Definition df-termc 49220

Description: Definition of the proper class (termcnex 49314) of terminal categories, or final categories, i.e., categories with exactly one object and exactly one morphism, the latter of which is an identity morphism (termcid 49232). These are exactly the thin categories with a singleton base set. Example 3.3(4.c) of [Adamek] p. 24.

As the name indicates, TermCat is the class of all terminal objects in the category of small categories (termcterm3 49261). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 49266). See also dftermc3 49277 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 49277, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 49237.

Followed directly from the definition, terminal categories are thin (termcthin 49224). The opposite category of a terminal category is "almost" itself (oppctermco 49251). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 49286).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 49247) and terminal objects (dftermo4 49248). The universal properties provide an alternate proof of initoeu1 18011, termoeu1 18018, initoeu2 18016, and termoeu2 49018. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 49262).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17687, 0thincg 49207, and 0funcg 48943.

(Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
df-termc	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

Distinct variable group:   𝑥,𝑐

Detailed syntax breakdown of Definition df-termc

Step	Hyp	Ref	Expression
1		ctermc 49219	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1538	. . . . . 6 class 𝑐
4		cbs 17215	. . . . . 6 class Base
5	3, 4	cfv 6528	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1538	. . . . . 6 class 𝑥
8	7	csn 4599	. . . . 5 class {𝑥}
9	5, 8	wceq 1539	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1778	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 49166	. . 3 class ThinCat
12	10, 2, 11	crab 3413	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1539	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is referenced by:  istermc  49221