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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1answer
20 views

Multicategories with out-arities

Basically, my question is: Why the emphasis on domains in the notion of multicategory? I will now give the formal framework to state it correctly. Passing from categories to multicategories ...
0
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1answer
32 views

Opposite directions of adjunction between direct and inverse image in $\mathsf{Set}$ and $\mathsf{Sh}(X)$

Let $f:X\rightarrow Y$ be an arrow in $\mathsf{Set}$. $f$ induces three functors: $$\exists (f),\forall (f):\mathcal P(X)\rightarrow \mathcal P(Y),\; \mathsf I(f):\mathcal P(Y)\rightarrow \mathcal ...
12
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1answer
90 views

Grothendieck's yoga of six operations - in relatively basic terms?

I'm reading about the basic interactions between sheaves over topological spaces and arrows in $\mathsf{Top}$, in particular, about the inverse/direct image functors $f^\ast \dashv f_\ast$, the proper ...
1
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1answer
25 views

Inverse image sheaf functor: proving $(f^\ast P)_x=P_{f(x)}$

Let $P$ be a $\mathsf{Set}$-valued presheaf and let $f^\ast:\mathsf{PSh}(Y)\rightarrow \mathsf{PSh}(X)$ be the (topological) inverse image sheaf functor, defined on objects as the filtered colimit ...
3
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1answer
42 views

Does the concept of “cograph of a function” have natural generalisations / extensions?

First, definitions: The graph of a function $f : A \to B$ is a subset of $A \times B$, namely the set $\{(x,y) : x \in A, y \in B, f(x) = y\}$. The cograph of a function $f : A \to B$ is the ...
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2answers
59 views

Prove that the free group generated by two elements is a coproduct of the integers by itself in Grp

That's a problem from Algebra: Chapter $0$ by P. Aluffi, p.78, ex.5.6. One needs to prove that the group $F(\{x,y\})$ is a coproduct $\mathbb Z*\mathbb Z$ of $\mathbb Z$ by itself in the category ...
2
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1answer
24 views

Coproduct of rooted posets

The questions Currently, I'm working with "Category Theory" by Steve Awodey (which is quite readable, by the way). Their, on page 68 he gives an example for a coproduct, the coproduct in rooted ...
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1answer
68 views

Equal Categories

Let $\infty$ be the "category" of all categories, where the objects are categories and the morphisms are functors. I am trying to motivate the definition of equal categories by doing it as follows. ...
12
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7answers
310 views

Why associativity $h \circ (g \circ f) = (h \circ g) \circ f$ is required in composition?

An introduction into category theory says that A category is a quadruple $A = (O, \mathrm{hom}, \mathrm{id}, \circ)$ consisting of blah-blah and is subject to the following conditions: (a) ...
1
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0answers
28 views

Understanding Tabulation in Rel Category

Tabulation in Allegories is a structure that is defined over the shape $A \rightarrow B$. In a specific Allegory, namely Rel category (Category of Sets and Relations) this seems to be the reification ...
4
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0answers
44 views

The ordinals as a free monoid-like entity

Let $\kappa$ denote a fixed but arbitrary inaccessible cardinal. Call a set $\kappa$-small iff its cardinality is strictly less than $\kappa$. By a $\kappa$-suplattice, I mean a partially ordered set ...
5
votes
3answers
371 views

Monoids in Category Theory

I don't have a strong math background (engineering math) so I am at a bit of a disadvantage here but I have been trying to learn the broad strokes of Category Theory to help get a fuller picture of ...
3
votes
1answer
61 views

A functor preserves a product of $A$ and $B$ iff $F(A \times B) \cong F(A) \times F(B)$?

Let $F \colon \mathbf A \to \mathbf B$ be a functor, and let $A, B \in \mathbf A$. Assume that there exists a product $A \times B$, with projections $p \colon A \times B \to A$ and $q \colon A \times ...
2
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1answer
19 views

Category with zero

From "An Introduction to Ring Theory", Paul Cohn: "Let $\mathcal{A}$ be any category and define $\mathcal{A}''$ as the category obtained from $A$ by adjoining one object $Z$with a single morphism ...
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0answers
32 views

Why is the trivial group a zero object for the category of groups, but the empty set isn't a zero object for the category of sets? [duplicate]

I understand that the zero ring can't be a zero object for the category of rings, because in that case the 'arrows' are ring homomorphisms which, by definition, but maintain the unit. But in the ...
1
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1answer
37 views

Distributivity of pullbacks

If we consider morphisms $A\rightarrow C\leftarrow B$ in a category $\mathcal{C}$, then we denote their pullback by $A\leftarrow A\times_{C}B\rightarrow B$. The question is the following: is it true ...
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0answers
26 views

How is a connection the same as Morphism between different pullbacks to the first infinitesimal neighbourhood

This question arose from the first pages of Delignes Equations Differentielles. There he defines a connection on a vector bundle $V$ on an complex analytic space $X$ as the following Data: For all ...
3
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1answer
80 views

Properties preserved under equivalence of categories

I would like to ask about properties that are preserved under equivalence of categories. To be more specific, is it true that equivalences preserve limits? Why?
2
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2answers
84 views

Is defining an “$S$-scheme” this way just a stylistic choice?

I have a potentially pretty silly question, involving the definition of an "$S$-scheme". For a fixed scheme $S$, the definition of the category of $S$-schemes is a category with: objects: scheme ...
2
votes
2answers
55 views

How can you take the dual of a category whose objects are Sets?

Let's say I have a category with two objects A {1, 2} B {3} I have the following morphisms ...
1
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1answer
32 views

Fourier-Mukai kernels of mutations?

if I have an exceptional object E (on say the derived category of a smooth and projective variety) then I can define the left and right mutation functors. These are typically defined in terms of ...
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0answers
101 views

Is there a special name for functors from a category C to a subcategory of C?

Is there a special name for functors from a category $C$ to a subcategory $S$ of $C$ where $S \ne C$? I'm using $\ne$ pretty informally above. I'd be happy with anything that reasonably captures the ...
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1answer
23 views

How should I think about morphism equality?

I am studying category theory, and I think I often struggle between the fuzzy lines between Set as a category and then the category theoretic abstractions. For example, take monomorphisms. I totally ...
1
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1answer
43 views

Is this correspondence covariant or controvariant?

I'm new to category theory and am trying a basic exercise. Is the correspondence from $S$ to $\mathcal{P}(S)$, which assigns to $f:S\rightarrow T$ the mapping $\mathcal{P}(S)\rightarrow ...
2
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0answers
64 views

Categories that differ in morphisms

Let $C_1$ and $C_2$ be two (small) categories defined over the same set of objects: that is, $C_1$ and $C_2$ differ only in their hom-sets. Specifically, $Hom(C_1) \neq Hom(C_2)$, and $Hom(C_1) \cap ...
0
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1answer
22 views

Is every bijective pseudograph homomorphism a pseudograph isomorphism?

The term pseudograph describes a graph that may have parallel edges and loops. Formally this is a triple $G = (V,E,\delta)$ with $V,E$ sets and a map $\delta \colon E \to (V \times V)/\sim$, where ...
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1answer
27 views

Category of Sets and Bag-valued functions

I asked here about the Category of sets and set-valued functions, and it turns out it to be equal to REL (Category of sets and Relations),so a good studding point to study that category. Now, It ...
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1answer
76 views

Contracted version of “isomorphic”

Had a look around and I can't find a word which acts as a contraction for "isomorphic" in the same way that "monic/epic" is a contraction of "monomorphic/epimorphic". For some reason this strikes me ...
7
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1answer
64 views

Group action on a category

Motivating example: We get a functor from the category of real vector spaces to the category of complex vector spaces by complexifying (i.e. tensoring over $\mathbb{R}$ with $\mathbb{C}$). Let ...
2
votes
1answer
46 views

Equivalent conditions for equivalence of categories (Proposition 7.26 in Awodey)

I'm trying to understand the proof of the following proposition in Steve Awodey's "Category Theory". Let $\mathbf{C}, \mathbf{D}$ be categories and let $F: \mathbf{C} \to \mathbf{D}$ be a functor. ...
1
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1answer
42 views

Can a contravariant functor be adjoint to a covariant one?

I am a bit confused about the definition of adjoint functors, since everywhere the definitions found (see example wikipedia https://en.wikipedia.org/wiki/Adjoint_functors) seem to not specify if we ...
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1answer
39 views

Direct (inductive) limit of groups

Let $(I,\prec)$ be a directed poset and $\{G_i\}_{i \in I}$ groups with group homomorphisms $f_{ij}:G_i \to G_j$ whenever $i \prec j$. Is is true that the direct limit of this system is given by $$ ...
7
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2answers
147 views

Expand $\binom{xy}{n}$ in terms of $\binom{x}{k}$'s and $\binom{y}{k}$'s

Motivated by this question, I want to find a complete set of relations for the ring of integer-valued polynomials, where the generators are the polynomials $\binom{x}{n}$ for $n\in \mathbb{N}$. The ...
0
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1answer
19 views

Compact objects in Ind-categories

Assume a category $\mathcal{D}$ admits small filtrant inductive limits. Then call an object $Y$ of $\mathcal{D}$ compact, if $\hom_{\mathcal{D}}(Y,\cdot )$ commutes with these small filtrant inductive ...
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2answers
293 views

Example of a forgetful functor that is not faithful.

In the forgetful functor Wikipedia article I read that "[Forgetful] Functors that forget the extra sets need not be faithful; distinct morphisms respecting the structure of those extra sets may ...
2
votes
1answer
31 views

Category with coproducts generated by an endomorphism

Let's call a category with arbitrary coproducts a $\coprod$-category. A $\coprod$-functor is a functor which preserves coproducts. An example is $\mathsf{Set}$, and this is in fact the universal ...
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0answers
51 views

when will homology and direct limit commute?

Question: Let a sequence of maps between topological spaces $$ X_1\to^{f_1}X_2\to^{f_2}X_3\to^{f_3}\cdots $$ The mapping telescope is denoted by $T$. Under what conditions will $H_*(T)$, the ...
3
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0answers
59 views

Monomorphisms and injectivity predicates

This is a curiosity question which struck me at a less-than-optimal moment; I apologize for not having thought much about it. Motivation. It is well-known that monomorphisms in a concrete category ...
0
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1answer
25 views

Equivalence of group objects in set and groups as one object categories.

There are (at least) two definitions of groups in category theory: As a group object (in a catgory $C$ with finite products, e.g. $C$ = Sets). This is a tuple $(G,m,inv,e)$ with the following data ...
4
votes
1answer
58 views

Two modules are isomorphic in the stable module category iff they are projectively equivalent

Let $R$ be a (not necessarily commutative) ring. Let ${\text{mod-}R}$ be the category of finitely generated right $R$-modules. Let $\underline{\text{mod-}R}$ be the stable module category, with the ...
3
votes
1answer
41 views

Zero-section as homomorphism of rings

Let $s : X \to E$ be the zero section of a vector bundle $E$ over a scheme $X$. Zariski-locally this corresponds to a homomorphism $Sym_A(M) \to A$ of $A$-algebras where $M$ is a finitely generated ...
4
votes
1answer
72 views

Problem based category-theory book.

I love problem based textbooks like all of those by R. P. Burn and Halmos' Linear Algebra Problem Book, etc. Are there any problem based Category Theory textbooks, I know that the first chapters of ...
3
votes
1answer
35 views

base change of an equivalence relation of fppf sheaves

Let $S$ be a scheme, $R,U$ be $S$-schemes and $s,t : R \to U \times_S U$ be an equivalence relation i.e. it's a monomorphisme such that for every $S$-scheme $T$, $R(T) \to U(T) \times U(T)$ is and ...
2
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1answer
86 views

Why composition is so important in category theory?

I'm reading "Category: The Essence of Composition" As a software developer, I understand why composition is important in programming. It's allows you to get complex components from simple ...
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0answers
34 views

Duality of Projective and Inductive Limit

Could someone please explain to me in what sense the projective and inductive limits are "dual" to one another?
5
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1answer
111 views

Why is the full subcategory consisting of simply connected spaces not complete?

Let $\mathbf{Top}_*$ be the category of pointed topological spaces and $\mathbf{Top_1}$ the full subcategory of simply connected spaces. $\mathbf{Top}_*$ is complete and cocomplete. I am trying to ...
3
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1answer
62 views

How does one define a group with commutative diagrams?

I am currently working through McLarty's book on Elementary Categories, Elementary Toposes. In Chapter 3, he considers a group as an object in a category with a unit map, a multiplication and an ...
5
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1answer
51 views

Natural Transformation: Direct Products

I have result that tells me $$\displaystyle \varphi : \text{Hom}_R \bigg(A, \prod_{i \in I} B_i \bigg) \to \prod_{i \in I} \text{Hom}_R(A, B_i)$$ is a $Z(R)$-isomorphism. The next result tells me that ...
3
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2answers
74 views

Category of sets and multi-valued functions

I would like to study category of sets and multi-valued functions: A category whose objects are sets and morphisms are multi-valued functions. By a multi-valued function $f:A\rightarrow B$, from set ...
3
votes
1answer
55 views

What's so special about the grounded poset of cardinality $2$?

By a grounded poset, I mean a poset $P$ with a bottom element $0_P$. Arrows between grounded posets are monotone mappings that preserve the basepoint. This gives a self-enriched category; lets call it ...