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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Compact objects in categories of comodules.

Are the finite dimensional comodules the compact objects in the category of comodules over a Hopf algebra? If yes, is there a reference? If no, which are the compact objects? Here by compact I mean ...
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81 views

Why is the Quillen model structure so painful to find?

Proving that the category of simplicial sets carries the Quillen model structure is undoubtedly difficult; the book by May and Ponto "A more concise course in algebraic topology" makes a considerable ...
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Why are split coequalizers “contractible”?

In the book Toposes, Triples and Theories by Barr and Wells, the authors define a contractible coequalizer (elsewhere known as a split coequalizer) to be a commutative diagram: $A ...
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A proof on limit preserving functors.

Proposition Consider a category $\mathscr{D}$ with initial object $0$ and a functor $F:\mathscr{D}\longrightarrow\mathscr{A}$. Let us write $0_D:0\longrightarrow D$ for the unique arrow from $0$ to ...
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Show that homology is a functor in a pure categorical way.

Let $\mathscr{A}$ be an abelian category i want to show that $\mathcal{H^i}$ ( the i-th homology group) is a functor from the category of complexes of $\mathscr{A}$ to $\mathscr{A}$. I showed this for ...
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Regarding $\mathscr{O}_X$ Modules and stalks

Suppose that we have a topological space $X$ together with a sheaf of rings $\mathscr{O}_x$ such that for a sheaf $\mathscr{F}$ of abelian groups we can define a $\mathscr{O}_x$ module. Can the ...
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60 views

Squares of adjunctions / Galois correspondences

$ \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\PS}{\mathcal P} \newcommand{\Idl}{\operatorname{Idl}} $ There are many situations where one encounters a square of things which are related by ...
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47 views

“Natural” labeling of triangles

The angles of a triangle are (capital) $A,B,C$ and the lengths of the sides are (lower-case) $a,b,c$. At your mother's knee, you were taught that the side whose length is called (lower-case) $a$ ...
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If weak-equivalences in a model category are generated by an equivalence relation $\sim$ on hom-sets…

Suppose that $C$ is a bicomplete category, and $\sim$ is an equivalence relation defined on every hom-set of $C$, compatible with composition. Then we can define the quotient category $C/\sim$ whose ...
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79 views

If two monoids have equivalent action categories, are they isomorphic?

If $G,H$ are groups such that $G\mathsf{-Set} \simeq H\mathsf{-Set}$, then $G \cong H$; see math.SE/1375309 for a proof by Zhen Lin. Question. Does this also hold when $G,H$ are monoids? Since there ...
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Unorthodox definition of semi-abelian category

I recently stumbled upon the book Derived Functors in Functional Analysis by Wengenroth. In it, he defines semi-abelian categories quite differently from the nlab: An $\mathsf{Ab}$-category ...
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42 views

Topological reflection of pretopological closure operator

Given a pretopological space $(X,\mbox{cl})$ where $\mbox{cl}$ is a pretopological closure operator. How does one find the topological reflection of $(X,\mbox{cl})$? I know of a way namely by ...
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51 views

Large categories

I'm learning theory category in class, and I learned that small categories are the categories for which the class of all the objects is a set. I also learned that locally small categories are the ...
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90 views

“Lossy” v.s. “Lossless” Monads

Let us look at two monads on $\bf Set$. The first will be the finite sequence monad (from the free forgetful adjunction with $\bf Mon$.) $$\eta(x)=[x]$$ $$\mu([[a,b, \dots, z],[\alpha,\beta, \dots, ...
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Is there a “properly categorical” description of Eilenberg-Moore algebras on a relative monad?

A relative monad is defined in Altenkirch et al. essentially as a pair of functors $J,F:\mathcal{C}\to\mathcal{D}$, a natural transformation $\eta:J\to F$, and an operation ...
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62 views

Looking for info on power set functor

I was reading here about the various functors which take a set $S$ to its power set. In particular, there is the normal contravariant one, and two covariant ones, which the article calls $\exists$ and ...
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178 views

A construction with homotopy colimits and homotopy pullbacks for descent.

I have some troubles in trying to give a meaningful interpretation to the following property which is stated in this preprint by professor Rezk (see Definition 6.5) as part of the requirement for a ...
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79 views

Is the projection from the cyclic nerve to the simplicial nerve a cyclic map?

I’m trying to understand Loday’s proof of Theorem 7.3.11 in “Cyclic Homology” (2nd ed 1998). I have a problem right at the beginning where it says: The projection map $\text{proj}:\Gamma_\bullet G ...
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159 views

Arrow kernel in category theory and generalized equivalence relation

let $F : \mathcal{C} \rightarrow \mathcal{D}$ be a functor from two categories. It looks like that there are various notions of kernels one could define for a functor. One could define the arrow ...
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128 views

Associativity of the tensor product of dendroidal sets

For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$. ...
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93 views

Category theory and complexity classes

Is there any interesting way to make the set of computational complexity classes into a category? Almost every interesting mathematical class of objects forms a category after all.
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Definition of Formally Smooth from Stack Project

$T, T'$ are affine schemes. What is meant by $F\leftarrow T$ (or $G \leftarrow T')$
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184 views

What is the “correct” way of making $\mathcal{P}(X)$ into a topological space?

If $X$ is a topological space, I want to know the "correct" way of making the powerset $\mathcal{P}(X)$ into a topological space. So let $\mathsf{SupLat}$ denote the infinitary Lawvere theory whose ...
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48 views

The bijection between central characters and linkage classes over a semisimple Lie algebra

I have a question about the modules over a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. Let $\mathfrak{h} \subset \mathfrak{g}$ be a Cartan subalgebra. For a given $\lambda \in ...
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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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65 views

Functors that preserve the subset relation on constructs

Is there a name for functors between concrete categories over Set that preserve '$\subset$'? Example, for a functor $F:\mathbf {Top}\to \mathbf {Grp}$, if $Y$ is a subspace of $X$ then $F(Y)$ ...
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What are coquantales?

A quantale can be defined as a monoid in the monoidal category $\mathbf{Sup}$ of complete join semilattices, equipped with tensor product. What are examples of non-diagonal comonoids in ...
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283 views

Subcategories of $\mathbf{Top}$ which are closed under arbitrary products and coproducts

Is there a classification of those (full) subcategories of the category $\mathbf{Top}$ of topological spaces that are closed under arbitrary products and arbitrary coproducts in $\mathbf{Top}$? EDIT: ...
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44 views

Equivalence of Modules

The category of modules over a ring can be viewed as an enriched version of an action of a monoid on a set (see nLab entry). Moreover, if $R$ is a commutative ring, the category of modules over it is ...
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67 views

When do (universal-algebraic) varieties have (enough) injectives?

Following Zhen Lin's suggestion in my previous question, I ask this question separately. Suppose $\mathcal{V}$ is a variety in the sense of universal algebra. considered as a category. When does ...
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187 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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What is the standard foundation for category theory?

Reference : http://arxiv.org/abs/0810.1279 For 2 days, I have searched what possible foundations for category theory are there and I think the reference link explains possible foundations nicely. The ...
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Few questions about compactly generated Hausdorff spaces and Yoneda lemma.

It is a well known fact that in CGWH spaces we have following homeomorphism: $$\mathrm{map}(X \times Y,Z)\cong\mathrm{map}(X, \mathrm{map}(Y,Z)).$$ The proof starts like this: We are dealing with ...
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Monoids, Semigroups, and a Reflective Subcategory.

The following is reflective in the category $\mathbf{Sem}$ of semigroups and their homomorphisms: we add a new neutral element to each semigroup, even monoids; then the reflections are identity ...
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84 views

Is a pushout of monics still monic?

I am sort of stuck with this problem and I have no idea if the answer is obvious and I am just too dumb to see it or if the question is really difficult. What i would like to prove is a particular ...
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168 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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191 views

Commuting with kernels implies left exactness in Abelian category

I'm following Vakil's notes - chapter on category theory. One issue that is unclear in the notes is the conclusion that right adjoint functors are left exact. The notes define a left exact functor as ...
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43 views

What is the restricted product categorically?

The restricted product is a construction for locally compact abelian topological groups. Let $I$ be an indexing set, with $J$ some finite subset. Let $G_i$ be a locally compact topological group ...
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89 views

What is the desirable function identification when setting up arrows in the category of types?

My question is which functions can not be allowed in a statically typed programming language, so that the "canonical" category is less coarse than what you get if you define it's arrows to be ...
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74 views

What are the algebraic and topological properties of a tree (graph theory), and what are any possible connections?

I'm a non-mathematician working with trees (mostly rooted and oriented trees). Typically, I understand them as join-semilattices, so they have an implicit algebraic structure: they form a subgroup ...
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108 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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191 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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Motivation of Strong Monics

Strong monics are defined as: A monic $m$ is strong iff every commutative square $mu=ve$, in $E$ with $e$ epi, has a diagonal i.e. there is a morphism $t$ such that $u=te$ and $v=mt$ in $E$. (where ...
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114 views

Weil restriction - from abstract nonsense to a practical procedure

Let $L/K$ be a finite field extension. Let $X$ be a $L$-scheme, that is $\exists$ a morphism of schemes $\pi : X \to \operatorname{Spec} L$. The Weil restriction of $X$ is a $K$-scheme $W_{L/K}X$ ...
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Topological characterisations of freeness and separability?

According to wikipedia, a spectral space is defined to be homeomorphic to the spectrum of some commutative ring. They form a category $Spec$ where we take morphisms to be those whose preimage of open ...
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(Almost) co-free objects?

another naming question from me, which comes up because I try to use category theory as a compass in developing some (new or not?) order-theoretic notions. According to my book (The Joy of Cats, ...
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For which categories do injections induce surjections in cohomology?

I'll ask a specific question first, but I believe my question might have a rather immediate abstraction, with which I'll finish. Let $H,G$ be finite affine group schemes over an algebraically closed ...
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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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71 views

Existence of a natural transformation

Let $G$ be a group, $S$ a set and $\chi$ an action of $G$ on $S$. This defines a functor $F$ from the group as a single object category to the category of sets. Let $\phi$ be an automorphism of $G$, ...
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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 ...