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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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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1answer
112 views

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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44 views

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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1answer
47 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 ...
3
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2answers
58 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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76 views

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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1answer
39 views

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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41 views

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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55 views

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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1answer
43 views

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" ...
4
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3answers
101 views

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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30 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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29 views

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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42 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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53 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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1answer
27 views

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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1answer
43 views

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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1answer
35 views

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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39 views

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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53 views

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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1answer
97 views

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, ...
3
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1answer
81 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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1answer
26 views

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 ...
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63 views

Does the ring of continuous functions determine $\mathbb R^n$?

I have two related questions which are just making the question asked in the title more specific: (a) Is every ring homomorphism (or maybe $\mathbb R$-algebra homorphism) between rings of the form ...
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105 views

Properties of the functor $(X, \mathcal{O}_X) \mapsto (X(k), \mathcal{O}_{X(k)})$

Let $k$ be an algebraically closed field. In Görtz and Wedhorns book one can read about an equivalence of categories $\{\text{integral schemes of finite type over } k\} \to \{\text{prevarieties over ...
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1answer
84 views

Why do dagger categories supposedly capture the structure of a Hilbert space?

A dagger functor is a contravariant endofunctor $(\;)^\dagger$ satisfying $X^\dagger = X$ on objects and $f^{\dagger\dagger}$ on morphisms. It is supposed to model adjoint maps on Hilbert spaces, and ...
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1answer
53 views

Weibel's book, Page 8. $\text{Tot}(C)$. What is the sum of the horizontal and vertical differentials in a bicomplex?

... define the total complexes $\text{Tot}(C) = \text{Tot}^{\Pi}(C)$ and $\text{Tot}^{\oplus}(C)$ by $\prod_{p+q = n} C_{p,q}$, and $\bigoplus_{p + q = n}C_{p,q}$. The formula $d = d^h + d^v$ ...
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1answer
44 views

Relation between fixed point and retraction theorem

There is this particular exercise in Lawvere/Schanuels book "Conceptual Mathematics: A first introduction to categories" that I've worked on, but I'm not entirely sure if I'm correct. Plus, I'm a ...
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Weibel exercise 1.2.2.: kernels, monics, and monomorphisms are the same in $R$-Mod.

See image below. I just want help proving that all kernels in $R$-Mod are monics. My attempt: Let $f : A \to B$ be a map in $R$-Mod. Suppose $i$ is a kernel of $f$, that is: $fi = 0$ and ...
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32 views

What does “universal w.r.t. this property” mean? (kernel of a morphism in an additive category)

In an additive category $\mathcal{A}$ a kernel of a morphism $f: B\to C$ is defined to be a map $i : A \to B$ such that $fi = 0$ and that is universal with respect to this property. This is ...
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1answer
46 views

Bridgeland stability conditions: The heart satisfies the Harder-Narasimhan property

Given a stability condition $(Z,\mathcal{P})$ on a triangulated category $\mathcal{D}$. Take $\mathcal{A}=\mathcal{P}((0,1])$. Then $\mathcal{A}$ is the heart of a bounded t-structure on ...
3
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1answer
124 views

Category theory: Enough that polygonal diagrams commute

I've read somewhere that for a categorical diagram to commute, it is enough that all its polygonal subdiagrams commute. I want a reference and a detailed proof of this. Please also give a formal ...
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1answer
40 views

Labeled commutative diagram

Consider a commutative diagram. For example the following diagram in $\mathbf{Set}$: $$ \begin{array}{ccc} & \overset{+1}{\longrightarrow} &\\ \mathbb{Z} & & \mathbb{Z} \\ & ...
2
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1answer
26 views

Algebraic theories and canonical algebras

I am reading Bodo Pareigis-Categories and functors (Pure and Applied Mathematics, Vol. 39). Let $\mathcal{U}$ be an algebraic theory (in the sense of Lawvere theories). A product-preserving functor ...
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77 views

Is this a better way to think about Groups as Categories?

I asked a bit ago how to reconcile the category theoretic and set theoretic definitions of groups (groupoid which is a monoid vs the set theoretic definition), and I got the answer I was looking ...
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1answer
115 views

Properties of Pushouts in the category of topological spaces

I have following question: $X, X_0, X_1$ and $X_2$ are topological spaces. Furthermore, $\mu_i:X_0\rightarrow X_i$ and $\alpha_i:X_i\rightarrow X$ morphisms in the category of topological spaces, ...
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81 views

A misleading commutative diagram

Let $U$ be a set, let $\phi$ be an involutive bijection of $U$ with itself. Let $A$, $B$ be subsets of $U$. Consider the commutative diagram $A \overset{\phi}{\leftrightarrow} B$ describing a ...
3
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1answer
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Prove that all cycles are identities

In the following commutative diagram: $$ \begin{array}{ccc} A & \longleftrightarrow & B \\ \updownarrow & & \updownarrow \\ C & \longleftrightarrow & D \end{array} $$ all ...
2
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1answer
67 views

Inclusions of Vector Spaces vs Sets

I have a conjecture relating statements about inclusions of sets to corresponding statements about inclusions of linear subspaces. More specifically, consider a formula \begin{equation*} \phi \equiv ...
4
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1answer
65 views

An endomorphism $f$ such that $f\circ f=1$

What is the name for an endomorphism $f$ of a category such that $f\circ f=1$? Note that I work with category $\mathbf{Set}$.
3
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1answer
53 views

Is a locally $\mathcal{C}$ space a direct limit of $\mathcal{C}$ spaces?

Let $\mathcal{C}$ be a class of topological spaces (for example Hausdorff spaces, compact spaces, connected spaces, finite spaces, etc...), and let $X$ be a topological space. Is the following true? ...
6
votes
1answer
63 views

Binomial rings closed under colimits?

A binomial ring is a ring (for the purposes of this question all rings are commutative and unital) which is torsion-free and has, for each $n$, a binomial function $\binom{x}{n}$ satisfying ...
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properties of pullback diagrams

Suppose you have a commutative diagram: $\require{AMScd}$ $\begin{CD} A @>>> B\\ @VVV @VVV \\ C @>>> D \\ @VVV @VVV \\ E @>>> F \end{CD}$ Let $T$ be the top "square", $B$ ...
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1answer
39 views

The identity elements of two multiplication satisfying interchange law.

Let $M$ be a set, and $\circ_1,\circ_2$ two binary operations defined on $M$ satisfying that both $(M,\circ_1),(M,\circ_2)$ are semigroups and $(a\circ_1 b)\circ_2(c\circ_1 d)=(a\circ_2 ...
2
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1answer
66 views

Universal Property for isomorphic objects

Suppose $A,B$ are isomorphic objects of some category $C$. Suppose that A satisfies a given universal property does B satisfy the same? If not can you provide a counterexample for an elementary ...
3
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1answer
66 views

Why should we expect duality to give useful concepts in category theory?

Why should we expect the abstract notion of flipping arrows in a category to generate useful concepts from other useful ones? What exactly does flipping the direction of arrows mean and why is it ...
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1answer
61 views

How to construct a tensor product of two preadditive categories in pure categorical fashion?

Let $\mathsf C$ and $\mathsf D$ be two preadditive categories (by an preadditive category I mean a category together with compatible abelian group structure on every hom-set). I thought of ...
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1answer
74 views

Category theory with objects as logical expressions over $\{\vee,\wedge,\neg\}$ and morphisms as?

I am wondering if there is a standard definition for a category with objects as first order logical (FOL) expressions e.g. $\neg x \vee y$. It seems to me that these logical expressions would be part ...
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1answer
61 views

Definition of a Cartesian Closed Category

I would like to check with someone that the following of a Cartesian Closed Category is correct and that I did not make a mistake when translating to a notation with predicate quantifiers. Let ...
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51 views

Category of torsion free abelian groups not abelian

In this article it says that the category of torsion free abelian groups is not abelian since the map $\mu: \mathbb Z \to \mathbb Z: k \mapsto 2k$ is not a kernel. I have trouble showing this: If ...