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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Categorical proof that subgroups of free groups are free?

Is there a categorical proof that the subgroups of free groups are free? Also that abelian subgroups of free abelian groups are free.
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112 views

Tensor Product Construction, Solution Set Condition.

I am developing the basic properties of tensors, using categories. Let $R$ be a commutative ring. Fix $A, B\in R-Mod$ and define $K:R-Mod\rightarrow Set$ by $KC=\left \{ \beta :\left | A \right ...
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69 views

Categorical description of Alexander horned sphere?

Looking at the construction of the Alexander horned sphere: Is it possible to describe as some kind of direct limit of spaces? How can one formally see this?
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33 views

Structural / design / meta optimization - is there mathematical theory. Optimization over categories?

There is huge branch of mathematical optimization theory, but it mostly considers the finding optimal parameter values for the predefined structures. There are variational calculus and optimal control ...
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34 views

pseudonatural vs natural

By a general result of steve lack's article, I know that there is a nice adjunction between the 2-category of 2-functors (and 2-natural transformations) and the 2-category of 2-functors and ...
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52 views

Does the five lemma hold true for Lie algebras?

According to wikipedia, the Five Lemma is true in Abelian categories. But the category of Lie algebras is not Abelian. Then is the Five Lemma still true for Lie Algebras?
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96 views

Solutions to questions in “Categories for the Working Mathematician”

Does anyone have solutions for the exercises in "Categories for the Working Mathematician"? I'm working my way through them, and want to check my answers.
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56 views

Monomorphisms in a concrete category

Let $\mathcal{C}$ be a concrete category, i.e., a category which admits a faithful functor $C:\mathcal{C}\rightarrow \mathsf{Set}$. It is certainly not the case that $f$ a mono in $\mathcal{C}$ ...
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41 views

How to prove that a particular (sub-)category has a projective generator.

Suppose that $\mathcal{C}$ is an abelian $k$-linear category ($k$ a field) in which every object is of finite length and every $k$-vector space $\text{Hom}(X,Y)$ is finite dimensional. How does one ...
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59 views

Adjoints with a Parameter

I am tryng to prove a theorem in McLane without looking at his proof and I am stuck on one point. Suppose $F:X\times P\rightarrow A$ is a bifunctor, that for each $p\in P $, the functor ...
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48 views

A question regarding Yoneda's lemma.

Suppose you have two objects $A$ and $A'$ in a category $\mathfrak{C}$, and morphisms $i_C:\text{Mor}(C,A)\to\text{Mor}(C,A')$ for any object $C\in\mathfrak{C}$. Show that the $i_C$ are induced ...
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58 views

Commutative diagrams with antiparallel arrows

A diagram in category theory is said to commute when for all objects $A$ and $B$ in it, every the composite morphism resulting from a possible path from $A$ to $B$ are the identical. Does that mean ...
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33 views

The closure of the graph of a certain composite in a topos.

Let $\mathcal{E}$ be an elementary topos with subobject classifier $\Omega$ and let $j\colon \Omega\to\Omega$ be a Lawvere-Tierney topology on it. Assume that, for an object $C$ of $\mathcal{E}$, each ...
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81 views

Transfinite composition and $\kappa$-categories

I wonder whether the following ideas make sense, and whether something in that direction has been written down somewhere; do you know a reference? The last definition is supposed to be a ...
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34 views

Why aren't *weak* test categories enough?

Short version : What is the motivation behind the local condition in the definition of a test category ? Why do we want the slice categories $A/a$ to be weak test ? Long version : I start ...
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84 views

If models of set theory can be construed as categories, can notions of forcing be construed as functors?

Consider the following passage from Blass and Scedrov's paper "Complete topoi representing models of set theory"(Annals of Pure and Applied Logic, vol. 57 (1992),PP. 1-26) where 'set theory' in this ...
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113 views

Must diagrams be commutative?

Given a category C and a function $ \Theta : Mor(\textbf{C}) \times Mor(\textbf{C}) \longrightarrow Mor(\textbf{Rel}) $ and suppose that the relation, with $(r,s)\in\Theta(u,\bar{u})$ and so forth, ...
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49 views

Category of Hilbert Spaces

Is it possible to triangulate the category of Hilbert spaces and bounded linear operators? I assume that one candidate for triangulation is the double dual space. What is a fact is that this ...
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24 views

Does the lax Gray tensor product preserve fully faithful 2-functors?

The question is in the title. By fully faithful 2-functor, I mean 2-functors such that the maps on the hom categories are isomorphism, and by preserve, I mean in each variable. I have an argument ...
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43 views

Kernels of induced maps in cohomology

Let $k$ be a field, and let $A$ and $B$ be two commutative $k$-algebras. Suppose $\varphi,\psi:A\to B$ are maps of $k$-algebras such that there are algebra automorphisms $G:A\to A$ and $F:B\to B$ ...
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29 views

Map of monads and left adjoints

Let $(T,\eta,\mu)$, $(T',\eta',\mu')$ be two monads on a category $X$. Let $\theta:T\Rightarrow T'$ be a map of monads. Then, we have an induced functor $X^\theta:X^{T'}\rightarrow X^T$ (from the ...
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65 views

Right Kan extension along a diagonal functor.

Denote $\newcommand{\simpcat}{\boldsymbol\Delta} \simpcat$ the simplex category (category of finite ordinals with non decreasing maps). The diagonal functor $\delta \colon \simpcat \to ...
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49 views

Relative topological tensor product

I know that for a pair of topological rings $R$ and $R'$ (perhaps with some additional topological/functional-analytic hypothesis on them?) there exist two topological tensor produts, the injective ...
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104 views

Category theory as a foundation for mathematics

Can Category theory form a foundation for mathematics like set theory and mathematical logic and if it can is there a way to know if that theory will be both consistent and complete
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64 views

Initial Algebras in a Category with ω-limits.

I am trying to prove the following: Let $C$ be a category with an initial object, and ω-limits. Suppose $F$ is an ω-continuous endofunctor on $C$. Then $F$ has an initial algebra. I know the result ...
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165 views

Profinite completion of a partial order

In Johnstone's Stone Spaces it is proved that the category of profinite partial orders is (equivalent to) the category of ordered Stone spaces (also called Priestley spaces) and that the obvious ...
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57 views

How to prove that $\mathcal{Y}_X$ is a sheaf by using epimorphic families

I'm trying to prove this: Let $(\mathcal{C}, J)$ be a site and suppose that for every $X\in ob(\mathcal{C})$ and $R\in J(X)$ the family $\{ \bar{f}_Y: \mathcal{Y}_Y\rightarrow \mathcal{Y}_X \}_{Y\in ...
2
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112 views

Algebraic theory and definition of multiplication: tensor product v.s. Cartesian product

In a monoidal category, one can define multiplication on a object $M$ as a morphism \begin{equation} M\otimes M\longrightarrow M, \end{equation} or a morphism \begin{equation} M\times ...
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95 views

What are the prerequisites for reading SGA 1?

My question concerns, basically, scheme theory. If there is someone who has actually read SGA 1, I would really like to hear what their opinion is on that. For example, is EGA in its entirety a ...
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40 views

Notation and shorthand of cobordism G=O, SO, U

This is really a basic question: From Wikipedia: "The basic examples are G = O for unoriented cobordism, G = SO for oriented cobordism, and G = U for complex cobordism using stably complex ...
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41 views

Should a left module be an enriched functor or enriched presheaf?

In this page http://ncatlab.org/nlab/show/module#InEnrichedCategory, They defined the left module over a monoid object A as an enriched presheaf. However, consider the modules over a ring, the left ...
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53 views

dense generators and left Kan extensions

Here is an exercise from Borceux, Handbook of Categorical Algebra I, p 174: Consider a category $\mathfrak{C}$, a family $(G_i)_{i \in I}$ of objects of $\mathfrak{C}$ and the corresponding full ...
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92 views

How can I make peace with contravariance?

My question is a bit vague, but I hope it can be answered in a good way. Various arguments involving contravariance sometimes trip me up when coming up with proofs in algebraic geometry and related ...
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65 views

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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80 views

Adjoints to Forgetful Functor

Suppose $C$ is a category, $X\in C$. I want to find minimal conditions on $C$ for which the forgetful functor $U:C/X\rightarrow C$ has a left adjoint. edit: As pointed out in the comments, $U$ has ...
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50 views

An equality of some equalizers of simplicial sets.

$\newcommand{\cosimp}[1]{{#1}^\bullet} \newcommand{\sSet}{\mathsf{sSet}} \newcommand{\Set}{\mathsf{Set}}$ [I realize that the post is imposing.However, all the content of the problem is in the gray ...
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37 views

Pullback in specific category

can someone help me with the following Let $C$ be the category with objects subsets of $\mathbb{N}$, and arrows functions $f:A \to B$ such that preimage of each point is a finite set i.e. for every ...
2
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51 views

Exact functors in the category of left R-modules - “Fun for the whole family”

The following question has proved troublesome and prompted some deeper questions which I will elaborate on. Our definition of a left exact functor is one which takes exact sequences: $$0 \rightarrow ...
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64 views

Push-out of product of push-out diagrams

Let $\pi(U\cup V)$ be the push-out of the diagram $\pi(U)\leftarrow \pi(U\cap V) \rightarrow \pi(V)$ that appears when we apply Vam-Kampen Theorem to the open sets $U,V$ in a topological space $X$. ...
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45 views

Push-out of product of diagrams

Im working on the category of groups. Let $D$ be the push-out of the diagram $B\leftarrow A \rightarrow C$. Let $D'$ be the push-out of the diagram $B' \leftarrow A' \rightarrow C'$. It is possible to ...
2
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44 views

Category theory inquiry: limits

excuse me but what is the best way to prove that the functor $F: \mathcal{C}^{2} \to \mathcal{C} \times \mathcal{C}$ creates limits, where $F(f:A \to B) = (A,B)$? Not a homework question - just trying ...
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67 views

An isomorphism of categories

This is a corrected follow up to When do finite sets embed in a category? Let $C$ be an (finite) extensive category with terminal object $1$. Let $I$ be an index category. Let $j: \mathrm{FinSet}\to ...
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41 views

Concrete categories possessing many forgetful functors

Given a set $X$, is there a name (like $X$-concrete category) for those categories $\mathbf{C}$ equipped with a forgetful functor $F_x : \mathbf{C} \rightarrow \mathbf{Set}$ for each $x \in X$? The ...
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73 views

Correct Definition of Concrete Category over Set

In the text Joy of Cats, a concrete category over $Set$ is simply a pair $\langle \mathcal C, U \rangle$ consisting of a category $\mathcal C$ and a faithful functor $U\colon \mathcal C \to Set$. But ...
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58 views

How does one characterise the reals without points?

The Real number line is defined upto unique isomorphism in the category $Fld$ as a Dedekind complete ordered field. One can view the Reals geometrically with its usual topology. In pointless ...
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Unipotent Group Scheme

Can someone give me a reference of why this is true? Let A an associative k-algebra in wich every element is nilpotent, (non unital) and free of finite rank k-module, then for every commutative ...
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56 views

Canonical “orientification” of a manifold? Canonical complexification of a manifold?

Maybe this is a silly question(i'm pretty new to both geometry and category theory) but i was wondering: 1)Consider the category of orientable smooth manifold on $\mathbb{R}$, if you forget the ...
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17 views

Internal additions in additive categories agree with given ones

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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52 views

Cauchy completion in ordinary category theory

In Borceux and Dejean's paper Cauchy completion in category theory, they conclude the smallness of $\bar{C}$, the full subcategory of $[C^{op},\mathrm{Set}]$ spanned by all the retracts of the ...
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35 views

Duality in abelian categories.

Let $C$ be an abelian category, with a projective separator $k$. Assume that $C$ has a duality, that's a functor $\ast:C\to C^{\text{op}}$ together with a natural isomorphism $\tau:1\to \ast\ast$ such ...