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One way to formalize the idea is to say the following: the category of classical propositional theories is a reflective subcategory of the category of intuitionistic propositional theories. In order to make sense of that claim, it's best to pass from "theories" to "algebras." Each classical theory corresponds to a Boolean algebra, and each intuitionistic ...

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If $S$ has a categorification $\mathrm{Decat}(\mathcal{C}) \to S$ and $T$ has a categorification $\mathrm{Decat}(\mathcal{D}) \to T$, then a categorificiation of a map $S \to T$ is a functor $\mathcal{C} \to \mathcal{D}$ such that the induced map $\mathrm{Decat}(\mathcal{C}) \to \mathrm{Decat}(\mathcal{D})$ makes the diagram \begin{array}{cc} ...

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I've seen the notation $\mathsf{CAlg}_R$ in many places. I prefer $\mathsf{CAlg}(R)$ in order to stress the functoriality. This is also used in Yves Diers' work. Many papers restrict to commutative rings and algebras in the first place and therefore just write $\mathsf{Alg}_R$ (which might be confusing - but this is just a local notation). More generally, ...

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Check out the new book (amazon-link) Tom Leinster, Basic Category Theory, Cambridge Studies in Advanced Mathematics, Vol. 143, 2014

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The polynomial ring $A [x]$ is the coproduct of $A$ and $\mathbb{Z} [x]$. One way to see this is to note that the category of (commutative unital) $A$-algebras is isomorphic to the slice category $^{A /} \mathbf{CRing}$, and the forgetful functor $^{A /} \mathbf{CRing} \to \mathbf{CRing}$ has a left adjoint, namely $B \mapsto A \otimes_{\mathbb{Z}} B$. ...

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The trick is to embed the category of abstract simplicial complexes inside the category of symmetric simplicial sets (= functor $\mathbf{F}^\mathrm{op} \to \mathbf{Set}$, where $\mathbf{F}$ is the category of positive finite cardinals): this can be done by sending an abstract simplicial complex $X$ to the symmetric simplicial set ...

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Notice that $V$ and $V'$ are orthogonal with respect to $b \oplus b'$, i.e. we have $(b \oplus b')(v \oplus v')=0$ and $(b \oplus b')(v' \oplus v)=0$ for all $v \in V$, $v' \in V'$. In fact, $(V \oplus V',b \oplus b')$ represents the subfunctor of $\hom((V,b),-) \times \hom((V',b'),-)$ which consists of those pairs of morphisms $f : (V,b) \to (W,c)$, $g : ... 2 Assuming we're working with small categories$\mathcal{C,D}$, the problem with images not being categories requires, for a functor$F$, the map of sets$F: Ob(\mathcal{C}) \to Ob(\mathcal{D})$not being injective (though this isn't sufficient to insure a problem). The issue that arises is that there may be objects and morphisms$f:W \to X, g:Y \to Z$in ... 2 The source of the problem, as described by JHance, can be realized as the smallest 'free' example of a functor$F\colon C\to D$whose image is not a category. Let$C$be the category with four (distinct) objects$w$,$x$,$y$,$z$and the two arrows$w\to x$and$y\to z$(besides the identities), and let$D$be the category with the objects$0$,$1$,$2$and ... 1 I have no idea what kind of generalizations are you looking for, but the above statement is completely false even in case of ordinary categories (and you can easily find a counterexample). It is true, however, under some additional assumptions (like, finite completeness). Check the following question: ... 1 Well, sets and maps. In set theory, a map from$A$to$B$is a certain subset of$A \times B$. If the map factors through some subset$B'$of$B$, we get two maps$A \to B$and$A \to B'$which are equal as sets. For example, for a set theorist,$\emptyset$is a map from$\emptyset$to any set. But for a category theorist, a morphism has to have a specified ... 1 You can define a "simplex" as having an orientation, thus getting an easier answer. A$k$-simplex is the convex hull of a set of$k+1$points. But what does it mean for a set to have$k+1$points? That there is a bijection from$\{1,2,\dots,k+1\}$. So simply define "$k$-simplex" in terms of a map$\{1,2,\dots,k+1\}\to\mathbb R^n$and you can pick an ... 1 Just don't call it a pullback. Yes this is ok. 1 Your idea is absolutely right. The map$\varphi_{(S,Q)}$maps a morphism$\psi \colon C(S) \rightarrow Q$(in$\mathrm{CompInfDist}_\vee$) to the morphism$\psi \circ \iota_S$(in$\mathrm{InvSem}$), where$\iota_S \colon S \hookrightarrow C(S)$is the map you denoted as$\iota$. 1 Introduction to Category Theory by Harold Simmons is a nice and gentle way to get into category theory with plenty of exercises (and full solutions!). I'm an undergrad as well, and I worked through this book before moving on to Categories for the Working Mathematician because it is more leisurely. More to the point of your question, Intro to Category Theory ... 1 Let$n \in \mathbb{N}_{\geq 1}$. The simplicial category$\mathfrak{C} [\partial \Delta^n]$has as objects the set$\{0,\ldots,n\}$, and the simplicial set$Hom_{\mathfrak{C}[ \partial \Delta^n]}(i,j)$, for$0 \leq i < j \leq n$and$(i,j) \neq (0,n)$, is$N(P_{i,j})$, the nerve of the partially ordered set of subsets$I \subseteq \{ i,\ldots,j \}$with ... 1 Let's do part of the task, to get you the feeling how to proceed with the other parts:$\emptyset$is initial in the category of sets (and maps): If$A$is any set, we need to exhibit a map$f\colon \emptyset\to A$and show that it is unique. To produce a map we need to specify an element$f(x)$for each$x\in\emptyset\$. But there is nothing to be done! ...

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