Lawvere and Tierney therefore formulated axioms for a topos that assumed a sub-object classifier, and some limit conditions ( to make a cartesian-closed category, at least ).

" Adrift on the seven seas: the medieval topos of exile at sea ", Florilegium 12 ( on-line text ; pdf file )

In the early 1960s Artin spent time at the IHÉS in France, contributing to the SGA4 volumes of the Séminaire de géométrie algébrique, on topos theory and étale cohomology.

In his treatise on the Topics, Aristotle does not explicitly define a topos, though it is " at least primarily a strategy for argument not infrequently justified or explained by a principle.

Likewise, the Argead conquest of Macedonia may be viewed as a commonly used literary topos in classical Macedonian rhetoric: tales of migration served to create complex genealogical connections between trans-regional ruling elites, whilst at the same time used by the ruling dynasty to legitimize their rule, heroicize mythical ancestors, and distance themselves from their subjects.

Walter Goffart believes it is probable that in this narrative Paul was making use of an oral tradition, and is sceptical that it can be dismissed as merely a typical topos of an epic poem.

This recognition of a religious apparition from likeness to an image was also a characteristic of pagan pious accounts of appearances of gods to humans, and was a regular topos in hagiography.

Physicist Lee Smolin writes in Three Roads to Quantum Gravity that topos theory is " the right form of logic for cosmology " ( page 30 ) and " In its first forms it was called ' intuitionistic logic '" ( page 31 ).

The topos was particularly characteristic of the later Middle Ages.

A frequent topos in the legends is that a defeated knight would, after revealing his name, ask the victor what his name was: if the victor turned out to be a much stronger and more famous knight ( e. g. one of Arthur's knights ) the loser actually saved face, because he was beaten by a knight already held to be stronger than him, and thus there was no shame in defeat.

It became somewhat of a topos in Romantic literature, and figures in the poem Der Schweizer by Achim von Arnim ( 1805 ) and in Clemens Brentano's Des Knaben Wunderhorn ( 1809 ) as well as in the opera Le Chalet by Adolphe Charles Adam ( 1834 ) which was performed for Queen Victoria under the title The Swiss Cottage.

Several times in his writings he suggests that his was a humble background, though this may just be a topos or convention of autobiographical writing.

In the light of later work ( c. 1970 ), ' descent ' is part of the theory of comonads ; here we can see one way in which the Grothendieck school bifurcates in its approach from the ' pure ' category theorists, a theme that is important for the understanding of how the topos concept was later treated.

Such a definition of a topos was eventually given five years later, around 1962, by Grothendieck and Verdier ( see Verdier's Bourbaki seminar Analysis Situs ).

The characterisation was by means of categories ' with enough colimits ', and applied to what is now called a Grothendieck topos.

The theory rounded itself out, by establishing that a Grothendieck topos was a category of sheaves, where now the word sheaf had acquired an extended meaning with respect to the idea of Grothendieck topology.

For a while this notion of topos was called ' elementary topos '.

One outcome was the eventual definition adopted in topos theory of geometric morphism, to get the correct notion of surjectivity.

As a set piece or topos of 18th century satire, the " Battle of the Books " was a standard, short-hand for both the Quarrel of the Ancients and Moderns and the era of Swift's battle with William Wotton.

Starting in the 1980s, a diagram-style was developed, with the detailed diagrams of the routes, called " topos " ( probably from French ).

The literary theme of Homo novus, or " how the lowly born but inherently worthy man may properly rise to eminence in the world " was the topos of Seneca's influential Epistle XLIV.

" However, the boorishness and gluttony of athletes was a topos of Greek comedy and even a hero like Hercules was the butt of many jokes.

Critics suggest Boccacio was simply putting down elements from the oral tradition, notably the popular topos of the ordeal, but the text was open enough to allow very misogynistic interpretations, giving Griselda's passivity as the norm for wifely conduct.

His Chicago lectures on categorical dynamics were a further step toward topos theory and his CUNY lectures on hyperdoctrines advanced categorical logic especially using his 1963 discovery that existential and universal quantifiers can be characterized as special cases of adjoint functors.

During the latter part of the 1950s, the foundations of algebraic geometry were being rewritten ; and it is here that the origins of the topos concept are to be found.

Here is an example of Kyrill ’ s humility topos taken from " A Tale of a layman, and on monasticism, and on the soul, and on repentance "; by the most sinful monk Kirill, for Vasilij, abbot of the Caves: “( 52 ) And me: I beg you, do not spurn me like a dog, but remember me even here in your prayers, and there throw me scraps from that holy table, and may all Christians be judged worthy of that life, through Jesus Christ our Lord, to whom glory with the Father and with the Holy Spirit, now and ever .” And another one from: A Sermon for Low Sunday by the unworthy monk Kirill in praise of the resurrection, and concerning the paschal bread, and concerning Thomas ’ s resting of the Lord ’ s ribs: “( 1 ) The Church requires a great teacher and a wise interpreter to adorn the feast.

His creation of topos theory has had an impact on set theory and logic.

Debates concerning the nature, essence and the mode of existence of space date back to antiquity ; namely, to treatises like the Timaeus of Plato, or Socrates in his reflections on what the Greeks called khora ( i. e. " space "), or in the Physics of Aristotle ( Book IV, Delta ) in the definition of topos ( i. e. place ), or even in the later " geometrical conception of place " as " space qua extension " in the Discourse on Place ( Qawl fi al-Makan ) of the 11th century Arab polymath Alhazen.

1041 ) presented a thorough mathematical critique and refutation of Aristotle's conception of place ( topos ) in his Risala / Qawl fi ’ l-makan ( Treatise / Discourse on Place ).

The later work on étale cohomology has tended to suggest that the full, general topos theory isn't required.

The depiction of monarchs and deities as seated on chairs is a common topos in the iconography of the Ancient Near East.

The internal logic of an elementary topos is based on the Heyting algebra of subobjects of the terminal object 1 ordered by inclusion, equivalently the morphisms from 1 to the subobject classifier Ω.

new approach to quantum theory based on topos theory.

Three hundred and sixty Archontes then of the Adamas, having refused to believe in the mystery of light, are assigned a dwelling-place in a still lower region, that of the air ( topos aerinos ), beneath the sphaira, or on the way of the mid-region, in via medii.

* Updates, topos, and more on the climbing in Romsdal

# Sheaves on topoi: If E is a topos and S is an object in E, the category E < sub > S </ sub > of S-objects is also a topos, interpreted as the category of sheaves on S. If f: T → S is a morphism in E, the inverse image functor f < sup >*</ sup > can be described as follows: for a sheaf F on E < sub > S </ sub > and an object p: U → T in E < sub > T </ sub > one has f < sup >*</ sup > F ( U )

Perhaps this is why topos theory has been treated as an oddity ; it goes beyond what the traditionally geometric way of thinking allows.

For example, Curtius notes the common observation in the ancient classical world that “ all must die ” as a topos in consolatory oratory ; that is, one facing one ’ s own death often stops to reflect that greater men from the past died as well.

Nishida, the school's founder, is most known for his groundbreaking work An Inquiry into the Good and later for his elucidation of the " logic of basho " ( Japanese: 場所 ; usually translated as " place ," or the Greek topos )-which brought him fame outside of Japan, and contributed largely to the attention later paid to philosophers from the Kyoto School.

The upper world, the kingdom of light, finds its completion in the twenty-fourth or last mystery, which again itself produces twelve subordinate mysteries and emanations ; beneath this is the magnum lumen kharagmes luminis, which again divides itself into five karagmai luminis, the primum praeceptum ( statutum ), which is divided into seven mysteries, the magnum lumen luminum, the five great Helpers ( parastatai, proegoumenoi ), which serve to conduct the energies of light into the lower regions, and finally the topos kleronomion luminis, the destined habitation of redeemed souls.

In the case of Greek endings, the plurals sometimes follow the Greek rules: phenomenon, phenomena ; tetrahedron, tetrahedra ; crisis, crises ; hypothesis, hypotheses ; stigma, stigmata ; topos, topoi ; cyclops, cyclopes ; but often do not: colon, colons not * cola ( except for the very rare technical term of rhetoric ); pentathlon, pentathlons not * pentathla ; demon, demons not * demones ; climaxes, not * climaces.

( Like always, there are simple ways to get non-standard NNOs ; for example, if z = s z, in which case the category or topos E is trivial.

The most famous concept in Nishida's philosophy is the logic of basho ( Japanese: 場所 ; usually translated as " place " or " topos "), a non-dualistic concrete logic, meant to overcome the inadequacy of the subject-object distinction essential to the subject logic of Aristotle and the predicate logic of Kant, through the affirmation of what he calls the " absolutely contradictory self-identity ", a dynamic tension of opposites that, unlike the dialectical logic of Hegel, does not resolve in a synthesis, but rather defines its proper subject by maintaining the tension between affirmation and negation as opposite poles or perspectives.

" He characterises it in the Rhetoric thus: " I call the same thing element and topos ; for an element or a topos is a heading under which many enthymemes fall.

" Utopia " is derived from the Greek words ou (), " not ", and topos (), " place ", with the suffix-iā () that is typical of toponyms ; hence Outopía (; Latinized as Ūtopia, with stress on the second syllable ), " no-place-land ".