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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Is this correspondence covariant or controvariant?

I'm new to category theory and am trying a basic exercise. Is the correspondence from $S$ to $\mathcal{P}(S)$, which assigns to $f:S\rightarrow T$ the mapping $\mathcal{P}(S)\rightarrow ...
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Categories that differ in morphisms

Let $C_1$ and $C_2$ be two (small) categories defined over the same set of objects: that is, $C_1$ and $C_2$ differ only in their hom-sets. Specifically, $Hom(C_1) \neq Hom(C_2)$, and $Hom(C_1) \cap ...
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Is every bijective pseudograph homomorphism a pseudograph isomorphism?

The term pseudograph describes a graph that may have parallel edges and loops. Formally this is a triple $G = (V,E,\delta)$ with $V,E$ sets and a map $\delta \colon E \to (V \times V)/\sim$, where ...
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19 views

Category of Sets and Bag-valued functions

I asked here about the Category of sets and set-valued functions, and it turns out it to be equal to REL (Category of sets and Relations),so a good studding point to study that category. Now, It ...
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71 views

Contracted version of “isomorphic”

Had a look around and I can't find a word which acts as a contraction for "isomorphic" in the same way that "monic/epic" is a contraction of "monomorphic/epimorphic". For some reason this strikes me ...
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Group action on a category

Motivating example: We get a functor from the category of real vector spaces to the category of complex vector spaces by complexifying (i.e. tensoring over $\mathbb{R}$ with $\mathbb{C}$). Let ...
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Equivalent conditions for equivalence of categories (Proposition 7.26 in Awodey)

I'm trying to understand the proof of the following proposition in Steve Awodey's "Category Theory". Let $\mathbf{C}, \mathbf{D}$ be categories and let $F: \mathbf{C} \to \mathbf{D}$ be a functor. ...
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41 views

Can a contravariant functor be adjoint to a covariant one?

I am a bit confused about the definition of adjoint functors, since everywhere the definitions found (see example wikipedia https://en.wikipedia.org/wiki/Adjoint_functors) seem to not specify if we ...
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36 views

Direct (inductive) limit of groups

Let $(I,\prec)$ be a directed poset and $\{G_i\}_{i \in I}$ groups with group homomorphisms $f_{ij}:G_i \to G_j$ whenever $i \prec j$. Is is true that the direct limit of this system is given by $$ ...
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Expand $\binom{xy}{n}$ in terms of $\binom{x}{k}$'s and $\binom{y}{k}$'s

Motivated by this question, I want to find a complete set of relations for the ring of integer-valued polynomials, where the generators are the polynomials $\binom{x}{n}$ for $n\in \mathbb{N}$. The ...
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17 views

Compact objects in Ind-categories

Assume a category $\mathcal{D}$ admits small filtrant inductive limits. Then call an object $Y$ of $\mathcal{D}$ compact, if $\hom_{\mathcal{D}}(Y,\cdot )$ commutes with these small filtrant inductive ...
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Example of a forgetful functor that is not faithful.

In the forgetful functor Wikipedia article I read that "[Forgetful] Functors that forget the extra sets need not be faithful; distinct morphisms respecting the structure of those extra sets may ...
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30 views

Category with coproducts generated by an endomorphism

Let's call a category with arbitrary coproducts a $\coprod$-category. A $\coprod$-functor is a functor which preserves coproducts. An example is $\mathsf{Set}$, and this is in fact the universal ...
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when will homology and direct limit commute?

Question: Let a sequence of maps between topological spaces $$ X_1\to^{f_1}X_2\to^{f_2}X_3\to^{f_3}\cdots $$ The mapping telescope is denoted by $T$. Under what conditions will $H_*(T)$, the ...
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55 views

Monomorphisms and injectivity predicates

This is a curiosity question which struck me at a less-than-optimal moment; I apologize for not having thought much about it. Motivation. It is well-known that monomorphisms in a concrete category ...
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25 views

Equivalence of group objects in set and groups as one object categories.

There are (at least) two definitions of groups in category theory: As a group object (in a catgory $C$ with finite products, e.g. $C$ = Sets). This is a tuple $(G,m,inv,e)$ with the following data ...
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Two modules are isomorphic in the stable module category iff they are projectively equivalent

Let $R$ be a (not necessarily commutative) ring. Let ${\text{mod-}R}$ be the category of finitely generated right $R$-modules. Let $\underline{\text{mod-}R}$ be the stable module category, with the ...
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41 views

Zero-section as homomorphism of rings

Let $s : X \to E$ be the zero section of a vector bundle $E$ over a scheme $X$. Zariski-locally this corresponds to a homomorphism $Sym_A(M) \to A$ of $A$-algebras where $M$ is a finitely generated ...
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Problem based category-theory book.

I love problem based textbooks like all of those by R. P. Burn and Halmos' Linear Algebra Problem Book, etc. Are there any problem based Category Theory textbooks, I know that the first chapters of ...
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35 views

base change of an equivalence relation of fppf sheaves

Let $S$ be a scheme, $R,U$ be $S$-schemes and $s,t : R \to U \times_S U$ be an equivalence relation i.e. it's a monomorphisme such that for every $S$-scheme $T$, $R(T) \to U(T) \times U(T)$ is and ...
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86 views

Why composition is so important in category theory?

I'm reading "Category: The Essence of Composition" As a software developer, I understand why composition is important in programming. It's allows you to get complex components from simple ...
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Duality of Projective and Inductive Limit

Could someone please explain to me in what sense the projective and inductive limits are "dual" to one another?
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110 views

Why is the full subcategory consisting of simply connected spaces not complete?

Let $\mathbf{Top}_*$ be the category of pointed topological spaces and $\mathbf{Top_1}$ the full subcategory of simply connected spaces. $\mathbf{Top}_*$ is complete and cocomplete. I am trying to ...
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61 views

How does one define a group with commutative diagrams?

I am currently working through McLarty's book on Elementary Categories, Elementary Toposes. In Chapter 3, he considers a group as an object in a category with a unit map, a multiplication and an ...
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50 views

Natural Transformation: Direct Products

I have result that tells me $$\displaystyle \varphi : \text{Hom}_R \bigg(A, \prod_{i \in I} B_i \bigg) \to \prod_{i \in I} \text{Hom}_R(A, B_i)$$ is a $Z(R)$-isomorphism. The next result tells me that ...
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Category of sets and multi-valued functions

I would like to study category of sets and multi-valued functions: A category whose objects are sets and morphisms are multi-valued functions. By a multi-valued function $f:A\rightarrow B$, from set ...
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55 views

What's so special about the grounded poset of cardinality $2$?

By a grounded poset, I mean a poset $P$ with a bottom element $0_P$. Arrows between grounded posets are monotone mappings that preserve the basepoint. This gives a self-enriched category; lets call it ...
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Is there a “Coalgebra - Cogeometry” duality? Good opposite of a category of coalgebras?

So the category of affine schemes is dual to the category of commutative rings, Stone spaces are dual to Boolean algebras, localizable measurable spaces are dual to commutative Von Neumann algebras, ...
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Are projective limits always subobjects of a product and dualy, inductive limits quotient objects of a coproduct?

This question is motivated by the first few explicit examples I came across, e.g. Wikipedia, Inverse limit or this question or Wiki, Direct limit In order to answer that question, let's start with ...
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44 views

Essentially surjective property is closed under composition of functors.

I want to prove the essentially surjective property is closed under composition of functors. A functor $F: C \to C'$ is essentially surjective if for each $Y \in C'$ there's an $X \in C$ such that ...
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38 views

Projective (or inverse) limit of C*-algebras

(I think that the term "inverse limit" is used when the index set is directed) To begin with, I'd like to know if projective limits of C*-algebras (in the category of C*-algebras) always exist, and ...
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Monoid as a single object category

I'm struggling with comprehending what monoids are in terms of category theory. In examples they view integer numbers as a monoid. I think I get the set theoretic definition. We have a set and a ...
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Functions in the definition of Universal Mapping Property of a free monoid

In Awodey's Category Theory (http://www.amazon.com/dp/0199587361/?tag=stackoverfl08-20) p.19, what is $|\bar{f}|$? What is its relation to $\bar{f}$, which is in the existence statement? Is it ...
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Given bifunctor $F$, what is the name of the functor with switched arguments?

Sorry for the unspecific title. Here the actual question: Given categories $\mathcal{A},\mathcal{B}$, let $S$ be the canonical functor $\mathcal{B} \times \mathcal{A} \to \mathcal{A} \times ...
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Lemma 1.3.11. (Kashiwara & Schapira), Zorn's Lemma?

Lemma 1.3.11. Consider a functor $F: C \to C'$ and a full subcategory $C_0'$ of $C'$ such that for each $X \in C$, there exist $Y \in C_0'$ and an isomorphism $F(X) \simeq Y$. Denote by $\iota$ the ...
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Metric spaces as Cauchy complete categories, nlab entry, insight into a few of the constructions.

I'm having a bit of trouble making sense of some of the concepts in the "Metric space" section on nlab's entry on "Cauchy complete category" ...
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Concrete category with non-standard products

I am working on a paper in which I need to talk about what I call concrete categories with standard products. I write $U(X)$ for the underlying set of an object $X$ in a concrete category $\cal C$ and ...
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26 views

Product-preserving functors

Let $\bf{S}^0$ be the dual category of sets, let $\mathcal{U}_{\infty}$ be a category and $A_{\infty}:\bf{S}^0\longrightarrow\mathcal{U}_{\infty}$ be a functor which is bijective on the object ...
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If a functor $\varphi : C \to C'$ is full, then so is the functor $\varphi \circ$

Let $I, C, C'$ be three categories, $\varphi : C \to C'$ a functor. Then $\varphi$ determines a functor $\varphi \circ : \text{Fct}(I,C) \to \text{Fct}(I, C')$. I want to show that if $\varphi : ...
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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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51 views

What's the largest universe we use?

I know that the notion of a Grothendieck universe is used to deal with the fact that sometimes the categories of category theory are "too large". In general, how large of a universe is worked in? If ...
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Page 17. Kashiwara's Categories & Sheaves equivalent statement to $F: C \times C' \to C''$ is a bifunctor.

Here's the book. The book says: A functor $F: \mathcal{C}\times \mathcal{C}'\to \mathcal{C}''$ is called a bifunctor. This is equivalent to saying that for $X \in \mathcal{C}$ and $X'\in ...
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Unique isomorphisms and universal properties

Having not studied category theory, I'm trying to piece together without using category theory what is meant when an algebraic structure is said to possess a universal property or be unique up to ...
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How does the functor $\text{op}$ assign values to maps?

This has confused me before and now that I'm studying it again it still confuses me. There is a functor $\text{op}: C \to C^{op}$ for any category $C$. I have $\text{op} : \text{Hom}_C(X,Y) \to ...
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Sub(P) is complete?

Let $\mathcal{C}$ be some abstract category, and $P$ be any set valued presheaf on $\mathcal{C}$. I want to show that the set of subfunctors of $P$, $Sub(P)$, is actually a Heyting algebra. For two ...
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Isomorphic as sets. Do they mean bijective? (Kashiwara's Categories & Sheaves)

Here's the book. On page 10 it says: A set is called $\mathcal{U}$-small if it is isomorphic to a set belonging to $\mathcal{U}$. and on page 11 it says: A category $C$ is called a ...
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Going from Closed Categories to Monoidal Categories

EDIT: I trimmed down the exposition a bit. I really just wanted everyone to know what my approach has been, but what I had was a bit bloated. Suppose we have a closed category $V$ as defined here, ...
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how to show naturally isomorphic

I have a homological exam on Saturday , and I have some problem to understand of naturally isomorphic.my problem . the end of this theorem must proof naturally isomorphic $T_n $and ...
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78 views

Iterating until a diagram commutes

I'm coming across the following 'commuting' diagram a lot in my work, and I think it should have a neat categorical formulation. But I can't work it out for myself, and don't know what too google for. ...
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In an SES of chain complexes in an abelian category two of complexes exact implies the third is exact.

Consider a short exact sequence of chain complexes: $$0_{\cdot} \rightarrow A_{\cdot} \xrightarrow{f} B_{\cdot} \xrightarrow{g} C_{\cdot} \rightarrow 0_{\cdot}$$ If any two of ...