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 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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1answer
16 views

Covariant and Contravariant Functor of Fixed Set Question - Category of Sets

I am very new to Category Theory and am currently working on a simple question, I know I'm wrong, just wanted to know HOW wrong I am in my answer: Question: "Verify for Fixed set A, the operations ...
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29 views

split quotient, split subobject,category theory [on hold]

A split quotient (or retraction) is an arrow $e: K \rightarrow L$ with $e \cdot m = id_L$. $m: L \rightarrow K$ is then dually called a split subobject. $e$ is in particular a coequalizer of $id_K$ ...
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34 views

Prove Coproduct in Slice category [on hold]

In category theory, how can I prove that, if the category $C$ has co-product, also the slice category $C/I$ admit it? I need a depth demonstration. Thanks a lot!
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1answer
19 views

Elements and arrows in an abelian category.

Suppose to work in an abelian category $\mathcal{A}$, so in particular for every objects $A$ and $B$, we have that $Hom(A,B)$ is an abelian group - in particular a set. My questions are: Does it ...
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112 views

Does the forgetful functor from $\mathbf{TopGrp}$ to $\mathbf{Top}$ admit a left adjoint?

Let TopGrp be the category of topological groups (not necessarily $T_0$) and Top the category of topological spaces. Does the forgetful functor $U:\mathbf{TopGrp}\to\mathbf{Top}$ admit a left ...
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4answers
190 views

Elements and arrows in a category.

Suppose to have two objects $A$, $B$ in a fixed category and an arrow $\eta : A \to B$. Has an object "elements"? In the sense does the symbol $a \in A$ have sense? (in the most generic context, ...
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57 views

Stable epimorphisms of commutative rings

Recall that an epimorphism $f : A \to B$ in a category with fiber products is called stable (or universal) if for every morphism $C \to B$ the base change $A \times_B C \to C$ is an epimorphism. ...
2
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1answer
39 views

when a presheaf is a sheaf

I've seen a very natural definition when a presheaf $F:C^{op}\rightarrow Set$ is actually a sheaf. This definition used the functors $hom(-,-)$ and $F$ and notions of injective and surjective maps ...
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42 views

Classifying space infinite totally ordered set contractible

Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This should be the (possible infinite) simplex with vertices $X$, hence I ...
2
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1answer
28 views

free product with amalgamation is correspondingly a pushout

I'm trying to proof that the following diagram in the category of groups with $i_1$ and $i_2$ being inclusions is a pushout iff $G$ is the free product with amalgamation (up to isomorphism). It should ...
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58 views

Proof of the Coherence of Monoids in a monoidal category (Final Edit)

This is not Maclane's Coherence theorem; rather, a variant. I would like a critique of step 2 of my attempted proof. Let $B=\left ( B,\square, \alpha ,\lambda ,\varrho , \right )$ be a moinodal ...
4
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0answers
33 views

Concrete category of topological spaces over preordered sets: what are the initial morphisms?

I am reading The Joy of Cats to become more familiar with category theory and I came upon the following question on concrete categories. Let Top be the category of topological spaces and let Prost be ...
3
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1answer
47 views

Why is a cartesian morphism called cartesian?

I am reading about fibred categories. After reading the definition of "vertical" morphism, I can imagine why they are named like that. What about "cartesian" morphisms? What is cartesian about them? I ...
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52 views

Adjunction between topological and simplicial presheaf categories

For a small discrete category $C$, the singularization/realization adjunction $Top \leftrightarrows sSet$ induces an adjunction $Top^C \leftrightarrows sSet^C$. If I have a small topological category ...
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25 views

Characterization of epic morphisms in the category of rings.

Let $C$ denotes the category of rings (with identities). It's easy to prove that for any $R\in \text{ob }C$, any multiplicative set $S\subset R$ and any ideal $I\leqslant R$, both $R\to S^{-1}R$ and ...
2
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1answer
68 views

How to understand cocategories

$\newcommand\CC{\mathsf{C}}$The notion of a category is well-known. There are multiple equivalent definitions; small categories can be seen as an internal category in $\mathsf{Set}$, that is a ...
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2answers
40 views

Properties of the category of smooth vector bundles over a smooth manifold

I am wondering if there are any sources that discuss the properties of the category of vector bundles over a smooth manifold. It seems that most differential geometry texts I've looked at avoid ...
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2answers
39 views

Some lengthy question on natural transformations, category theory, and dual objects

So I am fairly new to algebra, and my instructor often uses terms from category theory (such as Hom-set, natural transformations, etc) that I am not familiar with. As an attempt to have a better ...
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0answers
56 views

Three theorems for the price of one? (like duality)

Why the notion of "duality" (when we get two theorems for the price of one) are ubiquitous in mathematics (order and lattice theory, category theory, group theory), but "triality" (three theorems for ...
2
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1answer
63 views

An equivalence of categories

Let $F: \Pi(X) \to \text{SET}$ be a functor, where $\Pi(X)$ is the fundamental groupoid of $X$. I have shown earlier that we can construct a covering $p: Y = \bigsqcup_{x\in X} F(x) \to X$ from the ...
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29 views

Existence of the functors between higher categories [closed]

How to prove the existence of the functor between two arbitrary higher categories ? Why is the functor from an ordinary category to a higher category important to study ? Is there any or class of ...
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0answers
44 views

Choice of a skeleton

Suppose we are in presence of a strong enough axiom of choice (e.g., choice for conglomerates). I know that any category has a skeleton, but I would like to know if I can choose a skeleton which ...
5
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1answer
64 views

Correct meaning of two spaces being homotopy equivalent under a space

Let $p_0 : A \to X_0 $ and $p_1 : A \to X_1$ be two maps. I am confused about what does it mean to say that '$X_0$ and $X_1$ are homotopy equivalent under $A$'. Which of the following statements is ...
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25 views

Definition of Schur Functors on morphisms

I've been learning about Schur functors on nLab: http://ncatlab.org/nlab/show/Schur+functor A definition is given, for $R$ some finite dimensional representation of $S_n$, by the formula $S_R(-)=R ...
3
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1answer
69 views

Adjoints functors in scheme theory

What's useful information available to us, when we state that : If $ f: X \to Y $ a morphism of schemes, and if $ \mathcal{F} $ denotes $ \mathcal{ O }_X $ - modules, and $ \mathcal { G} $ denotes $ ...
3
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2answers
61 views

Commuting of Hom and Tensor Product functors?

Let $V_i,W_i$ be finite dimensional vector spaces, for $i=1,2$. Assume we have homomorphisms $\phi_i:V_i\rightarrow W_i$. Then, there is an induced map $\widehat{\phi_1 \times \phi_2} \in Hom(V_1 ...
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0answers
30 views

Adjoint functors between $Set^I$ and $Set/I$ [closed]

I've defined the two functors $F$ and $G$ between the two categories $Set^I$ and $Set/I$, but now I have to prove that they are also adjoint left and right, i.e. : $Set/I(F((X_i)_{i \in ...
2
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1answer
80 views

Do finite products commute with colimits in the category of spaces?

Let $X$ be a topological space. The endofunctor $\_\times X$ of the category of all topological spaces does in general not possess a right adjoint, since the category is not cartesian closed. Is it ...
2
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0answers
31 views

Question on the coskeleton functor $\mathbf{coskel}'$ for pointed simplicial sets

There is an abstract construction of the coskeleton functor for simplicial set as follows: Fix an integer $n\geq 0$ and take the inclusion of the full subcategory $i_n\colon\Delta_{\leq ...
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38 views

Category of $\mathcal{L}$-structures

Given a purely relational language $\mathcal{L}$, the set of $\mathcal{L}$-structures forms a category under the definition of homomorphism found here. Call this category ...
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1answer
30 views

Equivalence of category of subsets and subobjects

I'm trying to show that the categories $\mathcal{P}(X)$ and $Sub(X)$ are equivalent. According to Steve Awodey's "Category Theory" I need to find two functors $ E: \mathcal{P}(X) \to Sub(X)$ $F: ...
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0answers
26 views

Empty Category and Trace [closed]

I would like to have your assistance regarding the mathematica rationale behind the ideas of empty category principle and trace as mentiones in the following link: ...
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1answer
92 views

The moduli space of stable maps

I am reading Katz' book Enumerative Geometry and String Theory. I have a few questions regarding the moduli space of degree $d$ genus $0$ stable maps into $\mathbb P^n$, denoted ...
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27 views
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How to prove that $\mathrm{Proj}\left(B/J\right)$ is isomorphic to $\mathrm{Proj}\left(A/J\right)$ if $I\subset J$?

Let $B$ be a graded ring with positive degrees, and let $I$ and $J$ be homogeneous ideals of $B$. We suppose that there exists $N$ such that $I\cap B_{n}=J\cap B_{n}$ for all $n\ge N$. How to show ...
2
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1answer
55 views

If an isomorphism can be expressed as a composition of morphisms, what can we say about its components?

Suppose $f:X\to Y$ is an isomomorphism, and $f=g\circ h$ where $h:X\to Z$ and $g:Z\to Y$. Can we infer that either of these component morphisms is an isomorphism as well? And does this change ...
3
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2answers
47 views

Is a topological space $X$ the colimit of an open cover $\cup U_i$ in this way? [duplicate]

Let $X$ be a topological space space and $X=\cup_{i\in I} U_i$ a covering of $X$ by open subsets $U_i\subseteq X$. Is it true that $$ \operatorname{colim}\left(\coprod_{(i,j)\in I\times I} ...
2
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0answers
66 views

$\mathsf{Top}$ with proper maps has products.

In I.M. James' General Topology and Homotopy Theory, he presents proper maps before introducing compact sets, by defining $\phi:X \to Y$ to be proper iff $\phi \times \text{Id}_T$ is closed for all $T ...
2
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0answers
53 views

Maclane's Coherence Theorem: why not just use the functors themselves?

I have a softball question on Maclane's proof. Suppose $B=\left ( B, \square , \alpha ,\rho ,\lambda \right )$ is a monoidal category. Fix $b\in B$. Define $W$, the (monoidal) category of binary ...
3
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1answer
29 views

Understanding the $\mathfrak{a}$-adic completion of an $A$-module as a functor

$\require{AMScd}$ I recently read the chapter 10 on Completions in Atiyah-MacDonald. They describe the $\mathfrak{a}$-adic completion $\hat{M}$ of an $A$-module $M$ as the inverse limit of an inverse ...
3
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1answer
35 views

integral domains and field of fractions

I've read about integral domains and their induced fields of fractions. For an integral domain $R$ its field of fractions $K$ is the "smallest" field that includes $R$, i.e. there is an injective map ...
3
votes
2answers
82 views

Definition of adjoint functor and locally small categories

In the definition of an adjoint pair of functors, is it implicit that the categories are locally small? I have searched for ages, and nowhere is this stated as an assumption, but the definition seems ...
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1answer
26 views

Show that w-complete Posets and continuous aplications between them form a category

I'm really lost with this thing that looks innocent but just can't figure out... can you help me? Show that $\omega$-complete Posets and continuous functions between them form a category. Thank ...
2
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2answers
139 views

How is the notion of adjunction of two functors usefull?

Is there a secret or an intuitive idea behind the fact of creating the concept of adjunction of two functors ( Functor - Adjoint Functor ) ? How is this notion of adjunction of two functors usefull ? ...
2
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0answers
56 views

Relation between being epic and having a “full” image.

Let $g: A \to B$ be a morphism in an abelian category. Is it true that $g$ is epic iff $im(g)=B$? Context: Given a short sequence in an abelian category $0 \to A \to B \to C \to 0$ with maps $f: A ...
3
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1answer
116 views

Definiton of Limit and Foundational problems.

I am new to Category theory and I have a quite strong foundational problem. For example, let's start from the definition of Limit stated by wikipedia ...
7
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1answer
129 views

The colimit of all finite-dimensional vector spaces

Let $\mathsf{iFinVect}_K$ be the category of finite-dimensional vector spaces with injective linear maps and $X : \mathsf{iFinVect}_K \to \mathsf{Vect}_K$ be the inclusion functor. Then ...
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0answers
38 views

A localization of a topos is still a topos

I am trying to see how a localization of a topos is a topos. That is, localization of a cartesian closed category is cartesian closed, and if a category has a subobject classifier any localization of ...
5
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38 views

Is the Groupoid of germs associated to an orbifold a Hausdorff proper Lie groupoid?

I was studying the book Introduction to Foliations and Lie Groupoids by I. Moerdijk and J. Mrcun and I have a doubt. On page 140 they give the following definition for a proper Lie groupoid. A ...
2
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1answer
60 views

Build sheaf from stalks

If I have a topological space $T$ and for each $p \in T$ I have an object $A_p$ in some category $\mathscr{A}$, then how can I define a sheaf out of this? In other words can I build a sheaf with ...