# Tagged Questions

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### Basic Notions of Categorification

In this post John Baez defines a categorification of a set $S$ as a map $$p:\DeclareMathOperator{Decat}{Decat}\Decat(\mathscr C)\to S$$ where $\Decat(\mathscr C)$ is the set of isomorphism classes ...
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### Exercises in category theory for a non-working mathematican (undergrad)

I'm trying to learn category theory pretty much on my own (with some help from a professor). My main information source is the good old Categories for the working mathematician by Mac Lane. I find the ...
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### Category Theory textbook (learning through guided discovery Dummit and Foote)

sorry for asking the same question in a slightly different angle, I want a book in Category Theory similar to Dummit and Foote's book in Abstract Algebra. I want it to have tons of examples and ...
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### Distinguishing sets according to more fine-grained notions than cardinality.

I'm interested in distinguishing sets according to more fine-grained notions than cardinality. Now I don't know a thing about computability theory, but it seems to me that considering sets up to ...
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Let $C$ be a small category. Let $D,E$ be cocomplete categories. Let us denote by $\hom$ (resp. $\hom_c$) the category of (cocontinuous) functors. Then there is an equivalence of categories ...
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### References for a notion of “restricted adjoint”

A construction that I've been finding all over the place in studying the category of NF (Quine's New Foundations) sets and functions is a situation like the following: there's a functor ...
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### Learning the topology needed for topos theory.

I have just started learning topos theory and I am going through Mac Lane and Moerdijk's book, "Sheaves in Geometry and Logic". I have, unfortunately, very little experience with topology. I started ...
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### When will a pointed category / arrow category be cartesian closed / a model category if the base category is cartesian closed / a model category?

For a category $\mathcal{C}$ with terminal object we have some construction on it : define the pointed category to be $*\downarrow \text{Id}$ the coslice category relative to the terminal object ; ...
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### Categorical proof of Pontrjagin Duality?

I would like to ask if there is any reference in which Pontrjagin Duality is proved in a categorical context: I started reading the Pontrjagin Dual entry in nLab, ...
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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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### The Category of Small Categories: a Zoo of Functors.

Wouldn't it be great if there was some website or something that visualized (some small portion of) the category of small categories(*)? Imagine you click on some categories from a list, say, ...
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### Studying Galois Cohomology from Category Theory.

I am a masters student with background in Category Theory - I also know some Topos Theory. I would like to ask if you think it would be possible to start studying Galois Cohomology without any ...
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### Additive functor is exact $\iff$ quasi-ismorphisms preserved?

While reading Weibel's Homological Algebra, on pg. 391 he considers an additive functor $F:\mathcal{A}\to\mathcal{B}$ between abelian categories, and writes "If $F$ is not exact, then the induced ...
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### From groups to groupoids.

Let $\mathcal{G}$ be a groupoid and $p$ an object in $\mathcal{G}.$ It is well known that the set ${\rm Mor}_{\mathcal{G}}(p,p)$ is a group. I would like to know if there is a way to recognize a ...
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### iterated dual vector spaces

Let $K$ be a field and $\mathcal U$ a universe such that $K\in\mathcal U$. (Here, "universe" means "uncountable Grothendieck universe".) Let $\mathcal C$ be the category of $K$-vector spaces belonging ...
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### Objects are finite sets, arrows are matrices. How is this a category?

I just started to read this book on category theory. How is this example below a category? I have difficulty imagining what this construct really is. Could someone please illuminate me ? I ...
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### Logic in closed symmetric monoidal categories; reference request.

Suppose we want an algebraic theory $T$ to be interpretable in any closed symmetric monoidal category $\mathbf{C}.$ I am thinking in particular of the case where $\mathbf{C}$ is the category of models ...
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### Categorical introduction to Algebra and Topology

I am self-studying Mathematics in my free time. At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book $\textit{Algebra}$ by Serge Lang. I ...
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### Some reference for categorical logic?

By "categorical logic" I mean category-theoretical models of logic. In particular, I am more interested in models of intuitionistic predicate logic with conjunction, disjunction, implication and ...
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### An interleaved path of arrows in a category/digraph

In what I am doing now, a slightly strange concept emerged; My objects are sequences of arrows $\{f_i\}_{i=1}^{n}$ in a small category such that for every $1\leq i\leq n-1$, there is an arrow from the ...
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Let $X$ and $Y$ be posets seen as a category. Let $F:X \to Y$ and $G: Y \to X$ be an adjunction. That is, $(G \circ F) (x) \ge_X x$ and $(F \circ G) (y) \le_Y y$. Now, $F$ is additive and $G$ is ...
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### A class of “internal” endofunctors in cartesian closed categories

I'm interested in a class of endofunctors on cartesian closed categories with a quite natural definition, and am wondering whether/where this class has been studied so far (and how it is called). Fix ...
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I know that if $\mathcal C$ is a semiadditive category, then for every two objects $A$, $B$, the set $\mathrm{Hom}_\mathcal{C} (A, B)$ is automatically endowed with a structure of commutative monoid, ...
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### Collections of Homomorphic (defined) structures via $f$

Long ago I read a text about a collection of algebraic sturctures all homomorphic (or isomorphic) via a unique homomorphism An Example similar to the construction I found was this: Lets take define ...
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### Distributivity in linear monoidal categories

Let $\mathcal{C}$ be a linear monoidal category, that is a monoidal category (with tensor product $\otimes$) enriched over $\mathbf{Vect}$. Now as far as I can tell the axioms for a linear category ...
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### Inductive limit of manifolds?

The inductive limit of a direct system of manifolds is a topological space (which I don't think needs be a manifold). But it seems like it should retain some of the structure of manifolds : for ...
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### The composition of functors and algebraic structures

An algebraic structure can be viewed as a finite-product-preserving functor $X : \mathbf{L} \rightarrow \mathbf{C},$ where $\mathbf{L}$ is a Lawvere theory and $\mathbf{C}$ is a category. Thus given ...
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### “Topologification” of a subcollection of a power set

Let $X$ be any set and consider any $\mathscr{S} \subseteq \mathcal{P}(X)$, where the latter is the power set. It is natural to ask if we make $X$ a topological space by the subcollection ...
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### Where does one learn the algebraic geometry needed for topos theory?

I am a masters student familiar with category theory. I have started learning topos theory from MacLane-Moerdijk's book "Sheaves in Geometry and Logic: A First Introduction to topos Theory". I get the ...