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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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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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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60 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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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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138 views

Direct limits and pullbacks

Suppose we have three directed sequences of $C^*$-algebras, say $(A_n,\varphi_n)$,$(B_n,\psi_n)$ and $(C_n,\theta_n)$ and $*$-homomorphisms $\alpha_n:A_n\rightarrow C_n$ and $\beta_n:B_n\rightarrow ...
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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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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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163 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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56 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 ...
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93 views

Algeraic 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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92 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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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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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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51 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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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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61 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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78 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 ...
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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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62 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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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 ...
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42 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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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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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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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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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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51 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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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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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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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 ...
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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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Vector-Lattices and “Approximating $\mathscr{L^1(\mathbb{R}^k)}$”.

In this question I asked whether $\mathscr{L}^1(\mathbb{R}^k)$ forms a category in any way. It was concluded that indeed it does not. I thought to myself, "well, could we at least approximate the ...
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When can we complete bundle morphisms when given just a morphism of total spaces?

Consider a bundle $E \rightarrow B$ in some category. The morphisms of it to some other bundle $E' \rightarrow B'$ is simply two morphism $E \rightarrow E'$ and $B \rightarrow B'$ which make the ...
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EXAMPLE on p.58, in Hungerford(GTM)

In line 6, there is an explanation, '$h$ is an equivalence in $E$ if and only if $h$ is an equivalence in $C$. ' Suppose that $h : B \rightarrow D$ is an equivalence in $C$. Then $h$ is an ...
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Criteria for a contravariant functor from Schemes to Sets to be representable by a scheme

I am trying to prove that a functor $\mathcal{F}:(Sch/S)^{op}\rightarrow (Sets)$ is representable by an $S$-scheme $F$. My intuition says that it is indeed representable. I have been reading Fulton's ...
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Every monoidal category is strictly equivalent to a strict monoidal category.

I am reading Joyal and Street's article "braided tensor categories". The following theorem is proved (Theorem 1.2). Let $ C $ be a category. Let $FC$ and $F_sC$ be respectively the free monoidal ...
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Do subhomomorphisms / subfunctors have a standard name, and where can I learn more?

Sorry for all the mistakes in the original! I think they're mostly fixed now. Thank you for your patience. Part 1. If $A$ and $B$ are models in $\mathrm{Pos}$ of an algebraic signature $\sigma$, ...
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A question about the category of co-elements

The category of elements of a $K: \mathcal{D} \rightarrow \mbox{Set}$, denoted by $1\downarrow K$ is defined to be a category which has as objects arrows of the form $d: * \rightarrow Kd$ i.e. the ...
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Definition for a bar resolution for a module over a dg category

Let $ \mathcal{A}$ be a dg category and define a right $ \mathcal{A}$ module to be a dg functor $ M: \mathcal{A}^{op} \rightarrow dif\ k$ where $dif\ k$ is the category of differential $k$ modules ...
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A full embedding in a finitely complete category

I know that each category can be full embedded in a complete quasicategory. My question is: a category can be full embedded in a finitely complete category? There is a universal such one (that's a ...
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Do colimits preserve free group actions?

Suppose $C$ is a small category, $M$ is a category and $G$ is a discrete group. Let $X$ be a $G$-object in $M^C$ and suppose that the action of $G$ on $X$ is object-wise free. Under what conditions is ...
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Endofunctor as a presheaf

This looks pretty obvious, but an endofunctor $F : Set \to Set$ seems to be a presheaf over $Set^{op}$. Is there any useful fact that can be learnt from this view of endofunctors?
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Why are there no naturality condition in definition of exponential in a category?

In chapter 6 "Exponentials" of "Category Theory" Second Edition, Steve Awodey, 2010, there is no naturality requirement in the definition of exponential in a category. Exponential is an adjoint and ...
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Is the category of topological spaces coregular?

Everything is in the title. The category Top of topological spaces and continuous mappings is not regular, but is it coregular ? Furthermore, Top isn't cartesian closed, but does it satisfy the dual ...
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categorical description of the Minkowski sum of polytopes

Consider the category $\textbf{Poly}$ of polytopes, where the objects are convex hulls of finite subsets of $\mathbb{R}^d$ for arbitrary $d \in \mathbb{N}$ and where the morphisms are affine maps ...
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Is the requirement that every morphism factors as an epi composed with a mono part of the definition of an abelian category?

Hilton and Stammbach require an abelian category to be an additive category in which 1) all kernels and cokernels exist 2) all monos are the kernel of their cokernel, all epis are the cokernel of ...
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Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
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Do graded modules form an additive category?

I got two conflicting references. On page 466 of Shastri's Algebraic Topology, he claims that graded modules with homogeneous morphisms form an additive cat, whereas Hilton and Stammbach say on page ...