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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225 views

When is a category isomorphic to its opposite?

I could verify that if Morph$(A, B)$ is in bijective correspondence with Morph$(B, A)$ for all objects $A, B$ in a category then one shall construct isomorphism between that category and it's ...
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3answers
62 views

Given a subset $S$, is there any universal construction (property) of $L(S)$, the linear span of $S$?

Given a subset $S$ of a vector space $V$, is there any universal construction (property) of $L(S)$, the linear span of $S$ (like the way we can construct the fraction field of an integral domain)?
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0answers
33 views

How does a symmetric graph indexing category work?

In Category theory for the sciences, section 4.2.1.20 it is explained how a graph is a functor from an indexing category to a set. I think I understand the basic concept: Ar is mapped to a set of ...
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2answers
79 views

Show that the categories $G$-mod and $\mathbb{Z}G$-mod are equivalent.

I have another basic question inspired from reading the sixth chapter of Weibel's "An Introduction to Homological Algebra". First version of the question: a bit ambiguous At the first paragraph, ...
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1answer
51 views

How to prove adjunctions compose (via units and counits)?

Familiar background (partly to fix notation). Suppose we have functors $F\colon \mathscr{A} \to \mathscr{B}$, $G\colon \mathscr{B} \to \mathscr{A}$ such that $F \dashv G$, and functors $F'\colon ...
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1answer
40 views

Posets as Categories and Direct Systems of Objects?

My algebra textbook brings the following example of category: Every poset $(X, \preceq)$ might be seen as a category as follows: Objects: Elements of $X$; Morphisms: $\textrm{Hom}(a, ...
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1answer
134 views

canonical map of a monoid to its classifying space

Every monoid $M$ is a category with one object $M$ and morphisms the elements of $M$. [Martin Brandenburg.] Every small category $C$ has a classifying space $BC$, defined as the geometric realization ...
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1answer
42 views

Associativity of the additive law in a category with finite biproducts

It is well known that if $\mathcal{C}$ is a category with finite biproducts, then we can define a binary operation "$+$" on every set of morphisms $Hom_{\mathcal{C}}(X,Y)$ using the diagonal ...
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2answers
244 views

Category of binomial rings

A binomial ring is a commutative ring $R$ such that (1) the additive group of $R$ is torsionfree and (2) $n!$ divides $x(x-1)\dotsc(x-n+1)$ for all $n \in \mathbb{N}$ and $x \in R$. We may then define ...
2
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1answer
55 views

Sheafification as a Kan Extension of the Identity?

How can the sheafification functor be described in terms of a Kan extension of the identity on the category of $\mathsf{Set}$-valued sheaves (over some topological space)? How about general $\mathsf ...
2
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1answer
91 views

How is the free group on $S$ generators a cogroup?

According to nLab: Cogroup objects in the category of groups are free groups, and to give a free group the structure of a cogroup object is the same a choosing a generating set. This is an old ...
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0answers
47 views

Unit and co-unit of Exponential Product and Sets

Let $\cal C$ be a category with binary products and let $Y$ and $Z$ be objects of $\cal C$. The exponential object $Z^Y$ can be defined as a universal morphism from the functor $–×Y$ to $Z$. ...
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0answers
44 views

Homotopy colimits preserve weak equivalences

It is well known that homotopy colimits of diagrams are constructed so that if one has weak equivalences between all objects of two diagrams (under the same indexing category) the induced map between ...
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1answer
38 views

When do evaluation functors reflect (co)limits?

It is a well known result that limits in functor categories are computed pointwise. In Tom Leinster's Basic Category Theory he phrases this result in terms of the evaluation functors ...
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1answer
76 views

what is the classifying space of a monoid

In the paper Homology Fibrations and the "Group-Completion". Theorem. McDuff, D.; Segal, G., 1976, the first line: A topological monoid $M$ has a classifying space $BM$. I do not understand this ...
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0answers
38 views

variation on the image of a functor

It is well known that the naive "image" of a functor $F\colon C\to D$, i.e., all the images of objects in C and all images of morhipsms in C, need not constitute a category. It is then natural to ...
3
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1answer
50 views

Direct Limit in HTop

I am currently trying to figure out the importance of homotopy direct limits in Top, which are of course different from direct limits in HTop. I have been told that the latter need not even exist, but ...
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0answers
49 views

Cocomplete base category implies cocomplete slice category

I'm having trouble with the final step of the proof that if $\mathsf C$ is a cocomplete category, so is each of its slice categories. Here's the proof given in Borceux's Handbook of Categorical ...
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1answer
48 views

Beck-Chevalley via coend-calculus

Consider the following commutative square in $\bf Cat$; strictness can be removed from the initial assumptions, but let's proceed gradually. Consider the induced diagram between presheaves ...
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95 views

What should this definition be?

This is from Advanced Linear Algebra by S.Roman. Definition (on page 356). Referring to the figure below, let A be a set and let $\mathcal S$ be a family of sets. Let $$ \mathcal F = \{g\colon ...
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0answers
50 views

Quotients vs equivalence relations

In a recent preprint I defined the category of quasi-frames (qframes for short) as follows: a qframe is a modular and upper continuous complete lattice; a morphism of qframes $f:L_1\to L_2$ is a map ...
4
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2answers
90 views

Objects that send only monomorphisms

I just re-learned that fields can have only 1-to-1 homomorphisms from them. Is this a common trait in other categories? Can we extend, for instance, many topological spaces to spaces that have only ...
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1answer
39 views

Is Module Category over Monoidal category Monoidal?

let $\mathcal{C}$ be a monoidal category and $\mathcal{M}$ a $\mathcal{C}$-module category. Does $\mathcal{M}$ need to be a monidal category? I know it is true for certain categories, but is it true ...
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0answers
24 views

Fibrations over topoi

Let $\mathcal{S}$ be an elementary topos. What is (exactly) the relation between $\mathcal{S}$-indexed categories and fibrations over $\mathcal{S}$? Where can I read about this? (Or even find the ...
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2answers
203 views

Are groups with all the same Hom sets already isomorphic?

I was thinking about the following: Say we got two groups $A$ and $B$, and we know that for any group C, there is a bijection $$Hom(A,C) \to Hom(B,C).$$ Are $A$ and $B$ already isomorphic? If the ...
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1answer
39 views

How to read “realize the mapping $x \cdot -: T \rightarrow T$”

This question is about Category theory for the sciences (by David Spivak). In Exercise 3.1.2.4-a the set $T = \{x \in \mathbb{R} \; | \; 0 \leq x < 12\}$ needs to be defined using a coequalizer. I ...
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2answers
96 views

Unique extendable functions… Is there a theory?

Motivation: Take a continuous function $f$ from a topological space $X$ to a hausdorff space $Y$. If $g$ is a continuous function from $X$ to $Y$ that coincides with $f$ in a dense set, then $g=f$. ...
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0answers
49 views

Monomorphism preservation by pullback

I'm working through Category theory for the sciences (by David Spivak) and I'm a bit stuck in section 2 - my problem is with Proposition 2.7.5.5 and Exercise 2.7.5.6, I hope someone can help :) The ...
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0answers
67 views

“Lossy” v.s. “Lossless” Monads

Let us look at two monads on $\bf Set$. The first will be the finite sequence monad (from the free forgetful adjunction with $\bf Mon$.) $$\eta(x)=[x]$$ $$\mu([[a,b, \dots, z],[\alpha,\beta, \dots, ...
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1answer
44 views

Categorical version of the Tietze Extension Theorem

In Donald Hartig's short paper An Important Functor in Analysis and Topology, Theorem 1 is preceded by the following statement: Since the spaces we are dealing with are compact, a one-to-one map ...
0
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1answer
60 views

Category of ordinal numbers

Let $\Delta$ be the category of finite ordinal numbers with order-preserving maps, i.e., $\Delta$ consists of objects strings $$ [n]: 0 \to 1 \to 2 \to \dots \to n. $$ A morphism $f:[n] \to [m]$ is ...
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1answer
64 views

Categories for the working mathematician exercises III 1

I'm currently reading Mac Lane's Categories for the working mathematician and I'm having some trouble with the two following exercises from part III. Find (from any given object) an universal arrow ...
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4answers
789 views

Why is the cartesian product so categorically robust?

The major "broad/natural" categories I encounter in daily life are: sets, groups, topological spaces, smooth manifolds, vector spaces over a fixed field $k$, $k$-schemes, rings, $A$-algebras for a ...
4
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2answers
42 views

Prove that a transformation of the identity functor of a Group $G$ (seen as a category) into itself is just an element of the center of $G$

I want to prove the follow: Suppose $G$ is a group seen as a category, prove that a transformation of the identity functor of $G$ into itself is just an element of the center of $G$. I'm not ...
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0answers
53 views

Description of free Lie algebra in Weibel's book

In Exercise 7.3.2 in Weibel's book An Introduction to Homological algebra the following description of the free Lie algebra over some $k$-module $M$ is given (where $k$ is any commutative ring): ...
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1answer
52 views

If there is in a category $\mathcal{A}$ finite products and equalizers then it has pullbacks

My homework consist in showing that "If there is in a category $\mathcal{A}$ finite coproducts and coequalizers then it has pushouts" based on the proof that "If there is in a category $\mathcal{A}$ ...
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0answers
19 views

Extending actions of Monads on Endofunctors.

Let $X^{X}$ be the category of endofunctors on a category $X$. Then if we define $\otimes :X^{X}\times X^{X}\rightarrow X^{X}$ by $R\otimes S=RS$ on objects and $\tau \otimes \sigma $ to be horizontal ...
2
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1answer
59 views

Definition of Category of Hypergraphs

I have a question concerning the definition of Hypergraphs in category theory, which I adopted from "A category-theoretical approach to hypergraphs" by W.Dörfler and D.A.Waller: ...
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1answer
52 views

Coproduct of groups explanation

Could someone please explain the following? "Let $G=\prod G_{i}$ be a direct product of groups. Then each $G_j$ admits an injective homomorphism into the product, on the j-th component, namely the map ...
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1answer
36 views

equivalence between subcategories in abelian categories keeps exact sequence

Let $A$ be an abelian category and $B$ a subcategory, not necessary abelian. Let $C^\bullet$ be a exact complex in $A$ with $C^i\in B$. Suppose there is another abelian category $A'$ and $B'$ a ...
2
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2answers
51 views

Is this a correct way to think about specific examples of groups using the category theory definition?

I'll say now, before anything else, that I probably don't know what I'm talking about. This is more me making a (hopefully) educated guess about a topic I'm not too familiar with. I recently started ...
4
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1answer
79 views

Does the product functor preserve quotient maps?

In Hatcher's Algebraic Topology, he presents a proof that if $(X,A)$ satisfies the homotopy extension property, and $A$ is contractible, then $X \simeq X/A$. Part of Hatcher's proof goes: Suppose ...
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1answer
90 views

Prove the isomorphism of categories $Fun(\mathcal{A}\times\mathcal{B},\mathcal{C})\cong Fun(\mathcal{A},Fun(\mathcal{B},\mathcal{C})),$

I'm a computational engineer starting with a course of Introduction to Category Theory, and perhaps is extremely basic what I'm asking but I'm trying to learn how to make proofs in category theory ...
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2answers
45 views

Fiber product of groups

Let $f: G \longrightarrow K$, $g: H \longrightarrow K$ be group homomorphisms. For the fiber product $G \times_K H$ to exist, is it necessary to require that $f$ and $g$ be surjective? I can't see ...
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2answers
55 views

Proving that disjoint unions of presentations are coproducts of groups

I'm working through Aluffi's Algrebra: Chapter 0 and I need some assistance with an excercise. Aluffi, Ex. II.8.7 Let $(A|R)$, resp. $(A'|R')$, be a presentation for a group $G$ in Grp, resp. ...
3
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1answer
31 views

The coproduct of a family of objects of a Preorder (seen as a category)

If the coproduct of a family of objects of a Poset (seen as a category) is the least upper bound, who is the coproduct of a family of objects of a Preorder (seen as a category)? My intuition ...
3
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1answer
59 views

Universality of the Simplex Category. Proving Functoriality of the Map.

Let $B$ be a strict monoidal category, and $\left \langle c,\mu ',\eta ' \right \rangle$ a monoid in $B$. Now suppose we consider the simplex category $\left \langle \triangle ,+,0 \right \rangle$, ...
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1answer
121 views

What is the mathematical difference between group and category?

This question is quite similar to the following link: Why learn Category Theory in order to study Group Theory? The above link is nice but I could not find the difference mathematically between ...
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0answers
34 views

Simplicial function space and homotopy colimits

I am currently reading the book by Bousfield and Kan, in particular Ch. XII, par. 2, and would like to understand why the functor $hocolim: Top_{+}^{I} \rightarrow Top_{+}$ is left adjoint to $hom(I ...
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1answer
38 views

The pushout of an epimorphism is an epimorphism

I'm reading the book "Handbook of Categorical Algebra: Volume 1, Basic Category Theory" by Francis Borceux, and in page 52 he states that "..."the pullback of a monomorphism is a monomorphism". ...