduate student currently working my way through an introductory course in category theory. A question I have for this theory is why it is usefull? Now i know this is a rich theory that is ...
M.v.Roozendaal's user avatar
10 votes
0 answers
171 views

Is the octahedral axiom really equivalent to the 4 x 4 lemma?

$\require{AMScd}$I make reference to this paper. I've recently become aware that there's an annoying variety of a priori distinct (but "known" to be equivalent up to adding other hypotheses) ...
FShrike's user avatar
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10 votes
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276 views

Indexed Yoneda in the Elephant

I've been trying to gather definitions and results to understand the internal Diaconescu theorem, for I wish to read Joyal-Tierney's An Extension of the Galois Theory of Grothendieck. My question ...
interregno's user avatar
10 votes
0 answers
139 views

Which (co)limits exist in the simplex category?

Is there a simple description of those colimits that exist in the simplex category $\mathbf{\Delta}$ (of finite linear orders and non decreasing maps) ? It is easy to find examples of diagrams for ...
dicemaster666's user avatar
10 votes
0 answers
248 views

Higher homotopies

It is a standard fact that two morphisms $f,g \in S_1$ (with the same vertexes) in an $\infty$-category $S$ are homotopic in the sense that there is a $2$-simplex $\sigma \in S_2$ such that $d_0\...
lola's user avatar
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10 votes
0 answers
420 views

Stone Duality: What are $\sigma$-Algebras Dual To?

Stone duality, one of many dualities between certain lattices and certain topological spaces, asserts that there is a contravariant categorical equivalence between the category $\text{Bool}$ of ...
user avatar
10 votes
0 answers
141 views

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 ...
Shaun's user avatar
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10 votes
0 answers
276 views

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 ...
Luke's user avatar
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10 votes
1 answer
374 views

Map $\langle X, Y\rangle \to \text{Hom}(\pi_n(X), \pi_n(Y))$ bijection?

I have two questions. How do I see that the map$$\langle X, Y\rangle \to \text{Hom}(\pi_n(X), \pi_n(Y)), \quad [f] \mapsto f_*,$$is a bijection if $X$ is an $(n - 1)$-connected CW complex and $Y$ is ...
user avatar
10 votes
0 answers
198 views

UCT and Künneth, Hom-Tensor adjunction

A few days ago, there was a similar question in an other context. Künneth: Consider the tensor product of modules $H_i(X;\mathbb{Z})\otimes A$, where $A$ is an abelian group, $X$ is a topological ...
Sabrina G.'s user avatar
10 votes
1 answer
187 views

Liars, adjunctions, and functions $f : S \rightarrow UFS$. Does this lead anywhere interesting?

A student of mine was recently given the following question: "At least one of us is lying," said Andrew. "Only one of us is lying," said Bertas. "Squeak, two of us are lying," said the ...
goblin GONE's user avatar
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10 votes
0 answers
514 views

Relative chinese remainder theorem and the lattice of ideals

Let $R$ be a ring with two-sided ideals $I,J$. The proof of the "absolute" chinese remainder theorem revolves around the fact that if $I,J$ cover $R$ in the lattice of ideals, i.e $I+J=R$, then the ...
Arrow's user avatar
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10 votes
0 answers
349 views

What are some topos-theoretic insights about $G$-sets?

Since a $G$-set is just a functor $G\longrightarrow \mathsf{Set}$, the category of $G$-sets seems to be a simple example of a topos. What are some topos-theoretic insights into $G$-sets? Insights ...
Arrow's user avatar
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10 votes
0 answers
306 views

Uniform Spaces: Completeness

Attention This thread has been generalized to uniform spaces as general metric spaces. Context The context was the equivalence: $$K\text{ compact}\iff K\text{ totally bounded, complete}$$ That is a ...
C-star-W-star's user avatar
10 votes
0 answers
985 views

Morita-invariance of Hochschild (co)homology.

Ok, I’m reading the paper Homology and cohomology of associative algebras. A concise introduction to cyclic homology by Christian Kassel, and on page 19 he says that Hochschild homology is Morita-...
L-A's user avatar
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9 votes
1 answer
130 views

Is there a functor from the category of commutative rings that is faithful but not conservative?

The category of commutative rings is not balanced (the inclusion $\mathbb{Z} \to \mathbb{Q}$ is monic and epic but not an isomorphism). So, is there a category $C$ and a functor $\mathbf{CRing} \to C$ ...
Geoffrey Trang's user avatar
9 votes
0 answers
187 views

Fundamental groupoid of a filtered union

Let $X$ be a topological space and let $(X_i)_{i\in I}$ be a filtered family of subspaces. Let $X =\bigcup_{i \in I} X^°_i$ be the union of the interiors of the $X_i$. I want to prove the following ...
Alice in Wonderland's user avatar
9 votes
0 answers
184 views

The Dual Of Exponentiation?

I tried dualizing the concept of exponentiation in a cartesian closed category. This lends a "coexponentiation". I know that exponentiation is like the internal hom or an abstract function ...
IsAdisplayName's user avatar
9 votes
1 answer
280 views

What is the point of (Lie) derivations?

Probably some very naive questions, but ... Definition Let $A$ be a vector space together with a bilinear map $\mu: A \times A \rightarrow A$. We call a map $D: A \rightarrow A$ a derivation if it ...
Margaret's user avatar
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9 votes
0 answers
152 views

A tensor category need not be isomorphic to a strict tensor category

I'm reading the book "Tensor categories" by Etingoff (and others). In remark 2.8.6 (posted below), it is claimed that the category $\mathcal{C}_G^\omega$ (defined in example 2.3.8, also ...
user avatar
9 votes
0 answers
242 views

What is the universal property of the prime spectrum of a commutative rig?

Let $A$ be a commutative rig, i.e. a commutative monoid equipped with a unital associative commutative bilinear multiplication and let $L$ be a distributive lattice. For the purposes of this question, ...
Zhen Lin's user avatar
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9 votes
0 answers
265 views

How are the topos-theoretic programs of Caramello and Lurie related?

It is clear that we are living in very exciting time for topos theory, with many exciting developments from different directions. Of course most notable are Lurie's work on higher categories and ...
Lova Bolding's user avatar
9 votes
0 answers
220 views

Rings and categories with zero Grothendieck group

I am interested in examples of rings (or triangulated categories) that have zero Grothendieck group but are somehow still interesting. More example, for what rings $R$ is the category of finitely-...
user39598's user avatar
  • 1,544
9 votes
0 answers
172 views

Can we "integrate" functors?

Let $F:\mathcal{C}\rightarrow \mathcal{C}'$ be a functor between "nice" (e.g. abelian with enough injectives) categories. If F is not exact we can form the derived functors $F',F'',...$ Is it possible ...
Takirion's user avatar
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