Abstract
The very handwavy definition of a topos is that of a category with extra properties. The implications of these extra properties are that they make a topos “look like” Sets, in the sense that any mathematical operation which can be done in set theory can be done in a general topos. (As mentioned in previous chapters, Sets is a topos, albeit a very special one.)
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Notes
- 1.
As mentioned in previous chapters, Sets is a topos, albeit a very special one.
- 2.
One usually writes D=A× C B.
- 3.
Strictly speaking one has two cones \(C\xleftarrow{j}D\xrightarrow{i}A\) and \(C\xleftarrow{j}D\xrightarrow{h}B\), but when composing the diagrams it turns out that j=g∘h=f∘i, thus we can omit the j map.
- 4.
One usually writes D=C+ A B.
- 5.
Given h:A→C and k′,k:D→A, with h∘k=h∘k′. Then consider g:C→B such that (g∘h)=f is monic. It follows that:
$$ (g\circ h)\circ k=g\circ (h\circ k)=g\circ \bigl(h\circ k'\bigr)=(g\circ h)\circ k'\quad \text{therefore}\quad k=k'. $$(7.27) - 6.
Recall that !:A→1 is the unique arrow from A to the terminal object 1.
- 7.
Essentially ↓A is the sieve which contains all possible \(\mathcal{C}\)-arrows, which have as codomain A, i.e. \(\downarrow\!\!A:=\{ f_{i}\in \mathcal{C}|\mathit{cod}(f_{i})=A\}\).
- 8.
For the sake of clarity we remind the reader that a Boolean algebra B is a distributive lattice with 0 and 1 in which every element x∈B has a compliment ¬x such that x∧¬x=0 and x∨¬x=1.
- 9.
The fact that in Sets the set of truth values is simply {0,1} implies that the logic which is derived is a Boolean logic, i.e. a classical logic. This is basically the logic that each of us adopts when speaking any western language. It is the logic of the classical world and the logic of the western way of reasoning.
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Flori, C. (2013). Topos. In: A First Course in Topos Quantum Theory. Lecture Notes in Physics, vol 868. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35713-8_7
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