Articles of category theory

Coproducts and pushouts of Boolean algebras and Heyting algebras

I am having trouble of find a reference explaining how to compute coproduts and pushouts in the category of Boolean algebras and in the category of Heyting algebras. To be precise I am looking for as a concrete description of these colimits as possible. In particular I hope to do better than just describing them […]

Question on inverse limits

1.7. Remark. The inverse limit of an inverse system of non-empty sets might be empty as the following example shows: Let $I:=\mathbb{N}$ and $X_n:=\mathbb{N}$ for every $n\in\mathbb{N}$. Let $\theta_{n\leftarrow > n+1}:\mathbb{N}\to\mathbb{N},k\mapsto k+1$. Now assume $(x_n)_{n\in\mathbb{N}}$ is contained in the inverse limit. Then $x_{n+1}=x_n-1$ for every $n\in\mathbb{N}$ yielding a contradiction. Reference My question is: How is […]

How do we describe the equivalence between $f:(I,\partial I)\to(X,x_0)$ and its “naturally equivalent” map $\tilde f:(S^1, s_0)\to (X,x_0)$

In the theory of fundamental groups and homotopy, we always regard $f:(I,\partial I)\to(X,x_0)$ and $\tilde f:(S^1, s_0)\to (X,x_0)$ as the same thing and change one from another freely. I wonder if there is an advanced terminology in category theory or somewhere that describes their equivalence.

Modules finitely generated and of finite type (categorical meaning)

An object $C$ in an additive category admitting all filtered direct limits $\mathcal{C}$ is called “of finite type” if the canonical map $$\underrightarrow{\lim} Hom_{\mathcal{C}}(C,F(i))\to Hom_{\mathcal{C}}(C,\underrightarrow{\lim}F)$$ is injective for every $I$ directed poset for every functor $F:I\to \mathcal{C}$ In the case $\mathcal{C}$=Mod-R prove that this definition is equivalent to the definition of “finitely generated” The exercise […]

On the bounded derived category of a finite dimensional algebra with finite global dimension

Let $A$ be a finite dimensional $k$-algebra with finite global dimension. How can I prove that the category $D^b(A)$ (bounded derived category of the category of left finitely generated $A$-modules) is equivalent to $K^b(_AP)$ (bounded homotopy category of complexes made with left projective modules) ?

When is the category of (quasi-coherent) sheaves of finite homological dimension?

Let say from the beginning that my background is category of modules over a ring. So I know that if we take a given (nice) scheme $X$, then category of sheaves on $X$ is Grothendieck, so it must have enough injective, but it may fail to have projective. The question is then, when it is […]

ancient principle of mathematics: figure = varying element

On page 83 in the book Conceptual mathematics by Lawvere et al. it says: An ancient principle of mathematics holds that a figure is the locus of a varying element. What does this quote mean? In particular, what is the “locus of a varying element” and in which sense is it the same as a […]

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 to require it. (Where we take the definition in terms of Hom isomorphism, for example here: )

How are the cardinalities of the object images of adjoint functors related?

Here is a very silly question: Adjoint functors satisfy $$\mathrm{hom}_{\mathcal{C}}(FA,B) \cong \mathrm{hom}_{\mathcal{D}}(A,GB).$$ I consider numbers $a,b$ and read this as $$b^{\,f(a)}=g(b)^a.$$ If the objects in the categories can be assigned cardinalities, do the functors actually fulfill a relation along those lines? $\bf Edit$: If e.g. $|B^{FA}|=|B|^{|FA|}$ does make sense, just taking the cardinalities of the […]

Is every equivalence of monoidal categories a monoidal equivalence?

Let $C$ and $D$ be monoidal categories. Let $T:C \to D$ be a functor that gives the equivalence of $C$ and $D$ as just categories. My question is whether such an equivalence $T$ is always a monoidal equivalence or not. If this is true, could you give me a proof? Thank you.