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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Based space, commuting in diagram up to homotopy.

Theorem. For any based space $Z$, the induced sequence$$\dots \to [Z, \Omega F f] \to [Z,\Omega X] \to [Z, \Omega Y] \to [Z, Ff] \to [Z, X] \to [Z, Y]$$is an exact sequence of pointed sets, or of ...
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0answers
13 views

Is there an algorithm for computing pushouts in finite sets?

Does anyone know of an algorithm for computing pushouts in finite set? That is, given finite sets $X$, $Y$ and $Z$ with maps (of sets) from $X$ to $Z$ and $Y$ to $Z$, is there an algorithm for ...
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1answer
26 views

Definition of adjoint functor similar to the definition of homotopy equivalence?

I am new at category theory and I haven't get the definition of adjoint functor. I have seen the another definition of natural transformation much similar to the definition of homotopy. Other ...
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0answers
18 views

Maximum n-category constructible with a 1-category

Let $\mathcal{C}$ be a $1$-category. One can show that the category of internal categories in $\mathcal{C}$, with internal functors and internal natural transformations produce a $2$-category. I ...
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0answers
42 views

Hom(A, B) Cathegory theory

What this construction Hom(A, B) means? I did not found anything that explains clearly. For example $Hom(N, N_{\bot})$ N is the set of natural numbers, $N_{\bot} = N \cup \bot$ It is a notation for ...
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1answer
35 views

A question about covariant functor

Let $X$ be a fixed set and define a functor $f\colon S \to S$ by $Y\mapsto X \times Y$. Then is $f$ a left adjoint of the covariant $\operatorname{Hom}$ functor $h_X=\operatorname{Hom}_s(X,-)$? First, ...
7
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4answers
110 views

The Galois connection between topological closure and topological interior

[Update: I changed the question so that $-$ is only applied to closed sets and $\circ$ is only applied to open sets.] Let $X$ be a topological space with open sets $\mathcal{O}\subseteq 2^X$ and ...
6
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1answer
74 views

Is it possible to delete undesired identifications in algebraic structures?

In algebraic topology, there is a notion of covering space, which essentially "de-identifies" points that look the same but which for certain purposes really shouldn't be considered the same. I was ...
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56 views

Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form: Proof sketch ...
3
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1answer
51 views

The sections of the projection $\bigsqcup_{i:I} X_i \rightarrow I.$

I just noticed something funky. Let $X$ denote an $I$-indexed family of sets. There is a projection $$\pi_X: \bigsqcup_{i:I} X_i \rightarrow I.$$ It isn't necessarily surjective, of course, because ...
5
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2answers
53 views

Categories with some but not all exponentials

The introductory examples typically given of exponential objects in categories in fact involve categories which have all exponentials. Are there not-too-esoteric examples of categories of ...
3
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0answers
22 views

Realization of simplicial sets

Consider a simplicial set $X$, i.e. a contravariant functor from $\mathbf{\Delta} \to \mathbf{Set}$ where $\mathbf{\Delta}$ has as objects $[n]:=\{0, \cdots, n \}$ for all $n \in \mathbb{N}$ and as ...
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1answer
57 views

Topological Version of First Isomorphism Theorem

Given a set $X$ and an equivalence relation $\sim$ on $X$, we can define the set $X_\sim=\left\lbrace\left[x\right]:x\in X\right\rbrace$ of equivalence classes, and we can define a projection map ...
4
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1answer
57 views

Commutative Diagram for group structure

I remember seeing once a commutative diagram that explained group structure. Where the associativity, identity element, inverse, multiplication and all was shown in a singular diagram, it is trivial ...
7
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3answers
169 views

In the definition of a functor, why is it necessary that $F(id_{A})=id_{F(A)}$?

A functor $F$ is defined to be a mapping from category $\mathcal{C}$ to $\mathcal{D}$ such that: (1) $F(f\circ_{\mathcal{C}} g)=F(f)\circ_{\mathcal{D}} F(g)$ (say, for a covariant functor). (2) ...
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0answers
64 views

Existence of tensor product via category theory

In my class of category theory, my professor stated (without prove it) that the existence of tensor products between modules over commutative rings follows from the following result: a category ...
3
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1answer
82 views

Pointfree probability theory

I must confess I hardly know anything about probability theory. Still, I'm interested in the following: Much like pointfree topology, where one basically replaces topological spaces by their locales ...
7
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1answer
70 views

How to recover multiplication of group elements from category of groups?

Motivating question: If we know everything about $\mathbf{Grp}$, do we know everything about groups? Background: Consider the category $\mathbf{Grp}$ abstractly, rather than as a concrete category. ...
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1answer
27 views

Closed under extensions without zero object

Suppose A is an abelian category and $\mathcal{B}$ is a full subcategory of A. If $\mathcal{B}$ is closed under extensions, must it be closed under isomorphisms? We require that $\mathcal{B}$ contains ...
3
votes
1answer
55 views

Full subcategory of abelian category is abelian

I'm trying to understand a proof in Rotman's 'Introduction to Homological Algebra', Proposition 5.92, p.310. Proposition: Let $\mathcal S$ be a full subcategory of an abelian category $\mathcal A$. ...
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0answers
39 views

What is the “internal language of a topos”?

What does the sentence "[...] these statements should be interpreted, of course, in the internal language of the topos $\mathcal{E}$" mean, in the context of, say, the definition of a groupoid in ...
4
votes
1answer
36 views

Is it impossible to recover multiplication from the division lattice categorically?

In this question it was asked if the division lattice (i.e., the preorder category $(\Bbb Z_{>0}, \mid)$) contains enough information categorically to recover the relation ...
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0answers
46 views

generalized affine scheme

I'm thinking about following theorem. For a finitaly algebraic theory $\mathbb{T}$, $\text{FP}\mathbb{T}$ denotes the full subcategory of $\mathbb{T}\text{-Alg(Set)}$ consisting of finitely presented ...
4
votes
1answer
141 views

Splitting field as a terminal object?

Let $f(x)\in K[x]$ be a polynomial over field $K$ and let $E$ be a splitting field. I would like to prove that $E$ is unique up to isomorphism by expressing the inclusion $K\to E$ as a terminal object ...
2
votes
2answers
60 views

$\mathbf{Set} \not \simeq \mathbf{Set}^*$ by considering $\{1, 2 \} \to \{1\}$

This answer gives a nice way of seeing why the category of sets is not isomorphic to its dual. I would like to know whether there is a proof from a certain different direction. When considering the ...
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0answers
33 views

Finitely generated abelian groups form an abelian subcategory of $\mathbb{Z}$-Mod

According to Weibel's Homological Algebra book a subcategory $\mathcal{B}$ of an abelian category $\mathcal{A}$ is called an abelian subcategory if it is abelian and an exact sequence in $\mathcal{B}$ ...
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0answers
23 views

Question about “immediate” observation about finitely presentable objects

The following is an excerpt from volume II of Borceux: Here $F$ is the left adjoint to the forgetful functor $U:\mathsf{Mod}_\mathcal{T}\longrightarrow \mathsf{Set}$. I'm confused and can't manage ...
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1answer
41 views

Proof forgetful functor $U:\mathsf{Mod}_\mathcal{T}\rightarrow \mathsf{Set}$ has a left adjoint

I'm confused by a proof in volume II of Borceux. First, two facts: Proposition 3.3.3 $\;\;\;$ Let $\mathsf{T}$ be an algebraic theory. Consider the functor ...
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0answers
44 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 ...
3
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0answers
45 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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0answers
46 views

How is the delooping of a groupoid constructed explicitly?

Let $G$ be a group, nlab's "delooping" page says that $G$ can be considered as a discrete groupoid in the $(\infty,1)$-topos $\infty$Grpd of $\infty$-grupoids, the delooping of $BG$ is then the ...
3
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0answers
78 views

May Algebraic Geometry be appropriate for me? [closed]

I am a student of Mathematics who have to choose its area of specialization. I am trying to obtain as more information as possible, by asking a lot of questions to more experienced people, trying to ...
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1answer
86 views

How to give “categorical” specifications of categories like Grp?

Certain types of categories (like abelian categories) are specified by listing a set of "categorical" properties that the category must have. For example, we might demand the category has finite ...
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votes
3answers
1k views

Motivation for different mathematics foundations

I've been studying an introductory book on set theory that uses the ZFC set of axioms, and it's been really exciting to construct and define numbers and operations in terms of sets. But now I've seen ...
2
votes
1answer
57 views

Categories like $\mathsf{FinSet}$, but with elements of $\mathbb{Z}$ or $\mathbb{Q}$ as objects?

Suppose, we do universal algebra in a "non-evil" fashion, such that every algebra carries around an equivalence relation $\cong$ that poses as equality and such that size issues don't matter (we may ...
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3answers
68 views

Prove that in any category a product(if exists) is associative up to isomorphism.

I have a question regarding an exercise(p.38, ex.5.9) from Aluffi's Algebra textbook. Let C be a category with products. Find a reasonable candidate for the universal property that the product $A ...
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vote
1answer
46 views

If $Y\to Z$ is a monomorphism then $X_1\times_Y X_2 \to X_1\times_Z X_2$ is an isomorphism.

This is an (easy) exercise from Vakil's textbook on Algebraic Geometry. We are working in an arbitrary category, let $Y\to Z$ be a monomorphism and we are given maps $X_1, X_2\to Y, X_1, X_2\to Z$. We ...
2
votes
1answer
62 views

Isomorphism between colimits.

I actually need something weaker than this but my hope is that this holds in its fullest generality. Let $I$ be a small diagram and $I'$ a full subcategory of $I$. Let $F: I\to {\rm vec}$ be a functor ...
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vote
2answers
50 views

Is it possible that the inclusion functor does not preserve limits?

Let $\mathcal{B}$ be a full subcategory of a category $\mathcal{C}$. Is it possible that the inclusion functor does not preserve the pullbacks (or any limits) that exist in $\mathcal{B}$? I read this ...
5
votes
1answer
123 views

Metric limit and limit in category

Is it possible to construct a category $\mathcal{C}$ with $\mathrm{Ob}\,\mathcal{C}=\mathbb{R}$ and for every diagram of the from $$a_0\leftarrow a_1\leftarrow\cdots a_n\leftarrow\cdots$$ the inverse ...
0
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0answers
31 views

Locally presentable categories of categories

Cat, the category of small categories, is locally finitely presentable. I want to relax the quality of presentability to find interesting categories as colimits. Has anyone seen a treatment of ...
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0answers
22 views

Composition of Dual Maps in (rigid) Monoidal Categories

If $X, Y$ are objects in monoidal category $\mathcal{C}$ which have left duals $X^∗, Y^∗$ and $f : X → Y $ is a morphism in $\mathcal{C}$, then the left dual map $f^∗ : Y^∗ → X^∗$ of $f$ is given by: ...
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1answer
24 views

Natural isomorphism of a monoidal category

The definition of a Monoidal Category from "Categories for the working mathematician" says that it is a category equipped with tensor products, associative up to a natural isomorphism. What does ...
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1answer
34 views

A question on a property of geometric morphisms related to locales.

Is the "localic reflection" of a geometric morphism between topoi the same thing as its "localic part"?
9
votes
2answers
145 views

Can the identity $ab=\gcd(a,b)\text{lcm}(a,b)$ be recovered from this category?

Define the category $\mathcal{C}$ as follows. The objects are defined as $\text{Obj}(\mathcal{C})=\mathbb{Z}^+$, and a lone morphism $a\to b$ exists if and only if $a\mid b$. Otherwise ...
3
votes
1answer
41 views

Name of this 2-categorical structure?

Let $\mathscr C$ be a 2-category. For each object $X$ in $\mathscr C$ suppose there is a category $\mathcal R(X)$, and for each pair of objects $X,Y$ in $\mathscr C$ there is an "action" functor $$ ...
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1answer
36 views

Functor $M\otimes_B -$ is left and right adjoint to $\operatorname{Hom}_A(M,-)$ when there is a symmetrizing form for $A$?

Suppose $A$ is an $R$-algebra over a commutative ring $R$, which is finitely generated and projective as an $R$-module. A symmetrizing form is a map $t\in\operatorname{Hom}_R(A,R)$ such that ...
6
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3answers
186 views

What is the sheafification of the presheaf of the one point compactification?

Okay, so I had this idea for a presheaf that is quite peculiar. Instead of being based on algebraic category (i.e. abelian groups), it is based on a topological one, the category of compact ...
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1answer
34 views

Reconstruct a category : Forgetful fuctor to underlying graph, free functor of graph and then a quotient

There is a forgetful functor that takes a category to its underlying graph. There is then an adjoint to this that takes the graph to its free category. Can we then take a quotient of this free ...
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1answer
34 views

Braided Hopf algebra - properties of braiding

Let $(H,\nabla,\Delta,S)$ be a Hopf algebra in a braided category. I'm trying to simplify the following $(\nabla\otimes \mathrm{id}\otimes \mathrm{id})(\nabla \otimes \Psi \otimes ...