# 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 ...
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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 ...
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### 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 ...
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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 ...
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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 ...
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### 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....
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### 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 ...
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### 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 ...
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### 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 $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
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### 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 ...
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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 $\mathsf{Med}$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 (...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) ...
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### 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 ...
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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 ...