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 morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seeming different areas of mathematics.

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How to construct a quasi-category from a category with weak equivalences?

Let $(\mathcal{C},W)$ be a pair with $\mathcal{C}$ a category and $W$ a wide (containing all objects) subcategory. Such a pair represents an $(\infty,1)$-category. One model for such gadgets is a ...
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Grothendieck's Galois theory: fundamental theorem

I've been learning about Grothendieck's Galois theory, and I just haven't been able to understand the fundamental theorem properly. Let's phrase the fundamental theorem in the case of fields: Let $...
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What does it mean to say that a particular mathematical theory is a foundation for mathematics?

We usually hear that set theory is a foundation for contemporary mathematics. Category theory is also another foundation of maths. There are other theories which deemed to be a foundation for maths. ...
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Category of Lie group representations equivalent to the category of representations of their Lie algebra

Let $G$ be a lie group and $\mathfrak{g}$ its lie algebra. Consider the category $Rep(G)$ of finite dimensional representations of $G$ and the category $Rep(\mathfrak{g})$ of finite dimensional ...
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Tensor Product and Physics

During lecture, my abstract algebra professor said that the exactness of the tensor product is "absolutely essential" to the existence of physical phenomena such as black holes and the big bang. Is ...
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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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Examples of categories where epimorphism does not have a right inverse, not surjective

An epimorphism is defined as follows: $f \in \operatorname{Hom}_C(A,B)$ is an epimorphism if $\forall Z, \forall h', h'' \in \operatorname{Hom}_C(B, Z)$ then $h' f = h'' f \; \Rightarrow \; h' = h''...
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Is every monomorphism an injection?

We say a morphism is a monomorphism if $fg=fh$ implies $g=h$. So if $f$ is a monomorphism, is it necessarily an injection? i.e. $f(x)=f(y)$ implies $x=y$. My approach is to consider a specific ...
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Is there some universal sense of -ification (eg, groupification) in category theory

I have three questions. 1: Does the groupification of a semigroup always exist? I believe this should be yes because for every $x$ in the semigroup one could just define an element $x'$ that should ...
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Introduction to sheaves using categorical approach

When I first started to learn about sheaves, it was a very geometric approach. This is nice, but it seems like knowing more abstract categorical approach is very useful. For example, sheafification $...
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Properties of $\mathbf{Cat}$

Let $\mathbf{Cat}$ be the category of small categories. (1) Is $\mathbf{Cat}$ complete? (2) Is $\mathbf{Cat}$ cocomplete? Remark(Jan. 11, 2013) Of course, the question is implicitly asking for a ...
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Why don't we study 'metric vector spaces' on their own?

I recently took a mastercourse on functional analysis and I was wondering why we 'skip' the metric structure on vector spaces. I have seen that $$\{\text{vector spaces}\}\subsetneq\{\text{topological ...
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In category theory: What is a fibration and why should I care about them?

I stumbled upon the "fibration of points" in the definition of a protomodular category and apparently this is indeed an instance of a fibration. What are fibrations intuition-wise and how are they ...
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Importance of 'smallness' in a category, and functor categories

I feel like, having spent a little time doing category theory now, this is probably a silly question, but I keep coming up to many things (definitions, examples etc.) where smallness is required. I ...
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Easy to understand examples of category theoretic theorems that are useful

Every now and then I hear a category theorist saying that category theory is a unifying language for mathematics, and that category theory proves general theorems that some people would prove ...
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Book for Algebraic Topology- Spanier vs Tom Dieck

A number of times, questions have been asked on this website about good books on Algebraic Topology and the responses have been very valuable. However I need some more specific advice in this matter. ...
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Can we rediscover the category of finite (abelian) groups from its morphisms?

It was a question on stackexchange approximately a month ago if in the category $(grp)^{fin}$ $|Hom(H,G_1)|= |Hom(H,G_2)|$ for all $H \Rightarrow G_1 \cong G_2$. Link to the previous question. So ...
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Why is it that homotopy is better described by weak equivalences than by homotopies?

I've been reading about (abstract or not) homotopy theory, and I seem to have understood (correct me if I'm wrong) that weak equivalences describe homotopy better than homotopies, in the following ...
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Philosophy or meaning of adjoint functors

I have a question about the definition of the (left) adjoint of a functor.I am trying to understand the philosophy and reason of the definition of adjoint functors. If I understand correctly the ...
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On the importance of natural transformations

In p. 18 of Categories for the working mathematician (2d ed.), Mac Lane remarks that ..."category" has been defined in order to define "functor" and "functor" has been defined in order to define "...
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The category of theorems and proofs

On a philosophy website, it said that you could have a category with theorems as objects and proofs as arrows. This sounds awesome, but I couldn't find anything on the web that has both "category" and ...
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What is a concrete example of why one wants to have a *derived category* in algebraic geometry?

My question asks for a concrete (and hopefully easy) example, why one wants to derive things in algebraic geometry. I heard, that a resolution of an object by free ones behaves much better than the ...
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What's the difference between a logic, an internal logic (language) of a category, an internal logic of a topos and a type theory?

maybe this question doesn't make sense at all. I don't know exactly the meaning of all these concepts, except the internal language of a topos (and searching on the literature is not helping at all). ...
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Understanding Hom functions

I am very new to category theory. Started learning about this Hom sets/functions. I read $\operatorname{Hom}(S,T)$ as set of all functions from $S$ to $T$ but somehow this is a overloaded definition ...
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$F$ is an equivalence of categories implies that $F$ is fully faithful and essentially surjective

I read in wikipedia that: One can show that a functor $F : C → D$ yields an equivalence of categories if and only if $F$ is full, faithful and essnetially surjective. I'm trying to prove this but I ...
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lambda calculus and category theory

I am not particularly knowledgeable in either lambda calculus or category theory, but I am starting to learn Haskell so I would like to ask: are there connections between category theory and lambda ...
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What is the best path to learn Category theory and Type theory?

I have little background in Programming in functional languages and wanted to learn type theory. I started with taking Homotopy type theory class Online videos of Robert Harper. I thought rather than ...
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What does **Ens** stand for?

Earlier someone was asking about the category "Ens" described in Categories for the Working Mathematician. My question is more basic: What does Ens stand for? Most of the categories have names that ...
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Is the Laplace transform a functor?

I may be oversimplifying, as I know very little about category theory, but: Does the Laplace transform, which—to my limited recollection—is a morphism between differential equations and algebraic ...
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Is there a suitable definition in categories for a closed continuous function in topology?

Working in the category of topological spaces is it possible to give a 'categorical' definition for 'a closed continuous function'? I mean something like: 'a closed continuous function' is an arrow in ...
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What are some examples of hard theorems in category theory?

I'm currently learning some category theory, but so far I've used it only as a handy way to talk about various related concepts in algebra and topology with some nice, easy-to-prove lemmas like "left ...
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When can stalks be glued to recover a sheaf?

Let $\mathcal{F}$ be a sheaf over some topological space. The stalks are $\mathcal{F}_x= \underset{{x\in U}}{ \underrightarrow{\lim}} \mathcal{F}(U)$. Is there a special name for a sheaf that ...
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Natural isomorphisms: what is the status now of “the Eilenberg/Mac Lane Thesis”?

The Church/Turing Thesis that we all know and love asserts that every algorithmically computable function (in an informally characterized sense) is in fact recursive/Turing computable/lambda ...
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The arrows from the initial object in a category are monomorphisms?

Let $\mathbb{A}$ be a category and $I \in \mathbb{A}$ its initial object ($\forall A \in \mathbb{A}. \exists ! f:I \longrightarrow A$). For $A \in \mathbb{A}$, prove that $f:I \to A$ is a ...
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Applications of Eckmann–Hilton argument

I am looking for applications of the Eckmann–Hilton argument. I found one application in Algebraic Topology where we show that the fundamental group is abelian in case of a topological group. Thank ...
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categorical interpretation of quantification

Many constructions in intuitionistic and classical logic have relatively simple counterparts in category theory. For instance, conjunctions, disjunctions, and conditionals have analogues in products, ...
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Books/lecture notes/videos on category theory for programmer

I want to learn category theory. I tried different books and had several problems with them: Books are for mathematicians and they use a lot of examples with which I am not comfortable, like ...
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Why is an empty set not a terminal object in categories $\mathsf{Top}$ and $\mathsf{Sets}$?

From Awodey: In any category $\mathsf{C}$, an object $0$ is initial if for any object $C$ there is a unique morphism $0 \to C$, an object $1$ is terminal if for any object $C$ there is a unique ...
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Learning about Grothendieck's Galois Theory.

I have background in category theory and I am familiar with the very basics of algebraic geometry - Chapters I and II of Hartshorne. What would be a recommended (self-contained, maybe?) text for ...
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Unique up to unique isomorphism

If an object $X$ has a non-trivial automorphism $g$, for any isomorphism $f$ with an object $Y$ there is another isomorphism $f \circ g$ between $X$ and $Y$, so there is not a unique isomorphism ...
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What is the difference between analytic combinatorics and the theory of combinatorial species?

Yesterday I asked the question Why should a combinatorialist know category theory?, where Chris Taylor suggested me to have a look at combinatorial species. I had heard the term before but I haven't ...
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Natural isomorphisms and the axiom of choice

The definitions of "natural transformation", "natural isomorphism between functors", and "natural isomorphism between objects" captures - among other things - the intuitive notion of "an isomorphism ...
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Etymology of Tor and Ext Functors

The names of the derived functors $\operatorname{Tor}$ and $\operatorname{Ext}$ seem quite cryptic to me. Does anyone know what these abbreviations stand for? I would be glad if someone could tell me ...
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“The Yoneda embedding reflects exactness” is a direct consequence of Yoneda?

Let $A,B,C$ be objects of a category of modules over a ring. It is not hard to see that the Yoneda embedding "reflects exactness" (as Weibel puts it, on p. 28), i.e. if $\hom(X,A)\stackrel{f_*}{\to}\...
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Example of a functor preserving only finite coproducts

What is an example of a functor $$F : \mathsf{Set} \to \mathsf{Set}$$ which preserves finite coproducts, but not infinite coproducts? The functors preserving infinite coproducts are given by $T \...
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Are there useful categorical characterisations of the topological separation axioms?

Tietze's extension theorem states: If $X$ is a normal space, and $A$ a closed subspace. Then any continuous function to the reals $f:A\rightarrow R$ has an extension to $f':X\rightarrow R$ that is $f=...
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Developing category theory inside ETCS

Trying to understand how Lawvere's [E]lementary [T]heory of the [C]ategory of [S]ets can be used as a foundation for mathematics alternative to ZFC, I am getting stuck with the question on how to ...
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Is the thingie/cothingie distinction absolute?

Is there some inherent quality of a mathematical object that marks it as being "naturally" a thingie or a cothingie? Suppose, for example, that two mathematical concepts, say, doodad and doohickey, ...
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What's the difference between an endofunctor and a morphism?

The question is in the title, really. I understand the answer in a general sense: Morphisms map objects, and functors map both objects and morphisms. So an endofunctor would map morphisms as well as ...
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Is the construction of Zariski topology from polynomial rings functorial?

Given a polynomial ring $k[X_1,...,X_n]$ over a field $k$, we can consider the space $k^n$ equipped with Zariski topology whose closed sets are exactly the algebraic sets. Is this construction ...

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