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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$F$ is an equivalence of categories implies that $F$ is fully faithful and essentially surjective

I read in wikipedia that: One can show that a functor $F : C → D$ yields an equivalence of categories if and only if $F$ is full, faithful and essnetially surjective. I'm trying to prove this but I ...
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55 views

Connected components functor for free coproduct cocompletions

Any extensive category admits a notion of connected object and hence a disconnected object. However, not all disconnected objects are presentable as disjoint unions of connected objects. Among ...
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1answer
88 views

Homology as categorification of Euler characteristic

I am trying to understand: "Thus, the homology of a manifold M can be seen, in a sense, as a categorification of its Euler characteristic: a more sophisticated and richly structured bearer of ...
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42 views

What is an L-Function and how can it be used for categorization?

On hackernews, I came across this article, describing the LMFDB, the database of L-functions, modular forms, and related objects which aims at the categorization of mathematical objects where ...
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1answer
40 views

Finitely generated projective modules form exact category

I am looking for a proof of the fact that the category of f.g. projective modules over some commutative ring is exact. Actually, I would already be happy to know why this category is closed under ...
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2answers
84 views

The rigid additive tensor category freely generated by an object

I've found myself to be absolutely mystified by something in Deligne and Milne's notes on Tannakian categories. Namely, on p. 16 they are showing that there is a rigid additive tensor category $\...
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46 views

Forgetful functor applied to a module

I try to find a left adjoint to the forgetful functor $U: R-Mod \longrightarrow Ab$. I considered a functor $F:Ab \longrightarrow R-Mod$ defined by $F(G)=Hom(U(R),G)$. I'm not so sure that in this ...
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1answer
39 views

equivalence of definitions for connected objects?

For an extensive category, the following conditions are equivalent for an object $C$. The representable copresheaf of $C$ commutes with coproducts. The $C=X\amalg Y\implies X\text{ or }Y$ is $0$ and ...
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41 views

The benefit of writing Banach space theory in categorical language!

I was wondering if there exists a special benefit of writing Banach space theory in categorical language? I mean does there arise a hint of the existence of a connection with other mathematical field ...
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49 views

Slick Definition of the Category of Cartesian Closed Categories

I can produce elementary definitions by just inspecting the definition on nlab, but is there a readily available abstract definition? I vaguely remember seeing that they could be defined as algebras ...
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47 views

Categorical Representation of the Set of All Strings

Let $A$ be a small preordered category. How would we define a preordered category $\mathcal A$ for all strings over $A$ (e.g., Kleene Closure) ordered by the subword order (Def'n $3.1$) $\leqslant$? ...
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20 views

cancellative property of coproducts in extensive categories?

In the final paragraph of the first section of this article, the following is written: Given an isomorphism $A+B\cong A+C$ compatible with the injections, one can construct an isomorphism $B\cong ...
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48 views

Exact adjoint functors of triangulated categories

Let $T$ and $S$ be triangulated categories. Let $F:T\rightarrow S$ and $G:S\rightarrow T$ be two adjoint functors. Assume that one of them is exact (i.e. sends exact triangles to exact triangles and ...
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3answers
221 views

Why does the name “epimorphism” refer to a surjective homorphism?

The wikipedia page talks about epimorphisms with category theory in mind, but I have no experience with this and ask this question from a group theory point of view (answers from any point of view are ...
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94 views

Non-injective monomorphisms

I am reading Borceux, vol. 1, I found this example at page 27: we consider the category whose object are the pairs $\langle X,x\rangle$ where $X$ is a topological space and $x$ a point of $X$ (base ...
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128 views

Is there some kind of deep relationship between substitution and recursion?

Define $\mathbb{N}$ as the initial object in the following category: Objects. Sets $X$ equipped with a function $S : X \rightarrow X$ and an element $0:X$. Morphisms. Functions that preserves ...
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27 views

Representable bifunctors

Is there a notion of representability for functors in the form $F:C^{op} \times C \to Set$? Can anyone please give me a reference? Thanks.
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1answer
48 views

Regular coimages in a finitely complete and finitely cocomplete category

According to the nLab, in a finitely complete and finitely cocomplete category $\mathcal{A}$, for every morphism $f : A \to B$ a regular coimage exists and is given by the coequalizer of $f$'s kernel ...
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1answer
37 views

Free cocompletion of thin categories

Let $C$ be a thin category. Is there a cocomplete thin category $C\to C'$ such that for every functor $C\to D$ into a cocomplete thin category there is a unique, up to isomorphism, cocontinous functor ...
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71 views

What would an infinite dimensional projective space look like as a scheme?

In topology, we can construct $\mathbb{CP}^\infty$ as the direct limit of $\cdots\rightarrow \mathbb{CP}^n \rightarrow \mathbb{CP}^{n+1}\rightarrow \cdots$ with the embedding given by $[x_0: x_1: x_2: ...
3
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1answer
45 views

Bounded derived category and hereditary categories

Let $\mathcal{A}$ be an abelian category with enough projectives (injectives). I tried to prove that if every element $M$ of $\mathcal{D}^{b}(\mathcal{A})$ satisfies $$ M \cong \bigoplus_{i} H^{i}(M)[-...
2
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1answer
42 views

Extension of field homomorphisms and pullback square

Let $E/k$ and $F/k$ be two subextension of a field extension $K/k$. The following square induced by restriction functions is always pullback square (in category of sets and functions)? $$\begin{...
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39 views

Lax cones for a monad

A monad $T$ on $\cal C$ is a lax functor $\mathbb T : \mathbf{1}\to\bf Cat$. The lax colimit and limit of $\mathbb T$ are the Klesili and EM categories ${\cal C}^{\mathbb T}$, ${\cal C}_{\mathbb T}$ ...
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32 views

How should automorphisms of monoid actions in $\mathbf{Rel}$ be defined?

To explain the question, I'd like to start by considering monoid actions in $\mathbf{Sets}$. In this case, a monoid action is simply a functor $S$ from a monoid $M$ to $\mathbf{Sets}$. Then, one can ...
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49 views

Transfinite composition

I am reading Chapter 10 of P.S.Hirschhorn book on model categories, and I have a question about Proposition 10.2.6. and 10.2.7. Proposition 10.2.7. gives some sufficient conditions for a map $f:P\to ...
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43 views

Characterisation of adjunction via initial object of a comma category

The following is a definition given in my lecture notes that I'm not sure is standard, so I'll write it out: Given a functor $G: \mathcal{D} \to \mathcal{C}$ and an object $x$ in $\mathcal{C}$, we ...
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72 views

Existence of some categories in the category of categories

I'm studying category of categories. I read that when there are categories $A,B$, it is allowed to define the product $A\times B$. Equalizers and coequalizers also exist. However, there are some ...
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1answer
34 views

Proofs of uniqueness up to isomorphism

I'm embarrassed about this question but here goes: Reading through the proof of Lambek's lemma (https://ncatlab.org/nlab/show/initial+algebra+of+an+endofunctor), I was stumped at the step where $\...
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1answer
33 views

Is there an adjoint functor to the contravariant hom functor in the category of A-modules.

I should start by saying that I don't know any category theory. However, I am reading Atiyah-MacDonald and have just learned that in the category of A-modules (where here A is a commutative unital ...
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2answers
54 views

Morphism in category theory

Morphisms in category theory map from one object to another object. E.g. for group, object is set, right? Homomorphism is morphism. For topplogy space, object is set. Homeomorphism is morphism. My ...
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1answer
41 views

Formal Notation for Finding Inverses of Functions

Generally in most introductory university courses, finding the inverses of functions, is done in what seems to be to be a very haphazard way. Given any scalar function $f : \mathbb{R^n} \to \mathbb{R}...
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67 views

Elegant formalism for gluing spaces over open subsets (Vakil 17.2.B.)

This question is about exercise 17.2.B. from Vakil's algebraic geometry notes. Let $X$ be a scheme. The following data is given: For each affine open set $U \subset X$ a scheme $\pi_U :Z_U \to U$. ...
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55 views

Coproduct in the category of metric spaces

While discussing categories without coproducts, we stumbled with the category $\mathbf{Met}$ that takes metric spaces as its objects and short maps as its morphisms. It is claimed that $\mathbf{Met}$ ...
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1answer
37 views

On essential surjectivity of the restriction functor

Consider the following setting: $P$ and $Q$ are two finite posets, and $i\colon P\to Q$ is a fully faithful embedding of $P$ in $Q$ (that is, $p\leq p'$ in $P$, if and only if $p\leq p'$ in $Q$). ...
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1answer
36 views

Does every surjective morphism from an uncountable into a countable monoid admit a homomorphic right inverse function

Let $M$ be a an uncountable monoid (like $\mathbb R$ with addition or multiplication) and $N$ be a countable monoid (like $\mathbb N_0$, or $\mathbb Z$ with addition or multiplication). Further ...
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1answer
32 views

Closed maps in terms of lifting properties (analogousy to formally étale morphisms)?

In continuation to this MSE question, where closed maps are characterized by "fiber thickenings", I trying to formulate this fiber thickening condition as some lifting property of $f$ against some ...
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2answers
56 views

Coproducts in Top preserved under pullbacks?

The statement which I'd like to prove is as follows. Let $A=\coprod_{i \in I} A_i$ be the coproduct of the sets $\left(A_i\right)_{i \in I}$ and suppose that we have for all $i \in I$ a pullback in ...
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1answer
80 views

Why does this exact sequence of sheaves imply the maps are $G$-equivariant?

I'm confused about something in Tamme's Introduction to Étale Cohomology (page 27). Let $G$ be a group and let $G$-$\mathsf{Set}$ denote the category of left $G$-sets with $G$-equivariant maps as ...
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32 views

What notation is usually used to denote the category of $R,S$-bimodules

I came across the notation: $\mathbf{Mod}_{(R,S)}$. But generally, when handwritten, long superscripts/subscripts are becoming clumsy. So I'm wondering whether there's an adopted alternative. Is ...
5
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1answer
63 views

Functor of points definition of a space modeled on a site

I'm trying to find a definition of a space modeled on a site which is: (i) plausible and natural in the context of general sites (ii) subsumes common examples. Let $(C,J)$ be a grothendieck site and $...
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1answer
79 views

Advanced examples of categories

I'm looking for some example of categories which requires some effort to prove that it is a category (For example it is straightforward to prove that $\mathbf{Set}$ is a category, I don't want that ...
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Converse to pullback pasting and local diffeomorphisms

The nlab page on local diffeomorphisms gives the following two equivalent conditions for a smoonth function to be a local diffeomorphism. The derivative is an isomorphism of tangent spaces $df:T_xX\...
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1answer
30 views

The canonical base point for Weil algebras

Kock defines, after (16.2), the canonical base point of a small object $\operatorname{Spec}_R(W)$ to be $$\mathbf 1\overset{\operatorname{Spec}_R (\pi)}{\longrightarrow}\operatorname{Spec}_RW$$where $\...
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1answer
99 views

Is there another way to describe the category $\mathbf{Top}$ of topological spaces?

The category $\mathbf{Top}_\ast$ of pointed topological spaces can be viewed as the comma category $(\bullet\downarrow\mathbf{Top})$. The objects of the category $\mathbf{Top}$ are topological spaces ...
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Localization of triangulated categories

I have been reading from the Stacks project, and Lemma 13.5.4. says: Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor of pre-triangulated categories. Let $$ S = \{f \in \text{...
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Duality of diagrams for fibration and cofibration

According to May's A Concise Course in Algebraic Topology, the diagrams in the following represent cofibration and fibration, respectively if there exists an arrow diagonally (to the upper right ...
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1answer
48 views

Does this property really characterize monomorphisms?

In the post Does this property characterize monomorphisms?, I do not see how the third condition is equivalent to the others. Specifically, I require that $k_0$ and $k_1$ be isomorphisms in order that ...
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41 views

How to write the real projective plane as a pushout of a disk and the mobius strip?

I heard in topology class that the real projective plane is obtained by gluing a disk along the boundary of the mobius strip. I was wondering - how can I write this as a pushout? Also, how can I ...
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1answer
43 views

Are the components of the unit of an adjunction monomorphisms?

Looking at the list of adjunctions (in CWM) I strongly get the impression that the components $\eta_x$ of the unit $\eta$ involved are all monomorphisms. But uptil now I have missed/overlooked any ...
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1answer
45 views

Equivalence of definition of product in a category

I was reading Mitchel book on categories and the following observation without proof is given: A family of morphisms given by $\lbrace p_{i}:A \to A_{i} \rbrace$ is the product of $A_{i}$ if and only ...