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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99 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 ...
3
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1answer
41 views

Coherence result for (braided) monoidal functors

Is there any coherence result for (braided) monoidal functors? (like Mac Lane's coherence theorem for monoidal categories) What I have in mind is a theorem like the following: Let $F$ be a ...
2
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1answer
21 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 ...
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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 ...
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1answer
92 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 ...
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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 ...
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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 ...
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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. ...
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49 views
+100

Properties of the functor $(X, \mathcal{O}_X) \mapsto (X(k), \mathcal{O}_{X(k)})$

Let $k$ be an algebraically closed field. In Görtz and Wedhorns book one can read about an equivalence of categories $\{\text{integral schemes of finite type over } k\} \to \{\text{prevarieties over ...
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2answers
60 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 ...
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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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324 views

Functionally structured spaces and manifolds

The question requires some definitions that I have listed below for your convenience. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups. On a ...
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 ...
5
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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 ...
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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) ...
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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 ...
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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 ...
3
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1answer
62 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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1answer
501 views

Gaining insight into the Inverse Image Sheaf

Let $f: X \rightarrow Y$ be a continuous map of topological spaces and let $G$ be a sheaf of sets on $Y$. I am trying to understand the definition of the inverse image sheaf $f^{-1}G$ on $X$. This is ...
2
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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 ...
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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 ...
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?
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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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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
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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 ...
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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
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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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 ...
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 ...
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246 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 ...
5
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2answers
175 views

What would be the equivalent of the “gluing axiom” for a cosheaf

A sheaf is a presheaf $F$ such that for all $U$ and for all covering $\{U_i\}_{i\in I} $ of $U$, $F(U)$ is the equalizer $$ F(U) \overset{f}{\longrightarrow}\prod_{i\in I} F(U_i) ...
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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
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
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 ...
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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 ...
2
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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. ...
2
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1answer
72 views

Going from Closed Categories to Monoidal Categories

EDIT: I trimmed down the exposition a bit. I really just wanted everyone to know what my approach has been, but what I had was a bit bloated. Suppose we have a closed category $V$ as defined here, ...
3
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1answer
93 views

Category theory: Enough that polygonal diagrams commute

I've read somewhere that for a categorical diagram to commute, it is enough that all its polygonal subdiagrams commute. I want a reference and a detailed proof of this. Please also give a formal ...
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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 ...
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 ...
0
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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 $$ ...
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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 ...
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Is Erdős' lemma on intersection graphs a special case of Yoneda's lemma?

Under which name is the following proposition filed actually: Every poset $P$ embeds fully and faithfully in the powerset of $P$, ordered by subset inclusion. Let me call it Dedekind's lemma. ...
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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 ...
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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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1answer
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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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1answer
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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 ...
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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 ...
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1answer
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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 ...