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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If $F : \mathbf{C} \to \mathbf{D}$ is an equivalence, does $\alpha_{GD} = G(\beta_D)$ hold in general?

Let $F : \mathbf{C} \rightleftarrows \mathbf{D} : G$ be an equivalence with natural isos $\alpha : 1_\mathbf{C} \to GF$ and $\beta : 1_\mathbf{D} \to FG$ witnessing the referred equivalence. I wonder ...
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23 views

When is a functor of bicategories part of an equivalence?

Assuming the axiom of choice (I think), $F: \mathcal{C} \to \mathcal{D}$ is part of an equivalence of categories iff $F$ is fully faithful and essentially surjective on objects. This is sometimes ...
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46 views

Why is the definition of subfunctor well-defined?

I'm reading the definition of subfunctor in a book and I disagree with it. On Wikipedia I have the same definition: Let $\mathbf{C}$ be a category, and let $F$ be a contravariant functor from ...
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66 views

Is there a relationship between Logic and Object-Oriented Programming language? [closed]

I read A lot of thing about set theories, class logic, category theory and I would like to translate a class as define in computer science to a formula in logic. I found too this thread about OOP/OOD ...
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36 views

linear characters on a Grothendieck ring of a modular category.

I am reading the paper "Rank-Finiteness for Modular Categories" by Bruillard,Ng, Rowell, and Wang. Let $C$ be a modular category and let $K_0(C)$ be the Grothendieck ring generated by simple objects ...
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42 views

Functor category between two small categories is not small?

Let $C,D$ be two small categories, i.e. small sets of objects $O_C, O_D$ and small hom-sets $\mathrm{Hom}_C(c,c')$ and $\mathrm{Hom}_D(d,d')$. The set of objects of $D^C$ is the set of all functors ...
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14 views

How do you prove that $Y^*(1_B) = 1_{Y^*B}$ given $Y^*(f) = \mathcal{A}(f, -)$?

Let $\mathcal{A}$ be a small category and define $Y^*: \mathcal{A} \to \text{Fun}(\mathcal{A}, \text{Set}), \ Y^*(A) = \mathcal{A}(A, -), \ Y^*(f) = \mathcal{A}(f, -)$. Then $Y^*(1_B) = 1_{Y^*B}$. ...
2
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1answer
45 views

What are some interesting cases where the two obvious definitions of “discrete object” diverge?

The nLab page defines "discrete object" as follows: Definition. [nLab] Let $\mathbf{C}$ denote a concrete category whose forgetful functor $U$ has a left adjoint $F$. Call the counit of this ...
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50 views

Why does it make sense to consider $\text{Fun}(\mathcal{A}, \text{Set})$ when category $\mathcal{A}$ is small and not otherwise?

Let $\mathcal{A}$ be a small category, ie. $\text{Ob}(\mathcal{A})$ is a set. $\text{Fun}(\mathcal{A}, \text{Set})$ is the category of functors from $\mathcal{A}$ to the category of sets, with ...
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43 views

Product of compacts is compacts using closedness of projection to second component?

A well known characterization of compactness says $X$ is compact iff for all spaces $Y$ the projection $X\times Y\rightarrow Y$ is a closed map. I'm wondering whether there's some simple formal way ...
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The étale fundamental group as a functor

The usual "topological fundamental group" $\pi_1 (X,x)$ of a pointed topological space $(X,x)$ is functorial in the sense that a pointed continous map $f: (X,x)\rightarrow (Y,y)$ induces a ...
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1answer
60 views

Awodey's Category Theory Exercise 9.9.2

I am having problems with this question in Awodey's Category Theory book p.248: Show that every monoid M admits a surjection from a free monoid $F(X) → M$, by considering the counit of the free ...
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33 views

Is uniform continuity a property of the category of completely regular spaces?

If $(X, U)$ and $(Y, V)$ are uniform spaces then one has the notion of a map $f : X \to Y$ to be uniformly continuous relative to $U$ and $V$. A uniform space $(X, U)$ induces a completely regular ...
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44 views

Functor between ordered sets.

(a) Let $f : K \rightarrow L$ be a map of sets, and denote by $f^* : \mathscr{P}(L) \rightarrow \mathscr{P}(K)$ the map sending a subset $S$ of $L$ to its inverse image $f^{-1 }[S] \subseteq K$. ...
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38 views

Yoneda Lemma, newbie question. How is $\theta_{F,A}(\alpha) = \alpha_A(1_A)$ an element of the set $FA$?

Part of Yoneda Lemma: There exists a bijective correspondence $\theta_{F,A} : \text{Nat}(\mathcal{A}(A,-), F) \xrightarrow{\simeq} FA$, where $\mathcal{A}$ is an arbitrary category, $A \in ...
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175 views

Definition of the Diagonal functor

The diagonal functor $\Delta_C^J:C \to C^J$ and the constant functors $\Delta_C^J(c):J\to C$ definitions are a bit too generous and lead to contradictions when applied to $J=0$ (the initial category). ...
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32 views

$V(1)$ generates the tensor category of representations of $sl_2(\mathbb{C})$ - what exactly does this mean?

Does anyone know what it means for a $V(1)$ to generate the tensor category of (finite dimensional) representations of $sl_2(\mathbb{C})$. Here $V(1)$ is the 2 dimensional irreducible Lie algebra ...
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49 views

Adjoints functors.

Let G be a group. (a) What interesting functors are there between $\textbf{Set}$ and the category $[G,\textbf{Set}]$ of left $G$-sets? Which of those functor are adjoint to which. (b) ...
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66 views

How to generalize “Seven trees in one” to labelled/colored trees?

In the famous paper Seven trees in one, Andreas Blass showed that there is "a particularly elementary bijection between the set $T$ of finite binary trees and the set $T^7$ of seven-tuples of such ...
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29 views

projective model structure on presheaves , hom-functors are always cofibrant

Why hom-functors are always cofibrant in the projective model structure in $[\cal T,\cal V]$? The claim is here on page 5.
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Proper maps in terms of projection from pullback

I've read that a continuous $f:X\rightarrow S$ is proper (inverse images of compacts are compacts) iff for all other continuous maps $g:Y\rightarrow S$ the projection $X\times _SY\rightarrow Y$ in the ...
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1answer
47 views

The category of (completable) rings has enough projectives in it

I am working on functors and projective resolutions and of course the issue of "Enough projectives" comes up. I know $R$-modules have enough but I am curious about the category of rings in general? ...
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Isomorphism between categories and how to prove non-isomorphism

Regarding to this topic, I have some ambiguities on the notion of isomorphism between categories, that I'll ask here: 1- That we can't allow the contravariant functors to be isomorphisms (as ...
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48 views

Monomorphisms of monoids are stable under coproducts

Let $M,N,K$ be three monoids (or even groups, if you like) and let $N \to K$ be an injective homomorphism. Then, the induced morphism $M \sqcup N \to M \sqcup K$ is also injective. This is easy to ...
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1answer
54 views

Homotopy colimit,weighted colimit, homotopy theory

Let's take the definition of $\mathbb{hocolim}$ as the representation of the representable functor like this: $\underline{\cal M}(\mathbb {hocolim}_{ \cal D} F,m)\cong \mathrm {{sSet}^{\cal ...
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1answer
32 views

If a Set of Cones is Representable, then there is a Limit

I am following Maclane's proof ot the existence of a pointwise right Kan extension, and I am stuck on a probably easy fact: Let $F:J\to A$ be a functor, and suppose that, for all $a\in A$, there is ...
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1answer
32 views

Given a (2,1) category is there a canonical way of constructing a 1-category?

Given a (2,1)-category (for example, the category of algebraic stacks over some fixed site), you can consider the 1-category with the same objects, but where morphisms between the objects are just ...
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37 views

Categories of étale coverings of elliptic curves

Let $(E,\mathcal{O})$ be an elliptic curve over a (perfect) field $K$ and let $\textbf{Cov}(E,\mathcal{O})$ denote the category of finite pointed étale covers of $E$ from smooth varieties where the ...
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How does any map have a “pseudosection” (assuming axiom of choice)?

In the Lawvere and Rosebrugh book, Sets for Mathematics, exercise 4.34 is to show that the following is equivalent to the axiom of choice (every epimap has a section aka right inverse): If $f:X\to ...
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1answer
64 views

What is the definition of a presheaf in EGA?

In EGA I, Grothendieck says he is not going to bother recalling the definition of a presheaf (on a given topological space $X$ with values in some category $\textbf{K}$). I was just wondering what ...
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35 views

Show that $A$ cartesian closed category it does not imply $A^{J}$ cartesian closed category

Hello I have been looking for one example of this problem If $A$ cartesian closed category then it does not imply $A^{J}$ is a cartesian closed category with $J$ any category. I do not have any idea ...
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Grothendieck's definition of a universal problem

In EGA I, Grothendieck writes (I'm paraphrasing): Let $\mathbf{K}$ be a category, $(A_{\alpha})_{\alpha \in I}, (A_{\alpha \beta})_{(\alpha,\beta) \in I \times I}$ two families of objects of ...
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2answers
47 views

Subobject of a direct sum in terms of components?

Let $\mathbf C$ be an abelian category containing arbitrary direct sums and let $\{X_i\}_{i\in I}$ be a collection of objects of $\mathbf C$. Consider a subobject $Y\subseteq \bigoplus_{i\in I}X_i$ ...
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1answer
48 views

Category Theory: Free Abelian Groups and Coproducts

This is probably obvious to someone familiar with Category Theory (I'm just starting) but why are the following two statements true? $$ \operatorname{Hom}(\operatorname{F}(S),-)\cong ...
3
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1answer
102 views

Equivalent bimodule categories

Let $A,B$ be two rings such that their categories of bimodules are equivalent: $$A\mathsf{-Bimod} \simeq B\mathsf{-Bimod}$$ What can we say about $A$ and $B$? Are they isomorphic? Are they ...
2
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1answer
41 views

Category of sets equivalent to category of finite dimensional vector spaces?

Suppose char(K) = 0. Prove or disprove: The category of sets is equivalent to the category of finite dimensional K-vector spaces. I've only just begun to learn about category theory, so I'm not ...
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31 views

Is $\Lambda$ essentially the unique solution to $F(M)\cong\frac{F(M\oplus R)}{F(M)}$?

Let $R$ be a commutative ring and let $F$ be a functor $\mathbf{Mod}_R\rightarrow \mathbf{Mod}_R$. Then for a module $M$ the split mono $M\rightarrow M\oplus R$ gives a split mono $F(M)\rightarrow ...
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Prove that $F$ is a functor between $C/c$ and $C$.

I'm totally new in category theory and am using Awodey's Category theory. He makes an example of a functor $F: C/c \to C$ defined so that $F(X \to C)=X$ and $F(a:(f:X \to C) \to (g:Y \to C))=a:X \to ...
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2answers
79 views

Why do natural transformations express the fact that a vector space is canonically embedded in its double-dual but not in its dual?

I've been struggling for quite a while to understand why a vector space is considered to be "canonically embedded" into its double dual, but not its dual. As has been remarked in many other places, ...
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476 views

Is Set “prime” with respect to the cartesian product?

(Motivated by Stefan Perko's question here) Suppose $C, D$ are two categories such that $\text{Set} \cong C \times D$. Is either $C$ or $D$ necessarily equivalent to the terminal category $1$? I ...
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Could there be an “$n$-th root” of the category $\mathsf{Set}$?

Here is a thought experiment: Suppose we did not know what sets and functions are. The general idea of a topos is, that it somehow serves as a foundation for mathematics. So let there be an alternate ...
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What are the generating cofibrations of the canonical model structure on Cat?

It sais here that the canonical model structure on $Cat$ is cofibrantly generated. I found out that a generating trivial cofibration is the functor $I:*\rightarrow E $, where $E$ is the category with ...
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What is a morphism of Tannakian categories?

In this question, a Tannakian category over $k$ is a $k$-linear rigid symmetric monoidal tensor category, with the property that it has a fibre functor to $\mathbf{Vect}_\ell$ for some field extension ...
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(Co-)limit characterization of subobjects mapping to subobjects?

My motivating example is the category of rings, (though this of course works the same for groups). Given a homomorphism $\phi : A \rightarrow B$, and a subring of A, given by the inclusion $ \iota : ...
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3answers
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Is there a relationship between isometry as defined on metric spaces and those on vector spaces?

I am taking a course on linear algebra and another on real analysis. In linear algebra we defined that two vector spaces are isomorphic if there existed a bijective and linear map between ...
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1answer
113 views

How can you actually do universal algebra with monads?

Instead of digging deep into "classical" universal algebra, it seems more interesting or fruitful to look at universal algebra categorically. This should be doable with monads, since every category of ...
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Has a natural transformation between functors with codomain $Cat$ that is an equivalence on each component a weak inverse?

Let $Cat$ be the category of small categories and $C$ another category. Suppose we have two functors $F,G\colon C\rightarrow Cat$ and a natural transformation $\eta\colon F\implies G$, that is an ...
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1answer
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Bullet notation

I'm just trying to make acquaintance with homological algebra. I see there the notation $(A_\bullet,b_\bullet)$ as a short notation for $(\dots,A_{-1},A_0,A_1,\dots,\dots,b_{-1},b_0,b_1,\dots)$. ...
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65 views

“Minimal upper bounds” in a categorical setting

It is well-known that partial orders can be seen as very simple categories (those where there is at most one morphism between every two objects). Then, the notion of "(binary) join of two elements ...
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1answer
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Kan extensions and double application of Yoneda lemma

If we are given functors $$ F: \mathcal{A} \to \mathcal{B}, \\ G: \mathcal{B} \to \mathcal{C}, \\ H: \mathcal{A} \to \mathcal{C}, $$ the natural transformations $$ \alpha: H \to GF $$ are in bijective ...