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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Zero direct limit of nonzero objects

Can anyone present to me kindly a directed set of nonzero objects with the zero direct limit? I first tried $$F(U)=\{f:U \to R \mid f\text{ is continuous}\}$$ in p.507 of "Advanced Modern ...
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54 views

Is there a way to phrase “there does not exist a universal set” in structural language?

Both ZFC (which is a good example of a material set theory) and ETCS (which is a good example of a structural set theory) prove the sentence "there is no set having maximum cardinality" as an easy ...
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38 views

Existence of Initial Object implies Existence of Terminal Object? [closed]

Consider an initial object always exists in a category. Reversing all arrows now. Does this guarantee the existence of terminal objects in that category?
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85 views

Proof: Categorical Product = Topological Product

Is there a nice way to prove that the categorical product equals the topological product? What I mean is the following: Starting with a given family of topological spaces $X_i$ and any topological ...
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1answer
22 views

Are presheaf categories well-copowered?

I am trying to proof the existence of a right adjoint to the functor $F_*\colon \mathrm{Set}^D \to \mathrm{Set}^C$ given by precomposing with a functor $F\colon C \to D$, where $C$ and $D$ are small ...
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30 views

Characteristic Property for Products of Smooth Manifolds

Does the characteristic property for products of smooth manifolds hold as well: $$f\text{ smooth}\iff \pi_i\circ f\text{ smooth ...where }f:M\to\prod_{i\in I} M_i\text{ and }\pi_i:\prod_{i\in I} ...
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171 views

What does Structure-Preserving mean?

A very basic definition in category theory is the definition of morphism between objects. If the category is a construct, i.e., a category $\mathcal C$ equipped with a faithful functor $U\colon ...
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42 views

Characteristic Property = Universal Property?

Problem They seem to be the same -almost! But are they really or is it just unlucky accident that they look so similar however describe totally different notions? Example I was trying to set the ...
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53 views

Do the circle groups have any interesting stand-alone descriptions?

By the circle groups, I mean firstly the circle group $\mathbb{T} \subseteq \mathbb{C}$ of all complex numbers having modulus $1$, and secondly the commutative group $\mathbb{S} = \mathbb{T} \cap ...
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81 views

Is there a first order theory for equivalences classes?

Question will be a bit naive, so please, be kind. Consider a first order theory, $\Gamma$ . Let $\mathcal{M}$ be the category of models for $\Gamma$. Consider $\sim$ an equivalence relation on ...
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36 views

Unique representation of a degenerate simplex

I recently learned about simplicial sets, and now I'm studying some basic properties. I've learned that every degenerate simplex is a degeneracy of a unique non-degenerate (nice) simplex. However, I ...
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88 views

Quotient Topology = Coproduct

Quotient topology seems to satisfy the universal property for coproducts at first glance. However, at second glance they seem to fail to fit into that frame in general since not every map passes to ...
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46 views

When do finite sets embed in a category?

Let $FSet$ be the category of finite sets. Let $C$ be a category with binary coproduct and terminal object $*$. My first question: when does $1\mapsto *$ extend to a functor $FSet\to C$ preserving ...
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42 views

What does $1a \in Hom(a, a)$ mean?

I am trying to read Algebra, Chapter 0 by Aluffi. In category section i found this sentence and cant find out what it exactly means. So my question is: What is meaning of this: $1a ∈ Hom(a, a)$ ?
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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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75 views

Algebraic Object in Topology

Common algebraic categories are group, ring, module and algebra. Some of them have the corresponding object in topology, like topological group and topological linear space. We define them by making ...
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48 views

Question about accessibility of category of free abelian groups.

I've read, that the accessibility of the category of all free abelian groups is independent on basic set theory (say ZFC). What is the reason for that? And how can I interpret this result? Does it ...
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41 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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Prime Ideals as Ring theoretic Ultrafilters

I am confused by the following statement in Awodey's Category Theory p. 35: Ring homomorphisms $A\to \mathbb Z$ into the initial ring $\mathbb Z$ play an analogous and equally important role [to ...
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153 views

A proof using Yoneda lemma

Some clever geezer Mister Brandenburg pointed out elsewhere in the comments that he could give a one line proof, using the Yoneda lemma, of $$\frac{\mathbf{C}[x_1,\ldots,x_{n+m}]}{I(X)^e+I(Y)^e} \cong ...
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1answer
28 views

Origin of the term “module” for profunctors

Why do they call profunctors "modules"? How do they exactly relate to modules over rings?
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47 views

Can this category be identified as the category of graphs?

Let $\mathbb I$ be a category with exactly $2$ objects and $4$ arrows. The $2$ arrows that are not identities are parallel and their (common) domain and (common) codomain are distinct. Looking at the ...
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39 views

The intersection of $u_{A} : A \longrightarrow A + B$ and $u_{B} : B \longrightarrow A+B$ is zero.

I am trying to show that the intersection of $u_{A}:A \longrightarrow A+B$ and $u_{B}:B \longrightarrow A+B$ is the zero map. Here, the $u_{A}$ and $u_{B}$ are the embedding maps into the coproduct of ...
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113 views

Two topologies over the same set which are carried to each other by a lattice isomorphism induce homeomorphic spaces?

This is a question that I've been turning in my head and haven't been able to come up with a way to proceed to either prove it or construct a counter-example: Let $X$ be a set and $\mathfrak{T}_X$ ...
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1answer
29 views

Proof that sheafification induces isomorphism on stalks using adjoints

Let $\mathcal{F}$ be a presheaf on some topological space $X$. It is not hard to prove directly that the map $\mathcal{F}\rightarrow \mathcal{F}^{sh}$ induces an isomorphism of stalks (Here ...
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function application order

In traditional mathematics, when we post-compose $x$ by $f$ we write $fx$ or $f(x)$, that is we prefix writing things right to left. I realize some might be used to it, and it is absolutely trivial, ...
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From the representation category of a Lie group and the representation on a homogeneous space, can we reconstruct the stabiliser subgroup reps? [migrated]

Given a Lie group $G$ and a transitive action $- \triangleright - : G \times X \to X$ on a homogeneous space, we can recover the stabiliser subgroup $H_x$ of a point $x \in X$. It is the subgroup of ...
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101 views

Size Issues in Category Theory

Barr and Wells state in their text Toposes, Triples and Theories (pdf link) It seems that no book on category theory is considered complete without some remarks on its set-theoretic foundations. ...
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30 views

Definition of locally presentable category

The standard book on locally presentable categories defines them as : cocomplete categories with a small set of $\lambda$-small objets generating objects of the category under $\lambda$-filtered ...
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34 views

When the points of coproduct are the coproduct of points

Let $C$ be a category with coproduct. The Yoneda functor $C^{op}\to Psh(C)$ preserves all limits but not colimits. Suppose that $C$ has a terminal object, say $*$. My question: can we say anything ...
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170 views

Categorical introduction to Algebra and Topology

I am self-studying Mathematics in my free time. At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book $\textit{Algebra}$ by Serge Lang. I ...
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is the abelianization functor (on groups) full?

By abelianization I mean, for any group $G$, its commutator subgroup is the subgroup $[G,G]$ generated by elements of the form $ghg^{-1}h^{-1}$ for $g,h\in G$. Then the abelianization of $G$ is ...
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26 views

Resources for Polyadic and/or Cylindric Algebra

I'm looking to learn a little bit about polyadic and cylindric algebras, as part of an investigation into algebraic approaches to logic. The only "text" that I can find for polyadic algebra is ...
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52 views

Prove the determinant map is a natural transformation

I'm trying to learn about natural transformations, but I'm lost trying to work through the details of one. Problem: Let $Comm$ be the category whose objects are commutative rings and whose maps are ...
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51 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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Group categories with only one object with a defined product

Do you know how to deal with this kind of problem? Let $G$ be a group and $\mathcal {G} $ be the category with one object $G$ and $\mbox {Mor}( {G} ; {G} ) = {G} $. Find all groups $G$ such that ...
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1answer
39 views

What is the cokernel of $\Bbb Q^{\text{disc}} \hookrightarrow \Bbb R$?

These two should be the standard examples for why Locally compact abelian groups are not an abelian categoty. The cokernel of any of these maps should is not a LCAG. $$\Bbb Q^{\text{disc}} ...
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Categorical formulation of linear transformations between vector spaces

My question regards the formulation of linear transformations between vector spaces as morphisms in an appropriate category. I know that any biproduct category admits a calculus of matrix. What I'm ...
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1answer
46 views

The injective objects in the category of algebras

What is the definition of the injective objects in the category of algebras?
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23 views

A question on characterization of concrete categories

All objects of a category are subsets of a set. Is the category (isomorphic to) a concrete category?
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48 views

How to inverse this logic definition of separator of Set category?

Lawvere's Sets for Mathematics defines a separator object $S$ for morphisms $f_1, f_2: X \rightarrow Y$ of the Set category as a logic statement $s$ $s$: $\forall x \left[ S \xrightarrow{x} X ...
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How to prove a Set category with at least one element is a separator?

I'm reading Lawvere's Sets for Mathematics and got stuck at Exercise 1.15 In the category of abstract sets S, any set A with at least one element $1 \xrightarrow{x} A$ is a separator. I can see ...
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30 views

A question about the category $C_{A,B}$

Take the category $C_{A,B}$. I'm afraid I don't know how to draw a commutative diagram, but imagine maps between $Z_1$ and $A$ and $B$, and $Z_2$ and $A$ and $B$. Let the morphism between $Z_1$ and ...
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3answers
119 views

Some reference for categorical logic?

By "categorical logic" I mean category-theoretical models of logic. In particular, I am more interested in models of intuitionistic predicate logic with conjunction, disjunction, implication and ...
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59 views

Showing a subcategory of $\mathbf{Top}$ is Cartesian-Closed

We start with some preliminary definitions (necessary because there is not much literature on this): a test map is a continuous function $\varphi:V\rightarrow X$ where $V$ is an open subspace of ...
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92 views

How unique is the exponential of sets?

Firstly I would like to thank everyone in this site for their valuable help. You have helped me a lot understanding the idea of an exponential. I would like to ask a final question that is the last ...
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1answer
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Existence of Boundary Homomorphisms for Cohomology

I am just starting to learn the basics of cohomology and am confused about the construction of the cohomology groups. So given a group $G$, the idea is you take a projective resolution of $P_0 = ...
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Understanding Limits and Colimits by Generalized Elements

We want to characterize the limit and colimit of a functor $D\colon J\to \mathcal C$ by generalized elements. The existence of limits theorem states that the limit of $D$ is the equalizer of ...
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107 views

Inverse limit of an inverse system of topological spaces

Given an inverse system $\mathcal G=\{X_i\}$ of topological spaces over some directed set $I$. If $X=\prod\limits_{i\in I}X_i$, the inverse limit $X^*=\varprojlim X_i$ of $\mathcal G$ is a subspace ...
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Connected index category and limit of constant functor

This is question 8 from chapter 4 section 2 of MacLane's Category for the Working Mathematician: If the category J is connected, prove for any $c \in C$ that $Lim \triangle c \cong c$ and $Colim ...